OFFSET
1,2
COMMENTS
Suppose there are two types of coins (genuine and counterfeit) with different weights, only one of the weights known, and n independent mints each making coins of only one of the two types. Then a(n) is the minimum number of coins needed to determine in two weighings which mints are making counterfeit coins. - Charles R Greathouse IV, Jun 16 2014
Guy and Nowakowski give a(6) <=38 and a(7)<=74. Li improves this to a(6) <=31 and a(7)<=64. a(6)=28 is given by exhaustive search of all variants up to 27 coins and the solution (0,1,2,1,8,10), (1,2,2,5,5,0) with 1+2+2+5+8+10=28 coins. David Applegate finds a(7)=51 with (12,12,7,7,1,2,0), (12,0,8,2,7,3,2). - R. J. Mathar, Jun 20 2014
The unique solution for a(8)=90 is (27,1,12,12,6,1,0,4), (3,15,13,3,7,6,6,4) as determined by exhaustive search. There are a total of three solutions for a(7)=51: the one given above, (15,10,6,1,2,1,0), (0,10,9,7,4,4,2), and (15,6,9,1,4,3,1), (0,10,6,7,4,4,2). - David Applegate, Jul 03 2014
REFERENCES
Hugh ApSimon, Mathematical Byways in Ayling, Beeling and Ceiling, Oxford University Press (1991).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. K. Guy and R. Nowakowski, ApSimon's mints problem, Amer. Math. Monthly, 101 (1994), 358-359.
Tanya Khovanova, Attacking ApSimon's Mints, arXiv:1406.3012 [math.HO], 2014.
Tanya Khovanova, ApSimon’s Mints, Math Blog, June 2014.
Tanya Khovanova, ApSimon’s Mints Investigation, Math Blog, December 2014.
Xue-Wu Li, A new algorithm for ApSimon's Mints Problem, J. Tianjin Normal University 23 (2) (2003) 39-42.
R. J. Mathar, ApSimon's mint problem with three or more weighings, arXiv:1407.3613 [math.CO], 2014.
EXAMPLE
A pair of coin vectors gives a solution if every nonempty subset sum has a different ratio. (1,2,1,0) and (4,0,1,1) is a solution for 4 mints using 4+2+1+1=8 coins because 1:4, 2:0, 1:1, 0:1, (1+2):(4+0)=3:4, (1+1):(4+1)=2:5, (1+0):(4+1)=1:5, (2+1):(0+1)=3:1, (2+0):(0+1)=2:1, (1+0):(1+1)=1:2, (1+2+1):(4+0+1)=4:5, (1+1+0):(4+1+1)=2:6, (2+1+0):(0+1+1)=3:2, (1+2+0):(4+0+1)=3:5, (1+2+1+0):(4+0+1+1)=4:6 are all distinct ratios.
CROSSREFS
KEYWORD
hard,nonn,more,nice
AUTHOR
N. J. A. Sloane, Robert G. Wilson v, Aug 01 1994
EXTENSIONS
Solutions for a(6) and a(7) from Robert Israel and David Applegate, Jun 20 2014
a(8) from David Applegate, Jul 03 2014
STATUS
approved