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
KEYWORD
nonn
AUTHOR
Tanya Khovanova and PRIMES STEP junior group, Mar 15 2024
STATUS
approved