A Collection of My 100 Posts from 'Math's Believe It Or Not' Group in Facebook

MATH’S BELIEVE IT OR NOT A Collection of 100 Math Discussions By DR. MOLOY DE demoloy@yahoo.co.in CONTENTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Introduction 1 Alphametics 1 Anti-Pigeonhole Conjecture 2 Armstrong Number 2 Arnold Sommerfeld 3 Barber Paradox 3 Bellows Conjecture 4 Bernoulli Brothers 4 Birth of the Idea of Computer 5 Birthday Problem 5 Breakthrough Prizes in Mathematics 6 Bremermann’s Limit 7 BSD Conjecture 7 Budhayana and Pythagorous 8 Carmichael’s Theorem 8 Collatz Conjecture 9 Common Knowledge 10 Complete Sequence 11 Computing π 12 Count of Polytopes 12 Cramer’s Conjecture on Prime Gap 13 Cramer’s Paradox 13 Curve of Constant Width 14 Deciphering Enigma 15 Differential Privacy 15 Diophantine Equation 16 Disquisitiones Arithmeticae - the Book by Gauss 17 Dots and Boxes 18 Euler Brick 19 Euler Line 20 Fermat Number 20 Fermat Point 21 Fermat Vs. Descarte 21 Fermat’s Spiral 22 Fibonacci Powers 22 Four Color Theorem 23 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 George E P Box God’s Algorithm Graeco-Latin Square Graffiti Conjectures Haberdasher’s Puzzle Happy Number Heawood Conjecture Hemchandra, the Indian Scholar Hippasus Illegal Number Imaginary Time Infinitudes of Primes Josephus Problem Kakeya Set Kaprekar’s Constant Kepler Triangle Knight’s Tour Lazy Caterer’s Sequence Liar Paradox Life Insurance - the Contributors Martin Gardner Mathematician’s Anger McNugget Numbers Minesweeper Morley’s Trisector Theorem N Queens Problem Nagel Point Napoleon’s Theorem Nine-point Circle Normal Number Oh! I Forgot Pandigital Prime Paul Erdos Pell’s Equation Penrose Tiling 24 25 26 27 27 28 29 30 30 31 32 32 33 34 35 36 37 38 39 40 40 41 41 42 43 43 44 45 46 47 48 49 50 51 52 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Perfect Number Problem of Apollonius Prof. Anil Bhattacharya Pythagoraen Theorem in Ancient India Quantum Computing Quaternion Question Mark Function ?(x) Ramanujan Prime Ramsey Number Random Walk in Multidimensions Regular Heptadecagon Rubik’s Cube Saccheri Quadrilateral Sierpinski Triangle Sieve of Sundaram Simson Line SP Number Spirograph Squaring the Plane St. Petersburg Paradox Stigler’s Law of Eponymy Sum of Four Cubes Taxicab Number The Book Three Body Problem and Chaos Theory Triangle Centers Truncatable Primes Ulam Spiral Unexpected Hanging Paradox Znam’s Problem 53 54 55 55 56 57 58 59 60 60 61 62 63 64 65 66 66 67 68 68 69 70 70 72 73 74 75 76 77 78 INTRODUCTION Facebook is a well established and popular social media now. In mid 2011 I started a public group in facebook named ‘Math’s Believe It Or Not’ with around twenty odd mathematically inclined friends. The number of members is reaching 15,000 now. The explosion of member-count makes me believe that there lies a general interest about mathematics in today’s society. This book presents selected excerpts/mathematical discussions published in the Facebook group. Most of the discussions are initiated by me and also most of them refer to Wikipedia, another successful initiative of current age, as the source. I thank my friends, colleagues and specially the members of ‘Math’s Believe It Or Not’ to stand by my endeavor. 1. ALPHAMETICS Verbal arithmetic puzzles are quite old and their inventor is not known. An 1864 example in The American Agriculturist disproves the popular notion that it was invented by Sam Loyd. The name ‘cryptarithmie’ was coined by puzzlist Minos (pseudonym of Simon Vatriquant) in the May 1931 issue of Sphinx, a Belgian magazine of recreational mathematics, and was translated as ‘cryptarithmetic’ by Maurice Kraitchik in 1942. In 1955, J. A. H. Hunter introduced the word ”alphametic” to designate cryptarithms, such as Dudeney’s, whose letters form meaningful words or phrases. Journal of Recreational Mathematics, had a regular alphametics column. 1 2. ANTI-PIGEONHOLE CONJECTURE Let X be any issue and let A1 , A2 , · · · , An be any collection of distinct faculty members. Then during a long enough period of email exchanges among the above faculty on X at least n + 1 opinions will be voiced. From the Math Blog ‘Godels Lost Letter and P=NP’ 3. ARMSTRONG NUMBERS Armstrong numbers are the sum of their own digits to the power of the number of digits. say, 153 = 13 + 53 + 33 . Each digit is raised to the power three because 153 has three digits. They are totalled and we get the original number again. Armstrong numbers are base dependent and they are certainly rare. They cannot have more than 60 digits in base 10, because for n > 60, n9n < 10n−1 . Since there is an upper limit to their size, it is theoretically possible to find all of them, given sufficient computer time. However, 1060 is an unimaginably huge number, so such a ‘brute force’ approach would be unwise. Luckily, D. Winter proved in 1985 that there are exactly 88 base-10 Armstrong numbers, and they must have 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38 or 39 digits. Of course, the one digit Armstrong numbers are somewhat trivial since clearly 11 1 = 1, 21 = 2 etc. The Armstrong numbers up to 10 digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, and 4679307774. The largest Armstrong number (in base 10) is the 39-digit beast: 115132219018763992565095597973971522401. 2 4. ARNOLD SOMMERFIELD During the 1901-1950 period, Arnold Sommerfeld was nominated for the Nobel Prize 84 times, more than any other physicist (including Otto Stern, who got nominated 81 times), but he never received the award. His PhD students earned more Nobel prizes in physics than any other supervisor’s, ever. He introduced the 2nd quantum number (azimuthal quantum number) and the 4th quantum number (spin quantum number). He also introduced the fine-structure constant and pioneered X-ray wave theory. 5. BARBER PARADOX The barber shaves all and only those men in town who do not shave themselves. Who shaves the barber? If the barber does shave himself, then the barber (himself) must not shave himself. If the barber does not shave himself, then the barber (himself) must shave himself. This paradox is often attributed to Bertrand Russell, e.g., by Martin Gardner. It was suggested to him as an alternative form of Russell’s paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber’s paradox was an instance of his own. 3 6. BELLOWS CONJECTURE Area of a triangle is completely determined by the lengths of its three sides. This is not true for a quadrilateral. Bellows Conjecture states that a flexible polyhedron has constant volume when it flexes, i.e. a polyhedral bellows is impossible. The conjecture is false in two dimensions, true in three dimensions, thought to be true in four dimensions and is wide open for five dimensions or more. 7. BERNOULLI BROTHERS Unusually in the history of mathematics, a single family, the Bernoullis, produced half a dozen outstanding mathematicians over a couple of generations at the end of the 17th and start of the 18th Century. The Bernoulli family was a prosperous family of traders and scholars from the free city of Basel in Switzerland, which at that time was the great commercial hub of central Europe.The brothers, Jacob and Johann Bernoulli, however, flouted their father’s wishes for them to take over the family spice business or to enter respectable professions like medicine or the ministry, and began studying mathematics together. 4 8. BIRTH OF THE IDEA OF COMPUTER In 1812 Charles Babbage was sitting in his rooms in the Analytical Society looking at a table of logarithms, which he knew to be full of mistakes, when the idea occurred to him of computing all tabular functions by machinery. The French government had produced several tables by a new method. Three or four of their mathematicians decided how to compute the tables, half a dozen more broke down the operations into simple stages, and the work itself, which was restricted to addition and subtraction, was done by eighty computers who knew only these two arithmetical processes. Here, for the first time, mass production was applied to arithmetic, and Babbage was seized by the idea that the labours of the unskilled computers could be taken over completely by machinery which would be quicker and more reliable. 9. BIRTHDAY PROBLEM In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions include the assumption that each day of the year (except February 29) is equally probable for a birthday. The history of the problem is obscure. W. W. Rouse Ball indicated (without citation) that it was first discussed by Harold Davenport. However, Richard von Mises proposed an earlier version of what we consider today to be the birthday problem. The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function. 5 10. 2015 BREAKTHROUGH PRIZES IN MATHEMATICS All is number, taught Pythagoras. Though modern mathematics encompasses far more than numbers alone, the principle remains true. Mathematics is the universal language of nature. Math is also fundamental to the growth of knowledge, as it is the scaffolding that supports all the sciences. Its relationship to physics is particularly intimate. From imaginary numbers to Hilbert spaces, what once seemed pure abstractions have turned out to underlie real physical processes. In addition, all fields in the life sciences today utilize the power of statistical and computational approaches to research. The Breakthrough Prize in Mathematics rewards significant discoveries across the many branches of the subject. The prize was founded by Mark Zuckerberg and Yuri Milner and announced at the 2014 Breakthrough Prize ceremony. 6 11. BREMRMANN’S LIMIT Hans-Joachim Bremermann was a mathematician and biophysicist. He is famous for a limit on computation, Bremermann’s limit, which is the maximum computational speed of a self-contained system in the material universe. He noted back then that certain problems were beyond any reasonable brute-force search. In his own words: ‘The experiences of various groups who work on problem solving, theorem proving and pattern recognition all seem to point in the same direction: These problems are tough. There does not seem to be a royal road or a simple method which at one stroke will solve all our problems. My discussion of ultimate limitations on the speed and amount of data processing may be summarized like this: Problems involving vast numbers of possibilities will not be solved by sheer data processing quantity. We must look for quality, for refinements, for tricks, for every ingenuity that we can think of. Computers faster than those of today will be a great help. We will need them. However, when we are concerned with problems in principle, present day computers are about as fast as they ever will be. We may expect that the technology of data processing will proceed step by step, just as ordinary technology has done. There is an unlimited challenge for ingenuity applied to specific problems. There is also an unending need for general notions and theories to organize the myriad details.’ 12. BSD CONJECTURE Bhargava & Shankar (2015) proved that the average rank of the MordellWeil group of an elliptic curve over Q is bounded above by 7/6. Combining this with the p-parity theorem of Nekovr (2009) and Dokchitser & Dokchitser (2010) and with the proof of the main conjecture of Iwasawa theory for GL(2) by Skinner & Urban (2014), they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by Kolyvagin (1989), satisfy the Birch and Swinnerton-Dyer conjecture. 7 13. BUDHAYANA AND PYTHAGOROUS Baudhayana, (800 BCE) was the author of the Baudhayana sutras, which cover dharma, daily ritual, mathematics, etc. He belongs to the Yajurveda school, and is older than the other sutra author Apastamba. He was the author of the earliest Sulba Sutraappendices to the Vedas giving rules for the construction of altarscalled the Baudhayana Sulbastra. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem. Sequences associated with primitive Pythagorean triples have been named Baudhayana sequences. These sequences have been used in cryptography as random sequences and for the generation of keys. 14. CARMICHAEL’S THEOREM Carmichael’s theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F (n) has at least one prime divisor that does not divide any earlier Fibonacci number. The only exceptions for n up to 12 are: F (1) = 1 and F (2) = 1, which have no prime divisors F (6) = 8 whose only prime divisor is 2 which is F (3) F (12) = 144 whose only prime divisors are 2 which is F (3) and 3 which is F (4) 8 15. COLLATZ CONJECTURE The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani’s problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse’s algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. Take any natural number n. If n is even, divide it by 2 to get n/2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ”Half Or Triple Plus One”, or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called ‘oneness’. Paul Erdos said about the Collatz conjecture: ”Mathematics may not be ready for such problems.” He also offered $500 for its solution. 9 16. COMMON KNOWLEDGE Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum. The concept was first introduced in the philosophical literature by David Kellogg Lewis in his study Convention (1969). It was first given a mathematical formulation in a set-theoretical framework by Robert Aumann (1976). Computer scientists grew an interest in the subject of epistemic logic in general and of common knowledge in particular starting in the 1980s. There are numerous puzzles based upon the concept which have been extensively investigated by mathematicians such as John Conway. The idea of common knowledge is often introduced by some variant of the following puzzle: On an island, there are k people who have blue eyes, and the rest of the people have green eyes. At the start of the puzzle, no one on the island ever knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island at dawn the next day. On the island, each person knows every other person’s eye color, there are no reflective surfaces, and there is no discussion of eye color. At some point, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement: ‘At least one of you has blue eyes’. The outsider, furthermore, is known by all to be truthful, and all know that all know this, and so on: it is common knowledge that he is truthful, and thus it becomes common knowledge that there is at least one islander who has blue eyes. The problem: assuming all persons on the island are completely logical and that this too is common knowledge, what is the eventual outcome? The answer is that, on the kth dawn after the announcement, all the blue-eyed people will leave the island. 10 17. COMPLETE SEQUENCE An integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, the sequence of powers of two 1, 2, 4, 8, · · · the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 1001012 = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include: (1) The even numbers; since adding even numbers produces only even numbers, no odd number can be formed. (2) Powers of three; no integer having a digit ‘2’ in its ternary representation (2, 5, 6, · · ·) can be formed. Below is a list of the some complete sequences. (1) The sequence of the number 1 followed by the prime numbers (studied by S. S. Pillai and others); this follows from Bertrand’s postulate. (2) The Fibonacci numbers, as well as the Fibonacci numbers with any one number removed. This follows from the identity that the sum of the first n Fibonacci numbers is the (n + 2)nd Fibonacci number minus 1. (3) The Lazy caterer’s sequence that gives the maximum number of partitions that a plane can be divided into, using n straight lines as dividers. (4) The Cookie cutter’s sequence that gives the maximum number of partitions that a plane can be divided into, using n circles as dividers. 11 18. COMPUTING Π For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jrg Arndt and Christoph Haenel, thirtynine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom. Despite this, people have worked strenuously to compute π to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world. They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π. 19. COUNT OF POLYTOPES In two dimensions there are infinitely many regular polygons: one for each whole number of sides from three onward. The Greeks proved that there are exactly five regular solids in three dimensions: tetrahedron, cube, octahedron, dodecahedron and icosahedron. In five or more dimensions there are only three regular polytopes (as they are called) and they are analogous to tetrahedron, cube and octahedron. However, in four dimensional space there are six regular polytopes. 12 20. CRAMER’S CONJECTURE ON PRIME GAP The prime number theorem implies that on average, the gap between the prime p and its successor is log(p). However, some gaps between primes may be much larger than the average. Cramr proved that, assuming the √ Riemann hypothesis, every gap is O( p log(p)). This is a case in which even the best bound that can be proved using the Riemann Hypothesis is far weaker than what seems true: Cramr’s conjecture implies that every gap is O((log(p))2 ), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramr’s conjecture. 21. CRAMER’S PARADOX Cramer’s paradox was first published by Maclaurin. Cramer and Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. It has become known as Cramer’s paradox after featuring in his 1750 book Introduction l’analyse des lignes courbes algbriques, although Cramer quoted Maclaurin as the source of the statement. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points and discussed the problem in his book Introductio in analysin infinitorum. The result was publicized by James Stirling and explained by Julius Plcker. 13 22. CURVE WITH CONSTANT WIDTH A curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each having at least one point in common with the shape’s boundary but none with the shape’s interior, is the same regardless of the orientation of the curve. More generally, any compact convex planar body D has one pair of parallel supporting lines in any given direction. A supporting line is a line that has at least one point in common with the boundary of D but no points in common with the interior of D. The width of the body is defined as before. If the width of D is the same in all directions, the body is said to have constant width and its boundary is a curve of constant width; the planar body itself is called an orbiform. The width of a circle is constant: its diameter. On the other hand, the width of a square√varies between the length of a side and that of a diagonal, in the ratio 1 : 2. Thus the question arises: if a given shape’s width is constant in all directions, is it necessarily a circle? The surprising answer is that there are many non-circular shapes of constant width. A nontrivial example is the Reuleaux triangle. To construct this, take an equilateral triangle with vertices ABC and draw the arc BC on the circle centered at A, the arc CA on the circle centered at B, and the arc AB on the circle centered at C. The resulting figure is of constant width. The Reuleaux triangle lacks tangent continuity at three points, but constantwidth curves can also be constructed without such discontinuities. Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides (triangle, pentagon, heptagon, etc.). 14 23. DECIPHERING ENIGMA An Enigma machine was any of several codesigned electro-mechanical rotor cipher machines used in the twentieth century for enciphering and deciphering secret messages. Enigma was invented by the German engineer Arthur Scherbius at the end of World War I. Early models were used commercially from the early 1920s, and adopted by military and government services of several countriesmost notably by Nazi Germany before and during World War II. Several different Enigma models were produced, but the German military models are the most commonly recognised. The movie ”The Imitation Game” is the biography of Alan Turing, who broke the Enigma Code and supposedly reduced World War II by two years. 24. DIFFERENTIAL PRIVACY A statistical database is a database used for statistical analysis purposes. It is an OLAP (online analytical processing), instead of OLTP (online transaction processing) system. Modern decision, and classical statistical databases are often closer to the relational model than the multidimensional model commonly used in OLAP systems today. Statistical databases often incorporate support for advanced statistical analysis techniques, such as correlations, which go beyond SQL. They also pose unique security concerns, which were the focus of much research, particularly in the late 1970s and early to mid-1980s. The notion of indistinguishability, later termed Differential Privacy, formalizes the notion of ”privacy” in statistical databases.Cynthia Dwork is a computer scientist who is a Distinguished Scientist at Microsoft Research. She has done great work in many areas of theory, including security and privacy. Her notion of differential privacy calls for attention. 15 25. DIOPHANTINE EQUATION A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied where an integer solution is a solution such that all the unknowns take integer values. A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it. The questions asked in Diophantine analysis include: (1) Are there any solutions? (2) Are there any solutions beyond some that are easily found by inspection? (3) Are there finitely or infinitely many solutions? (4) Can all solutions be found in theory? (5) Can one in practice compute a full list of solutions? These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth, in some cases, rather than treat them as puzzles. The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis. While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations, beyond the theory of quadratic forms, was an achievement of the twentieth century. 16 26. DISQUISTIONES ARITHMETICAE BY GAUSS The Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin[1] by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. The Disquisitiones covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. However, Gauss did not explicitly recognize the concept of a group, which is central to modern algebra, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the Disquisitiones Gauss describes the scope of the book as: ”The inquiries which this volume will investigate pertain to that part of Mathematics which concerns itself with integers.” 17 27. DOTS AND BOXES Dots and boxes, also known as Boxes, Squares, Paddocks, Pigs in a Pen, Square-it, Dots and Dashes, Dots, Line Game, Smart Dots, Dot Boxing, or, simply, the Dot Game, is a pencil and paper game for two players (or sometimes, more than two) first published in 1889 by douard Lucas. Starting with an empty grid of dots, players take turns, adding a single horizontal or vertical line between two unjoined adjacent dots. A player who completes the fourth side of a 1 × 1 box earns one point and takes another turn. The points are typically recorded by placing in the box an identifying mark of the player, such as an initial. The game ends when no more lines can be placed. The winner of the game is the player with the most points. The board may be of any size. When short on time, 2 × 2 boxes, created by a square of 9 dots, is good for beginners, and 5 × 5 is good for experts. In combinatorial game theory dots and boxes is an impartial game, and many positions can be analyzed using SpragueGrundy theory. However, dots and boxes is considered to be he mathematically richest popular child’s game in the world and so by a substantial margin. 18 28. EULER BRICK An Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. The smallest Euler brick, discovered by Paul Halcke in 1719, has edges (a, b, c) = (44, 117, 240) and face diagonals 125, 244, and 267. Some other small primitive solutions, given as edges (a, b, c) and face diagonals (d, e, f ), are: (85, 132, 720) and (157, 725, 732); (140, 480, 693) and (500, 707, 843); (160, 231, 792) and (281, 808, 825); (240, 252, 275) and (348, 365, 373). A perfect cuboid (also called a perfect box) is an Euler brick whose space diagonal also has integer length. In other words, the following equation is added to the system of Diophantine equations defining an Euler brick: a2 + b2 + c2 = g 2 , where g is the space diagonal. Thus (a, b, c, g) must be a Pythagorean quadruple. As of November 2012, no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, one of its edges must be greater than 3 × 1012 . Furthermore, its smallest edge must be longer than 1010 . 19 29. EULER LINE In some ways, Euler’s discovery of the Euler line is analogous to Columbus’s ”discovery” of America. Both made their discoveries while looking for something else. Columbus was trying to find China. Euler was trying to find a way to reconstruct a triangle, given the locations of some of its various centers. Neither named his discovery. Columbus never called it ”America” and Euler never called it ”the Euler line.” Both misunderstood the importance of their discoveries. Columbus believed he had made a great and wonderful discovery, but he thought he’s discovered a better route from Europe to the Far East. Euler knew what he’d discovered, but didn’t realize how important it would turn out to be. Finally, Columbus made several more trips to the New World, but Euler, as with his polyhedral formula and the Knigsberg bridge problem, made an important discovery but never went back to study it further. 30. FERMAT NUMBER If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. However the converse was incorrectly conjectured by Fermat in 1650 and disproved by Euler in 1732 showing the counter-example 5 F5 = 22 + 1 = 232 + 1 = 4294967297 = 641 × 6700417 20 31. FERMAT POINT The Fermat point of a triangle, also called the Torricelli point or FermatTorricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem is first raised by Fermat in a private letter to Evangelista Torricelli, who solved it. The Fermat point gives a solution to the geometric median and Steiner tree problems for three points. 32. FERMAT VS. DESCARTE Fermat (1601-1665) worked on many of the same problems as Ren Descartes (1596-1650). They independently discovered analytic geometry, but since Fermat seldom published anything, Cartesian coordinates bear the name of Descartes, not Fermat. Both tried to restore the lost books of Apollonius, and when Fermat discovered a pair of amicable numbers, Descartes retaliated by finding another pair. Both discovered techniques for finding the line tangent to a given curve at a given point, and Fermat showed how to find the area under a curve given by the equation y = xn , as long as n was not equal to −1. All of this was very important in setting the stage for the discovery of calculus, later in the 1600s. Fermat and Descartes did not like each other very much. In fact, some people describe their relationship as a feud, but it seems that Descartes resented Fermat more than Fermat disliked Descartes. They probably never met. 21 33. FERMAT’S SPIRAL In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spiralsideally. That is because Fermat’s spiral traverses equal annuli in equal turns. The full model was proposed by H Vogel in 1979. Fermat’s spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants. 34. FIBONACCI POWERS 1, 8 and 144 are the only powers (power of a positive integer) in the Fibonacci Sequence 1, 1, 2, 3, 5, ... See http://www-irma.u-strasbg.fr/ bugeaud/travaux/fibo.pdf for the proof and more. 22 35. FOUR COLOR THEOREM The four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not. The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken’s approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. Since then the proof has gained wider acceptance, although doubts remain. To dispel remaining doubt about the AppelHaken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software. 23 36. GEORGE E P BOX While one can certainly make precise formalisations of at least some aspects of the models, one should not be inflexibly wedded to a specific such formalisation as being ”the” correct way to pin down the model rigorously. To quote the statistician George Box: ”all models are wrong, but some are useful”. 24 37. GOD’S ALGORITHM Some well known games with a very limited set of simple well-defined rules and moves have nevertheless never had their God’s algorithm for a winning strategy determined. Examples are the board games chess and Go. Both these games have a rapidly increasing number of positions with each move. The total number of all possible positions, approximately 10154 for chess and 10180 (on a 19 × 19 board) for Go, is much too large to allow a brute force solution with current computing technology (compare the now solved, with great difficulty, Rubik’s cube at only about 4.3 × 1019 positions). Consequently, a brute force determination of God’s algorithm for these games is not possible. True, chess computers have been built that are capable of beating even the best human players, but they do not calculate the game all the way to the end. Deep Blue, for instance, searched only 11 moves ahead reducing the search space to only 1017. After this, each position is assessed for advantage according to rules derived from human play and experience. Even this strategy is not possible with Go. Besides having hugely more positions to evaluate, no one so far has successfully constructed a set of simple rules for evaluating the strength of a Go position as has been done for chess. Evaluation algorithms are prone to make elementary mistakes so even for a limited look ahead with the goal limited to finding the strongest interim position, a God’s algorithm has not been possible for Go. On the other hand draughts, with superficial similarities to chess, has long been suspected of being ”played out” by its expert practitioners. In 2007 Schaeffer et al. proved this to be so by calculating a database of all positions with ten or fewer pieces. Thus Schaeffer has a God’s algorithm for all end games of draughts and used this to prove that all perfectly played games of draughts will end in a draw. However, draughts with only 5 × 1020 positions and even fewer, 3.9 × 1013 , in Schaeffer’s database, is a much easier problem to crack and is of the same order as Rubik’s cube. 25 38. GRAECO-LATIN SQUARE A Graeco-Latin square or Euler square or orthogonal Latin squares of order n over two sets S and T , each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T , such that every row and every column contains each element of S and each element of T exactly once, and that no two cells contain the same ordered pair. In the 1780s Euler demonstrated methods for constructing Graeco-Latin squares where n is odd or a multiple of 4. Observing that no order-2 square exists and unable to construct an order-6 square, he conjectured that none exist for any oddly even number n = 2 (mod 4). Indeed, the non-existence of order-6 squares was definitely confirmed in 1901 by Gaston Tarry through exhaustive enumeration of all possible arrangements of symbols. However, Euler’s conjecture resisted solution for a very long time. In 1959, R.C. Bose and S. S. Shrikhande constructed some counterexamples (dubbed the ‘Euler Spoilers’) of order 22 using mathematical insights. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search on a UNIVAC 1206 Military Computer while working at the UNIVAC division of Remington Rand (this was one of the earliest combinatorics problems solved on a digital computer). In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler’s conjecture to be false for all n ≥ 10. Thus, Graeco-Latin squares exist for all orders n ≥ 3 except n = 6. Graeco-Latin squares are used in the design of experiments, tournament scheduling, and constructing magic squares. The French writer Georges Perec structured his 1978 novel ‘Life: A User’s Manual’ around a 10 × 10 GraecoLatin square. 26 39. GRAFFITI CONJECTURES Graffiti is a computer program which makes conjectures in various subfields of mathematics (particularly graph theory) and chemistry, but can be adapted to other fields. It was written by Siemion Fajtlowicz at the University of Houston. Research on conjectures produced by Graffiti has led to over 60 publications by other mathematicians. 40. HABERDASHER’S PUZZLE Henry Ernest Dudeney (10 April 1857 to 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country’s foremost creators of mathematical puzzles. One of Dudeney’s most famous innovations were his 1903 success at solving the Haberdasher’s Puzzle (Cut an equilateral triangle into four pieces that can be rearranged to make a square) in affirmative. 27 41. HAPPY NUMBER A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers). For example, 19 is happy, as the associated sequence is: 12 + 92 = 82 82 + 22 = 68 62 + 82 = 100 12 + 02 + 0 2 = 1 The 143 happy numbers up to 1,000 are: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000. The happiness of a number is unaffected by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number. 28 42. HEAWOOD CONJECTURE In graph theory, the Heawood conjecture or RingelYoungs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. It was formulated in 1890 by Percy John Heawood and proven in 1968 by Gerhard Ringel and Ted Youngs. One case, the non-orientable Klein bottle, proved an exception to the general formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood’s original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3,... If g = 12s + k, the genera fall into 12 cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11. To simplify the discussion, let’s say that case k has been established if only a finite number of g’s of the form 12s + k are in doubt. Then the years in which the twelve cases were settled and by whom are the following: 1954, Ringel: case 5 1961, Ringel: cases 3,7,10 1963, Terry, Welch, Youngs: cases 0,4 1964, Gustin, Youngs: case 1 1965, Gustin: case 9 1966, Youngs: case 6 1967, Ringel, Youngs: cases 2,8,11 The last seven sporadic exceptions were settled as follows: 1967, Mayer: cases 18, 20, 23 1968, Ringel, Youngs: cases 30, 35, 47, 59, and the conjecture was proved. 29 43. HEMCHANDRA, THE INDIAN SCHOLAR Hemachandra, following the earlier Gopala, presented an earlier version of the Fibonacci sequence. It was presented around 1150, about fifty years before Fibonacci (1202). He was considering the number of cadences of length n, and showed that these could be formed by adding a short syllable to a cadence of length n − 1, or a long syllable to one of n − 2. This recursion relation F (n) = F (n − 1) + F (n − 2) is what defines the Fibonacci sequence. 44. HIPPASUS Hippasus of Metapontum (5th century BC), was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered √ the irrationality of 2, which it is believed was discovered around the time that he lived. 30 45. ILLEGAL NUMBER An illegal number may represent some type of classified information or trade secret, legal to possess only by certain authorized persons. An AACS encryption key that came to prominence in May 2007 is an example of a number claimed to be a secret, and whose publication or inappropriate possession is claimed to be illegal in the United States. It allegedly assists in the decryption of any HD DVD or Blu-ray Disc released before this date. The issuers of a series of cease-and-desist letters claim that the key itself is therefore a copyright circumvention device, and that publishing the key violates Title 1 of the US Digital Millennium Copyright Act. In part of the DeCSS court order and in the AACS legal notices, the claimed protection for these numbers is based on their mere possession and the value or potential use of the numbers. This makes their status and legal issues surrounding their distribution quite distinct from that of mere copyright infringement. Any image file or an executable program can be regarded as simply a very large binary number. In certain jurisdictions, there are images that are illegal to possess, due to obscenity or secrecy/classified status, so the corresponding numbers could be illegal. In 2011 Sony sued George Hotz and members of fail0verflow for jailbreaking the PlayStation 3. Part of the lawsuit complaint was that they had published PS3 keys. Sony also threatened to sue anyone who distributed the keys. Sony later accidentally tweeted an older dongle key through its fictional Kevin Butler character. 31 46. IMAGINARY TIME ‘One might think that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?’ from Prof. Stephen Hawking Imaginary time is a concept derived from quantum mechanics and is essential in connecting quantum mechanics with statistical mechanics. 47. INFINITUDE OF PRIMES This is just a curiosity. We have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. It is nice to show what proofs are there and like to know more because it’s cool to see that something can be proved in so many different ways. 32 48. JOSEPHUS PROBLEM The Josephus Problem, or Josephus permutation, is a theoretical problem related to a certain counting-out game. There are people standing in a circle waiting to be executed. The counting out begins at some point in the circle and proceeds around the circle in a fixed direction. In each step, a certain number of people are skipped and the next person is executed. The elimination proceeds around the circle, which is becoming smaller and smaller as the executed people are removed, until only the last person remains, who is given freedom. The task is to choose the place in the initial circle so that you are the last one remaining and so survive. The problem is named after Flavius Josephus, a Jewish historian living in the 1st century. According to Josephus’ account of the siege of Yodfat, he and his 40 soldiers were trapped in a cave, the exit of which was blocked by Romans. They chose suicide over capture and decided that they would form a circle and start killing themselves using a step of three. Josephus states that by luck or possibly by the hand of God, he and another man remained the last and gave up to the Romans. 33 49. KAKEYA SET A Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero. A Kakeya needle set, sometimes also known as a Kakeya set, is a Besicovitch set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation. Again, the disk of radius 1/2 is an example of a Kakeya needle set. Besicovitch showed that there are Kakeya needle sets of arbitrarily small positive measure. 34 50. KAPREKAR’S CONSTANT 6174 is known as Kaprekar’s constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property: 1. Take any four-digit number, using at least two different digits. (Leading zeros are allowed.) 2. Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. 3. Subtract the smaller number from the bigger number. 4. Go back to step 2. The above process, known as Kaprekar’s routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 1467 = 6174. The only four-digit numbers for which Kaprekar’s routine does not reach 6174 are repdigits such as 1111, which give the result 0. 495 is the equivalent constant for three-digit numbers. For two-digit numbers, there is no equivalent constant; for any starting number with differing digits, the routine enters the loop (45, 9, 81, 63, 27, 45). For each digit length greater than four, the routine may terminate at one of several fixed values or may enter one of several loops instead. 35 51. KEPLER TRIANGLE A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571 to 1630), who first demonstrated that this triangle is characterised by a ratio between short side and hypotenuse equal to the golden ratio. Kepler triangles combine two key mathematical conceptsthe Pythagorean theorem and the golden ratiothat fascinated Kepler deeply, as he expressed in this quotation: ‘Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.’ Some sources claim that a triangle with dimensions closely approximating a Kepler triangle can be recognized in the Great Pyramid of Giza. 36 52. KNIGHTS’S TOUR A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Creating a program to find a knight’s tour is a common problem given to computer science students. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 8, as well as irregular (non-rectangular) boards. For any m×n board with m ≤ n, a closed knight’s tour is always possible unless one or more of these three conditions are true: (1) m and n are both odd and m and n are not both 1, (2) m = 1, 2, or 4 and m and n are not both 1, (3) m = 3 and n = 4, 6, or 8 37 53. LAZY CATERER’S SEQUENCE The lazy caterer’s sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle, a pancake or pizza is usually used to describe the situation, that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines and can be generalized to higher dimensions. The analogue of this sequence in 3 dimensions is the cake number. In two dimensions the maximum number p of pieces that can be created with a given number of cuts n, where n ≥ 0, is given by the formula p= n2 + n + 2 2 . 38 54. LIAR PARADOX On someone’s T-shirt it is written that ‘I am a liar’. The question is whether the statement is True or False. If the statement is True then the person wearing the shirt is a liar and the statement coming from a liar cannot be True. If the statement is False, then the person wearing the T-shirt is not a liar hence his words must be True. The Indian grammarian-philosopher Bhartrhari (late fifth century CE) was well aware of a liar paradox which he formulated as ‘everything I am saying is false’ (sarvam mithya bravimi). He analyzes this paradox together with the paradox of ‘unsignifiability’ and explores the boundary between statements that are unproblematic in daily life and paradoxes. In early Islamic tradition liar paradox was discussed for at least five centuries starting from late 9th century apparently without being influenced by any other tradition. Nasir al-Din al-Tusi could have been the first logician to identify the liar paradox as self-referential. 39 55. LIFE INSURANCE - THE CONTRIBUTORS ‘The history of mortality tables and life insurance is sprinkled with the names of people more famous for other things. American composer Charles Ives, who, like your columnist, worked in Danbury, Connecticut, also invented the insurance agency, so that insurance customers themselves no longer had to negotiate directly with the insurance companies. Edmund Halley, of comet fame, devoted a good deal of energy to calculating one of the earlier mortality tables. Henry Briggs, better known for his pioneering work with logarithms, calculated interest tables. Swiss religious reformer John Calvin preached that life insurance was not necessarily immoral usury, as some maintained at the time. Daniel Defoe, author of Gulliver’s Travels, proposed a national insurance scheme for England in 1697. We find other familiar names, DeMoivre, Fermat, Harriot, Hudde, Huygens, de Witt, van Shooten, Maclaurin, Maseres, Pepys and, of course, Euler.’ - From How Euler Did Even More by Ed Sandifer 56. MARTIN GARDNER ‘Mathematicians insist on analyzing anything analyzable.’ - Martin Gardner Gardner was best known for creating and sustaining general interest in recreational mathematics for a large part of the 20th century, principally through his Scientific American ‘Mathematical Games’ columns from 1956 to 1981 and subsequent books collecting them. He was an uncompromising critic of fringe science and was a founding member of CSICOP, an organization devoted to debunking pseudoscience, and wrote a monthly column, ‘Notes of a Fringe Watcher’, from 1983 to 2002 in Skeptical Inquirer, that organization’s monthly magazine. He also wrote a ‘Puzzle Tale’ column for Asimov’s Science Fiction magazine from 1977 to 1986 and altogether published more than 100 books. 40 57. MATHEMATICIAN’S ANGER If nobody understands a mathematical proof, does it count? Shinichi Mochizuki of Kyoto University, Japan, has tried to prove the ABC conjecture, a long-standing pure maths problem, but now says fellow mathematicians are failing to get to grips with his work. 58. MCNUGGET NUMBERS The McNuggets version of the coin problem was introduced by Henri Picciotto, who included it in his algebra textbook co-authored with Anita Wah. Picciotto thought of the application in the 1980s while dining with his son at McDonald’s, working the problem out on a napkin. A McNugget number is the total number of McDonald’s Chicken McNuggets in any number of boxes. The original boxes (prior to the introduction of the Happy Meal-sized nugget boxes) were of 6, 9, and 20 nuggets. According to Schur’s theorem, since 6, 9, and 20 are relatively prime, any sufficiently large integer can be expressed as a linear combination of these three. Therefore, there exists a largest non-McNugget number, and all integers larger than it are McNugget numbers. Namely, every positive integer is a McNugget number, with the finite number of exceptions: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. 41 59. MINESWEEPER Minesweeper is a single-player puzzle video game. The objective of the game is to clear a rectangular board containing hidden ‘mines’ without detonating any of them, with help from clues about the number of neighboring mines in each field. The game originates from the 1960s, and has been written for many computing platforms in use today. It has many variations and offshoots. The game is played by revealing squares of the grid by clicking or otherwise indicating each square. If a square containing a mine is revealed, the player loses the game. If no mine is revealed, a digit is instead displayed in the square, indicating how many adjacent squares contain mines; if no mines are adjacent, the square becomes blank, and all adjacent squares will be recursively revealed. The player uses this information to deduce the contents of other squares, and may either safely reveal each square or mark the square as containing a mine. In 2000, Richard Kaye published a proof that it is NP-complete to determine whether a given grid of uncovered, correctly flagged, and unknown squares, the labels of the foremost also given, has an arrangement of mines for which it is possible within the rules of the game. The argument is constructive, a method to quickly convert any Boolean circuit into such a grid that is possible if and only if the circuit is satisfiable; membership in NP is established by using the arrangement of mines as a certificate. If, however, a minesweeper board is already guaranteed to be consistent, solving it is not known to be NP-complete, but interestingly it has been proven to be co-NP-complete. 42 60. MORLEY’S TRISECTOR THEOREM In plane geometry, Morley’s trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles. There are many proofs of Morley’s theorem, some of which are very technical. Several early proofs were based on delicate trigonometric calculations. The first published geometric proof was given by M. T. Naraniengar in 1909. Recent proofs include an algebraic proof by Alain Connes extending the theorem to general fields, and John Conway’s elementary geometry proof. The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be similar to any selected triangle. Morley’s theorem does not hold in spherical and hyperbolic geometry. 61. N QUEENS PROBLEM In how many independent ways, say Q(N ), N Queens can be arranged in an N × N chess board so that none attacks another? Q(8) = 12 not considering the symmetries. In 1992, Demirrs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa. 43 62. NAGEL POINT The Nagel point is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. Given a triangle ABC, let TA , TB , and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively. The lines ATA , BTB , CTC concur in the Nagel point N of triangle ABC. The Nagel point is named after Christian Heinrich von Nagel, a nineteenthcentury German mathematician, who wrote about it in 1836. Another construction of the point TA is to start at A and trace around triangle ABC half its perimeter, and similarly for TB and TC . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments ATA , BTB , CTC are called the triangle’s splitters. The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line. The incenter is the Nagel point of the medial triangle; equivalently, the Nagel point is the incenter of the anticomplementary triangle. 44 63. NAPOLEON’S THEOREM Napoleon’s theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centres of those equilateral triangles themselves form an equilateral triangle. The triangle thus formed is called the Napoleon triangle (inner and outer). The difference in area of these two triangles equals the area of the original triangle. The theorem is often attributed to Napoleon Bonaparte (1769 to 1821). However, it may just date back to W. Rutherford’s 1825 question published in The Ladies’ Diary, four years after the French emperor’s death. Plainly there is no reference to Napoleon in either the question or the published responses, which appeared a year later in 1826, though the Editor evidently omitted some submissions. Also Rutherford himself does not appear amongst the named solvers. The first known reference to this result as Napoleon’s theorem appears in Faifofer’s 17th Edition of Elementi di Geometria published in 1911. 45 64. NINE-POINT CIRCLE The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: The midpoint of each side of the triangle The foot of each altitude The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes). The nine-point circle is also known as Feuerbach’s circle, Euler’s circle, Terquem’s circle, the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle. In 1822 Karl Feuerbach discovered that any triangle’s nine-point circle is externally tangent to that triangle’s three excircles and internally tangent to its incircle; this result is known as Feuerbach’s theorem. He proved that: ‘... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle...’ (Feuerbach 1822) The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point. 46 65. NORMAL NUMBER A normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2 , all b3 triplets of digits equally likely with density b−3 , etc. Intuitively this means that no digit, or combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is ‘favored’. While a general proof can be given that almost all real numbers are normal, in the sense that the set of exceptions has Lebesgue measure zero, this proof is not constructive and only very few specific numbers have been shown to be normal. For example, it is widely believed that the numbers √ 2, π, and e are normal, but a proof remains elusive. 47 66. OH! I FORGOT In 1868, Smith returned to the geometrical researches which had first occupied his attention. For a memoir on ”Certain cubic and biquadratic problems” the Royal Academy of Sciences of Berlin awarded him the Steiner prize. In February, 1882, Smith was surprised to see in the Comptes rendus that the subject proposed by the Paris Academy of Science for the Grand prix des sciences mathmatiques was the theory of the decomposition of integer numbers into a sum of five squares; and that the attention of competitors was directed to the results announced without demonstration by Eisenstein, whereas nothing was said about his papers dealing with the same subject in the Proceedings of the Royal Society. He wrote to M. Hermite calling his attention to what he had published; in reply he was assured that the members of the commission did not know of the existence of his papers, and he was advised to complete his demonstrations and submit the memoir according to the rules of the competition. According to the rules each manuscript bears a motto, and the corresponding envelope containing the name of the successful author is opened. There were still three months before the closing of the concours (1 June 1882) and Smith set to work, prepared the memoir and despatched it in time. Two months after Smith’s death, the Paris Academy made their award. Two of the three memoirs sent in were judged worthy of the prize. When the envelopes were opened, the authors were found to be Smith and Minkowski, a young mathematician of Koenigsberg, Prussia. No notice was taken of Smith’s previous publication on the subject, and M. Hermite on being written to, said that he forgot to bring the matter to the notice of the commission. 48 67. PANDIGITAL PRIME A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1223334444555567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by : 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689. No base 10 pandigital number can be a prime number if it doesn’t have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9. The first base 10 pandigital prime is 10123457689. 49 68. PAUL ERDOS Paul Erdos (Hungarian: 26 March 1913 to 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians of the 20th century, but also known for his social practice of mathematics with more than 500 collaborators and eccentric lifestyle. Erdos pursued problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. Possessions meant little to Erdos; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life as a vagabond, traveling between scientific conferences, universities and the homes of colleagues all over the world. He earned enough in stipends from universities as a guest lecturer, and from various mathematical awards to fund his travels and basic needs; money left over he used to fund cash prizes for proofs of ”Erdos problems” (see below). He would typically show up at a colleague’s doorstep and announce ”my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom to visit next. Because of his prolific output, friends created the Erdos number as a tribute. An Erdos number describes a person’s degree of separation from Erdos himself, based on their collaboration with him, or with another who has their own Erdos number. Erdos alone was assigned the Erdos number of 0 for being himself, while his immediate collaborators could claim an Erdos number of 1, their collaborators have Erdos number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdos number, and some have estimated that 90 percent of the world’s active mathematicians have an Erdos number smaller than 8 not surprising in light of the small world phenomenon. Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics have Erdos numbers as well. 50 69. PELL’S EQUATION Pell’s equation is any Diophantine equation of the form x2 − ny 2 = 1, where n is a given positive nonsquare integer and integer solutions are sought for x and y. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell’s equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. This equation was first studied extensively in India, starting with Brahmagupta, who developed the chakravala method to solve Pell’s equation and other quadratic indeterminate equations in his Brahma Sphuta Siddhanta in 628, about a thousand years before Pell’s time. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell’s equation and other quadratic indeterminate equations. Solutions to specific examples of the Pell equation, such as the Pell numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and to a similar date in India. The name of Pell’s equation arose from Leonhard Euler’s mistakenly attributing its study to John Pell. Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell. 51 70. PENROSE TILING A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry. A Penrose tiling has many remarkable properties, most notably: (1) It is non-periodic, which means that it lacks any translational symmetry. (2) It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through ‘inflation’ or ‘deflation’ and any finite patch from the tiling occurs infinitely many times. (3) It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order. Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes and coverings. 52 71. PERFECT NUMBER In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). This definition is ancient, appearing as early as Euclid’s Elements where it is called perfect, ideal, or complete number. Euclid also proved a formation rule whereby p(p + 1)/2 is an even perfect number whenever p is what is now called a Mersenne prime. Much later, Euler proved that all even perfect numbers are of this form. This is known as the EuclidEuler theorem. For example, the first four perfect numbers are generated by the formula 2p−1 (2p − 1), with p a prime number, as follows: for p = 2 : 21 (22 − 1) = 6 for p = 3 : 22 (23 − 1) = 28 for p = 5 : 24 (25 − 1) = 496 for p = 7 : 26 (27 − 1) = 8128. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. Settling the existence of odd perfect number is the oldest unresolved problem in today’s Mathematics. In 1888, Sylvester stated: ‘...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] its escape, so to say, from the complex web of conditions which hem it in on all sides would be little short of a miracle’. 53 72. PROBLEM OF APPOLLONIUS Apollonius’s problem is to construct circles that are tangent to three given circles in a plane. Apollonius of Perga (ca. 262 BC ca. 190 BC) posed and solved this famous problem in his work ‘Tangencies’; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8 = 23 . However the general statement of Apollonius’ problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. Solutions to Apollonius’ problem are sometimes called Apollonius circles, although the term is also used for other types of circles associated with Apollonius. 54 73. PROF. ANIL BHATTACHARYA In statistics, the Bhattacharyya distance measures the similarity of two discrete or continuous probability distributions. It is closely related to the Bhattacharyya coefficient which is a measure of the amount of overlap between two statistical samples or populations. Both measures are named after Anil Kumar Bhattacharya, a statistician who worked in the 1930s at the Indian Statistical Institute. The coefficient can be used to determine the relative closeness of the two samples being considered. It is used to measure the separability of classes in classification and it is considered to be more reliable than the Mahalanobis distance, as the Mahalanobis distance is a particular case of the Bhattacharyya distance when the standard deviations of the two classes are the same. Therefore, when two classes have similar means but different standard deviations, the Mahalanobis distance would tend to zero, however, the Bhattacharyya distance would grow depending on the difference between the standard deviations. 76. PYTHAGOREAN THEOREM IN ANCIENT INDIA ‘Another standard would involve,’ he adds, ‘the requirement of a document that explicitly states the Pythagorean theorem – the geometric theorem. That first occurs about 800 BC in India in the Shuba Sutra of Baudhayan.’ Dr Manjul Bhargava (Fields Medal winner and Princeton University Professor) 55 75. QUANTUM COMPUTING Quantum computing studies theoretical computation systems, quantum computers, that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses qubits (quantum bits), which can be in superpositions of states. A quantum Turing machine is a theoretical model of such a computer, and is also known as the universal quantum computer. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers. The field of quantum computing was initiated by the work of Yuri Manin in 1980, Richard Feynman in 1982, and David Deutsch. A quantum computer with spins as quantum bits was also formulated for use as a quantum spacetime in 1968. As of 2015, the development of actual quantum computers is still in its infancy, but experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Both practical and theoretical research continues, and many national governments and military agencies are funding quantum computing research in an effort to develop quantum computers for civilian, business, trade, and national security purposes, such as cryptanalysis. Large-scale quantum computers will be able to solve certain problems much more quickly than any classical computers that use even the best currently known algorithms, like integer factorization using Shor’s algorithm or the simulation of quantum many-body systems. There exist quantum algorithms, such as Simon’s algorithm, that run faster than any possible probabilistic classical algorithm. Given sufficient computational resources, however, a classical computer could be made to simulate any quantum algorithm, as quantum computation does not violate the ChurchTuring thesis. 56 76. QUATERNION Quaternion algebra was introduced by Hamilton in 1843. Important precursors to this work included Euler’s four-square identity (1748) and Olinde Rodrigues’ parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in threedimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j 2 = k 2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. 57 77. QUESTION MARK FUNCTION ?(x) The question mark function is a strictly increasing and continuous, but not absolutely continuous function. The derivative vanishes on the rational numbers. There are several constructions for a measure that, when integrated, yields the question mark function. One such construction is obtained by measuring the density of the Farey numbers on the real number line. The question mark measure is the prototypical example of what are sometimes referred to as multi-fractal measures. The question mark function maps rational numbers to dyadic rational numbers, meaning those whose base two representation terminates, as may be proven by induction from the recursive construction outlined above. It maps quadratic irrationals to non-dyadic rational numbers. It is an odd function, and satisfies the functional equation ?(x+1) =?(x)+1; consequently x →?(x) − x is an odd periodic function with period one. If ?(x) is irrational, then x is either algebraic of degree greater than two, or transcendental. The question mark function has fixed points at 0, 1/2 and 1, and at least two more, symmetric about the midpoint. One is approximately 0.42037. The graph of Minkowski question mark function is a special case of fractal curves known as de Rham curves. 58 78. RAMANUJAN PRIME A Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. In 1919, Ramanujan published a new proof of Bertrand’s postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: π(x)− π(x/2) = 1, 2, 3, 4, 5, · · · for all x = 2, 11, 17, 29, 41, · · · respectively, where π(x) is the prime-counting function, equal to the number of primes less than or equal to x. The converse of this result is the definition of Ramanujan primes: The nth Ramanujan prime is the least integer Rn for which π(x) − π(x/2) ≥ n, for all x ≥ Rn . The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently, Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2 for all x ≥ Rn . Note that the integer Rn is necessarily a prime number: π(x) − π(x/2) and, hence, π(x) must increase by obtaining another prime at x ≥ Rn . Since π(x) − π(x/2) can increase by at most 1, π(Rn ) − π(Rn /2) = n. 59 79. RAMSEY NUMBER The Ramsey number R(m, n) gives the solution to the party problem, which asks the minimum number of guests R(m, n) that must be invited so that at least m will know each other or at least n will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices v = R(m, n) such that all undirected simple graphs of order v contain a clique of order m or an independent set of order n. Ramsey’s theorem states that such a number exists for all m and n. Till recent times the exact values of the following Ramsey Numbers are known to us. m n R(m, n) Reference 3 3 6 Greenwood and Gleason 1955 3 4 9 Greenwood and Gleason 1955 3 5 14 Greenwood and Gleason 1955 3 6 18 Graver and Yackel 1968 3 7 23 Kalbfleisch 1966 3 8 28 McKay and Min 1992 3 9 36 Grinstead and Roberts 1982 4 4 18 Greenwood and Gleason 1955 4 5 25 McKay and Radziszowski 1995 80. RANDOM WALK IN MULTIDIMENSIONS Let p(d) denote the probability of eventual return in a d-dimensional symmetric random walk. Then, p(1) = 1, p(2) = 1, p(3) = 0.34, p(4) = 0.20, p(5) = 0.136, p(6) = 0.105, p(7) = 0.0858, p(8) = 0.0729 60 81. REGULAR HEPTADECAGON The regular heptadecagon is a constructible polygon, that is, one that can be constructed using a compass and unmarked straightedge, as was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years. Gauss’s proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of the number of sides are distinct Fermat primes, which are of the form Fn =22n +1. Constructing a regular heptadecagon thus involves finding the cosine of 2π/17 in terms of square roots, which involves an equation of degree 17, a Fermat prime. Constructions for the regular triangle and polygons with 2h times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. The only known Fermat primes today are Fn for n = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537. The first explicit construction of a heptadecagon was given by Johannes Erchinger in 1825. Another method of construction uses Carlyle circles. Based on the construction of the regular 17-gon, one can readily construct n-gons with n being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular n-gon with 2h times as many sides. 61 82. RUBIK’S CUBE In the mid-1970s, Erno Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. Although it is widely reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart. He did not realize that he had created a puzzle until the first time he scrambled his new Cube and then tried to restore it. Rubik obtained Hungarian patent HU170062 for his ‘Magic Cube’ in 1975. Rubik’s Cube was first called the Magic Cube (Buvs kocka) in Hungary. The puzzle had not been patented internationally within a year of the original patent. Patent law then prevented the possibility of an international patent. Ideal wanted at least a recognizable name to trademark; of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980. The original (3 × 3 × 3) Rubik’s Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an even permutation of the corners implies an even permutation of the edges as well. When arrangements of centres are also permitted, as described below, the rule is that the combined arrangement of corners, edges, and centres must be an even permutation. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities. So the total number of possible arrangements are 8! × 37 × (12!/2) × 211 = 43, 252, 003, 274, 489, 856, 000 which is approximately 43 quintillion. The puzzle is often advertised as having only ‘billions’ of positions, as the larger numbers are unfamiliar to many. To put this into perspective, if one had as many standard sized Rubik’s Cubes as there are permutations, one could cover the Earth’s surface 275 times. 62 83. SACCHERI QUADRILATERAL A Saccheri quadrilateral (also known as a KhayyamSaccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclides ab omni naevo vindicatus (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum. The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral. Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry. 1. The summit angles (at C and D) are equal and acute. 2. The summit is longer than the base. 3. The line segment joining the midpoint of the base and the midpoint of the summit is mutually perpendicular to the base and summit. 4. The line segment joining the midpoints of the sides is not perpendicular to either side. 5. The above two line segments are perpendicular to each other. 6. The line segment joining the midpoint of the base and the midpoint of the summit divides the Saccheri quadrilateral into two Lambert quadrilaterals. 7. Two Saccheri quadrilaterals with congruent bases and congruent summit angles are congruent (i.e., the remaining pairs of corresponding parts are congruent). 8. Two Saccheri quadrilaterals with congruent summits and congruent summit angles are congruent. 63 84. SIERPINSKI TRIANGLE The Sierpinski triangle, also with the original orthography Sierpinski, also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Waclaw Sierpinski described the Sierpinski triangle in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy, and other places of central Italy, for carpets in many places such as the nave of the Roman Basilica of Santa Maria in Cosmedin, and for isolated triangles positioned in rotae in several churches and Basiliche. In the case of the isolated triangle, it is interesting to notice that the iteration is at least of three levels. 64 85. SIEVE OF SUNDARAM Sundaram’s sieve is based on an array of numbers formed from arithmetic progressions, in other words, sequences of numbers in which successive numbers are a given fixed distance apart. We start with the infinite sequence in which successive numbers are exactly three steps apart, and which starts with the number 4: 4, 7, 10, 13, 16, 19, 22, 25, ... Next up is the sequence starting with the number 7 and with successive numbers exactly 5 steps apart: 7, 12, 17, 22, 27, 32, 37, 42, ... Now consider the sequence starting with the number 10 and with a difference of 7 between successive numbers: 10, 17, 24, 31, 38, 45, 52, 59, ... One can see the pattern emerging: the starting point of each sequence is three steps on from that of the previous one, and the distance between successive numbers is the distance of the previous sequence plus 2. Writing the sequences beneath each other, we get the following doubly infinite array: 4 7 10 13 16 .. 7 10 13 16 19 22 25 ... 12 17 22 27 32 37 42 ... 17 24 31 38 45 52 59 ... 22 31 40 49 58 67 76 ... 27 38 49 60 71 82 93 ... .. .. .. .. .. .. .. ... The array exhibits a lot of structure: the first row is equal to the first column, each row is an arithmetic progression, and the differences between entries in two consecutive rows form the sequence 3, 5, 7, 9, 11, 13, 15, 17, ... which is itself an arithmetic progression. Now for any number N in this array 2N+1 is not a prime, and conversely, for any number N not in the array, the number 2N+1 is prime. There are other methods for sieving out primes, including the famous sieve of Eratosthenes and the lesser-known visual sieve. These methods are altogether different, although their workings are equally mysterious and the mathematics involved is similarly elementary. 65 86. SIMSON LINE Given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P, named for Robert Simson. The concept was first published, however, by William Wallace in 1797. The converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on the circumcircle of the triangle formed by the three lines. Or in other words, the Simson line of a triangle ABC and a point P is just the pedal triangle of ABC and P that has degenerated into a straight line and this condition constrains the locus of P to trace the circumcircle of triangle ABC. 87. SP NUMBER In March 1999 issue of THE MATHEMATICAL GAZETTE, in an interesting article titled ”SP Numbers”, the author defines a positive integer n to be an SP number if it equals the product of the sum of its digits and the product of its digits. The only SP numbers are proved to be 1, 135 and 144. 66 88. SPIROGRAPH The mathematician Bruno Abakanowicz invented the spirograph between 1881 and 1900. It was used for calculating an area delimited by curves. Drawing toys based on gears have been around since at least 1908, when The Marvelous Wondergraph was advertised in the Sears catalog. An article describing how to make a Wondergraph drawing machine appeared in the Boys Mechanic publication in 1913. The Spirograph itself was developed by the British engineer Denys Fisher, who exhibited at the 1965 Nuremberg International Toy Fair. It was subsequently produced by his company. US distribution rights were acquired by Kenner, Inc., which introduced it to the United States market in 1966 and promoted it as a creative children’s toy. In 2013 the Spirograph brand was re-launched in the USA by Kahootz Toys and in Europe by Goldfish and Bison with products that returned to the use of the original gears and wheels. The modern products use removable putty in place of pins or are held down by hand to keep the stationary pieces in place on the paper. The Spirograph was a 2014 Toy of the Year finalist in 2 categories, almost 50 years after the toy was named Toy of the Year in 1967. 67 89. SQUARING THE PLANE In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares whose sizes are all natural numbers without repetitions, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his Scientific American column and appeared in several books, but it defied solution for over 30 years. In Tilings and Patterns, published in 1987, Branko Grnbaum and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares grew exponentially. Recently, James Henle and Frederick Henle proved that this, in fact, can be done. Their proof is constructive and proceeds by ”puffing up” an Lshaped region formed by two side-by-side and horizontally flush squares of different sizes to a perfect tiling of a larger rectangular region, then adjoining the square of the smallest size not yet used to get another, larger L-shaped region. The squares added during the puffing up procedure have sizes that have not yet appeared in the construction and the procedure is set up so that the resulting rectangular regions are expanding in all four directions, which leads to a tiling of the whole plane. 90. ST. PETERSBURG PARADOX St. Petersburg paradox takes its name from its resolution by Daniel Bernoulli, one-time resident of the eponymous Russian city, who published his arguments in the Commentaries of the Imperial Academy of Science of Saint Petersburg. However, the problem was invented by Daniel’s cousin Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort on September 9, 1713. 68 91. STIGLER’S LAW OF EPONYMY Stigler’s law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication ‘Stiglers law of eponymy’. In its simplest and strongest form it says: ‘No scientific discovery is named after its original discoverer.’ Stigler named the sociologist Robert K. Merton as the discoverer of ‘Stigler’s law’, so as to avoid this law about laws disobeying its very own decree. Historical acclaim for discoveries is often assigned to persons of note who bring attention to an idea that is not yet widely known, whether or not that person was its original inventor theories may be named long after their discovery. In the case of eponymy, the idea becomes named after that person, even if that person is acknowledged by historians of science not to be the one who discovered it. Often, several people will arrive at a new idea around the same time, as in the case of calculus. It can be dependent on the publicity of the new work and the fame of its publisher as to whether the scientist’s name becomes historically associated. 69 92. SUM OF FOUR CUBES Experiments led several mathematicians to conjecture that every integer is the sum of four integer cubes. As yet, no proofs exits, but the evidence is substantial and some progress have been made. It would be enough to prove the statement for all positive integers. Computer calculations verify that every positive integer up to 10 million is a sum of four cubes. And in 1966 V. Demjanenko proved that any number not of the form 9k+4 and 9k-4 is a sum of four cubes. It is even possible that with a finite number of exceptions, every positive integer might be the sum of four positive or zero cubes. In 2000 JeanMarc Deshouillers, Francois Hennecart, Bernard Landreau and I. Gusti Putu Purnaba conjectured that the largest integer that cannot be so expressed is 7,373,170,279,850. - Professor Stewart’s Incredible Numbers 93. TAXICAB NUMBER In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways. The name is derived from a conversation involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. ‘No’, he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.’ 70 So far, the following six taxicab numbers are known: T a(1) = 2 = 13 + 13 T a(2) = 1729 = 13 + 123 = 93 + 103 T a(3) = 87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143 T a(4) = 6963472309248 = 24213 + 190833 = 54363 + 189483 = 102003 + 180723 = 133223 + 166303 T a(5) = 48988659276962496 = 387873 + 3657573 = 1078393 + 3627533 = 2052923 + 3429523 = 2214243 + 3365883 = 2315183 + 3319543 T a(6) = 24153319581254312065344 = 5821623 + 289062063 = 30641733 + 288948033 = 85192813 + 286574873 = 162180683 + 270932083 = 174924963 + 265904523 = 182899223 + 262243663 71 94. THE BOOK Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Gnter M. Ziegler. The book is dedicated to the mathematician Paul Erdos, who often referred to ”The Book” in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdos said, ”You don’t have to believe in God, but you should believe in The Book.” Proofs from THE BOOK contains 32 sections (44 in the fifth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, analysis, combinatorics and graph theory. Erdos himself made many suggestions for the book, but died before its publication. The book is illustrated by Karl Heinrich Hofmann. It has gone through five editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian and Spanish. 72 95. THREE BODY PROBLEM AND CHAOS THEORY The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since Newton’s time. This was known originally as the three-body problem and later the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gsta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: Given a system of arbitrarily many mass points that attract each according to Newton’s law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly. In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincar, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, ”This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics.” The first version of Poincar’s contribution contained a serious error. The version finally printed contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s. 73 96. TRIANGLE CENTERS Even though the ancient Greeks discovered the classic centers of a triangle they had not formulated any definition of a triangle center. After the ancient Greeks, several special points associated with a triangle like the Fermat point, nine-point center, symmedian point, Gergonne point, and Feuerbach point were discovered. During the revival of interest in triangle geometry in the 1980s it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center. In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting. They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in Clark Kimberling’s Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992. In geometry the Morley centers are two special points associated with a plane triangle. Both of them are triangle centers. One of them called first Morley center (or simply, the Morley center) is designated as X(356) in Clark Kimberling’s Encyclopedia of Triangle Centers, while the other point called second Morley center (or the 1st MorleyTaylorMarr Center) is designated as X(357). The two points are also related to Morley’s trisector theorem which was discovered by Frank Morley in around 1899. As of 11 November 2014, Clark Kimberling’s Encyclopedia of Triangle Centers contains an annotated list of 6,102 triangle centers. 74 97. TRUNCATABLE PRIMES In number theory, a left-truncatable prime is a prime number which contains no 0, and if the leading (”left”) digit is successively removed, then all resulting numbers are prime. For example 9137, since 9137, 137, 37 and 7 are all prime. There are exactly 4260 decimal left-truncatable primes and the largest is the 24-digit 357686312646216567629137. A right-truncatable prime is a prime which remains prime when the last (”right”) digit is successively removed. For example 7393, since 7393, 739, 73, 7 are all prime. There are 83 right-truncatable primes and the largest is the 8-digit 73939133. There are 15 primes which are both left-truncatable and right-truncatable. They have been called two-sided primes. The complete list is 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397. 75 98. ULAM SPIRAL The Ulam spiral, or prime spiral, in other languages also called the Ulam Cloth, is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a ‘long and very boring paper’ at a scientific meeting. Shortly afterwards, in an early application of computer graphics, Ulam with collaborators Myron Stein and Mark Wells used MANIAC II at Los Alamos Scientific Laboratory to produce pictures of the spiral for numbers up to 65,000. In March of the following year, Martin Gardner wrote about the Ulam spiral in his Mathematical Games column; the Ulam spiral featured on the front cover of the issue of Scientific American in which the column appeared. In an addendum to the Scientific American column, Gardner mentions work of the herpetologist Laurence M. Klauber on two dimensional arrays of prime numbers for finding prime-rich quadratic polynomials which was presented at a meeting of the Mathematical Association of America in 1932more than thirty years prior to Ulam’s discovery. Unlike Ulam’s array, Klauber’s was not a spiral. Its shape was also triangular rather than square. 76 99. UNEXPECTED HANGING PARADOX The unexpected hanging paradox or hangman paradox is a paradox about a person’s expectations about the timing of a future event that he is told will occur at an unexpected time. The paradox is variously applied to a prisoner’s hanging, or a surprise school test. The paradox has been described as follows: A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the ”surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left - and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner’s door at noon on Wednesday which, despite all the above, was an utter surprise to him. Everything the judge said came true. Despite significant academic interest, there is no consensus on its precise nature and consequently a final correct resolution has not yet been established. One approach, offered by the logical school of thought, suggests that the problem arises in a self-contradictory self-referencing statement at the heart of the judge’s sentence. Another approach, offered by the epistemological school of thought, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge. Even though it is apparently simple, the paradox’s underlying complexities have even led to it being called a ‘significant problem’ for philosophy. 77 100. ZNAM’S PROBLEM One solution to the improper Znm problem is easily provided for any k, the first k terms of Sylvester’s sequence have the required property. Sun (1983) showed that there is at least one solution to the (proper) Znm problem for each k = 5. Sun’s solution is based on a recurrence similar to that for Sylvester’s sequence, but with a different set of initial values. It is known that there are only finitely many solutions for any fixed k. It is unknown whether there are any solutions to Znm’s problem using only odd numbers, and there remain several other open questions. 78