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A372265
a(n) = floor((2*n - 3 + sqrt(1 + 4*n!))/2).
1
0, 2, 4, 7, 14, 31, 76, 207, 609, 1913, 6327, 21896, 78922, 295272, 1143549, 4574158, 18859692, 80014850, 348776594, 1559776287, 7147792837, 33526120102, 160785623566, 787685471345, 3938427356638, 20082117944270, 104349745809099, 552166953567254, 2973510046012938, 16286585271694984
OFFSET
1,2
COMMENTS
Information-theoretic bound on the largest card deck with which one can perform an n-card trick, when the assistant chooses two cards to hide.
The bound is based on the following argument: The assistant has n choose 2 ways to pick the hidden cards and (n-2)! ways to arrange the rest of the cards. The number of strategies can't be smaller than the number of potential guesses by the magician which is d - n + 2 choose 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. 21.
Michael Kleber and Ravi Vakil, The best card trick, The Mathematical Intelligencer 24 (2002), 9-11.
EXAMPLE
For n=3, the equation on the deck size becomes the following: d-1 choose 2 can't exceed 3. Thus, a(3) = 4.
MATHEMATICA
Table[Floor[(2 n - 3 + Sqrt[1 + 4 n!])/2], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Tanya Khovanova and the MIT PRIMES STEP junior group, Apr 24 2024
STATUS
approved