A372265 a(n) = floor((2*n - 3 + sqrt(1 + 4*n!))/2).

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

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Apr 24 2024

Keywords

• nonn

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.

Examples

			For n=3, the equation on the deck size becomes the following: d-1 choose 2 can't exceed 3. Thus, a(3) = 4.
		

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.

Crossrefs

Cf. A370888, A371217, A372255, A372256, A372264.

Programs

Mathematica

Table[Floor[(2 n - 3 + Sqrt[1 + 4 n!])/2], {n, 30}]