A371217 The maximum deck size to perform Colm Mulcahy's n-card trick.

1, 4, 15, 52, 197, 896, 4987, 33216, 257161, 2262124, 22241671, 241476060, 2867551117, 36960108680, 513753523571, 7659705147976, 121918431264273, 2063255678027668, 36991535865656959, 700377953116334788, 13963866589144933461, 292421219327021540176, 6417047546280200867819

Offset: 1

Author

Tanya Khovanova and PRIMES STEP junior group, Mar 15 2024

Keywords

• nonn

Comments

With this card trick the magician's assistant gets n cards from a deck, hides one card, and displays the rest, where it is allowed to place some of the displayed cards face down. After that, the magician guesses the hidden card.

The trick for n = 4 was invented by Colm Mulcahy and is a variation of the Fitch Cheney trick. Surprisingly, the largest possible deck is the standard deck of 52 cards.

Examples

			Suppose the deck consists of 4 cards (1,2,3,4), and the assistant gets two cards. If the two cards contain 4, the assistant hides 4 and signals it with the other card face down. If there is no 4, then the cards are a and a+1 modulo 3. The assistant hides a+1, and signals it with a.
		

References

• Wallace Lee, Math Miracles, published by Seeman Printery, Durham, N.C., 1950.
• Colm Mulcahy, Mathematical card magic: fifty-two new effects, published by CRC press, 2013.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..450
• 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. 10.

Crossrefs

Cf. A000035, A030495, A275929, A370888.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, n*(n^2-2*n+2),
      ((11*n^2-66*n-61)*a(n-1) -(17*n^2-155*n+134)*a(n-2)
       +(n-3)*(n-81)*a(n-3) +(n-4)*(5*n+26)*a(n-4))/(11*n-72))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Mar 18 2024

Mathematica

Table[1 + (k - 1)(2 Sum[Binomial[k - 1, i] (i - 1)!, {i, 1, k - 1}] + 1), {k, 20}]

Python

from math import factorial
def A371217(n): return n+((n-1)*sum(factorial(n-1)//((i+1)*factorial(n-i-2)) for i in range(n-1))<<1) # Chai Wah Wu, May 02 2024

Formula

a(n) = 1 + (n-1)*(1 + 2*Sum_{i=1..n-1} (i-1)!*binomial(n-1, i)).

a(n) mod 2 = n mod 2 = A000035(n). - Alois P. Heinz, Mar 22 2024

a(n) ~ 2*exp(1)*(n-1)!. - Vaclav Kotesovec, Jul 27 2024