

A371217


The maximum deck size to perform Colm Mulcahy's ncard trick.


6



1, 4, 15, 52, 197, 896, 4987, 33216, 257161, 2262124, 22241671, 241476060, 2867551117, 36960108680, 513753523571, 7659705147976, 121918431264273, 2063255678027668, 36991535865656959, 700377953116334788, 13963866589144933461, 292421219327021540176, 6417047546280200867819
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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: fiftytwo new effects, published by CRC press, 2013.


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. 10.


FORMULA

a(n) = 1 + (n1)*(1 + 2*Sum_{i=1..n1} (i1)!*binomial(n1, i)).


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^22*n+2),
((11*n^266*n61)*a(n1) (17*n^2155*n+134)*a(n2)
+(n3)*(n81)*a(n3) +(n4)*(5*n+26)*a(n4))/(11*n72))
end:


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+((n1)*sum(factorial(n1)//((i+1)*factorial(ni2)) for i in range(n1))<<1) # Chai Wah Wu, May 02 2024


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



