OFFSET
2,1
COMMENTS
An information-theoretic bound on the largest card deck with which one can perform an n-card trick in which the audience chooses two cards to hide.
The bound is based on the following argument: The assistant has (n-2)! ways to arrange the cards. This number can't be smaller than the number of potential guesses by the magician which is binomial(d - n + 2, 2), where d is the deck size.
LINKS
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 20.
Michael Kleber, The best card trick, The Mathematical Intelligencer 24 (2002), 9-11.
EXAMPLE
For n=3, the constraint on the deck size becomes: binomial(d-1, 2) can't exceed 1!=1. Thus a(3) = 3.
MATHEMATICA
Table[Floor[(2 k - 3 + Sqrt[1 + 8 (k - 2)!])/2], {k, 2, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Tanya Khovanova and the MIT PRIMES STEP junior group, Apr 24 2024
STATUS
approved