A372266 a(n) = floor((2*n - 3 + sqrt(1 + 8*(n - 2)!))/2).

2, 3, 4, 7, 11, 21, 44, 107, 292, 861, 2704, 8946, 30964, 111611, 417574, 1617219, 6468832, 26671628, 113158082, 493244584, 2205856773, 10108505566, 47413093736, 227385209476, 1113955476453, 5569777382171, 28400403557955, 147572825753404, 780881994429038

Offset: 2

Author

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

Keywords

• nonn

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.

Examples

			For n=3, the constraint on the deck size becomes: binomial(d-1, 2) can't exceed 1!=1. Thus a(3) = 3.
		

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.

Crossrefs

Cf. A370888, A371217, A372255, A372256, A372264, A372265.

Programs

Mathematica

Table[Floor[(2 k - 3 + Sqrt[1 + 8 (k - 2)!])/2], {k, 2, 30}]