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

1,2

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.

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.

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

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A000035, A030495, A275929, A370888.

Sequence in context: A329253 A161125 A027295 * A208722 A057332 A230623

Adjacent sequences: A371214 A371215 A371216 * A371218 A371219 A371220

Keyword

nonn

Author

Tanya Khovanova and PRIMES STEP junior group, Mar 15 2024

Status

approved