OFFSET
1,1
COMMENTS
A 2n-card playing deck is shuffled and then revealed one-by-one to a player who guesses the color (red or black) of each card prior to its being revealed. The player earns one dollar for each card whose color he guesses correctly; there is no penalty for being wrong.
REFERENCES
Thane Plambeck and others, Posting to Math Fun Mailing List, Dec 26 2018.
LINKS
Michael Andreoli (proposer), Guessing Card Colors, Problem #630, College Mathematics Journal Vol. 30, No. 3 (May, 1999), pp. 234-235. Solution by John Henry Steelman.
FORMULA
The optimal payoff is n - 1/2 + 2^(2n-1)/binomial(2n,n).
EXAMPLE
3/2, 17/6, 41/10, 373/70, 823/126, 3565/462, 7625/858, 129293/12870, 272171/24310, 1139735/92378, 2376047/176358, ...
PROG
(PARI) a(n) = denominator(n - 1/2 + 2^(2*n-1)/binomial(2*n, n)); \\ Michel Marcus, Dec 28 2018
(Python)
from fractions import Fraction
from math import comb
def A322756(n): return (n-Fraction(1, 2)+Fraction(1<<(m:=n<<1)-1, comb(m, n))).denominator # Chai Wah Wu, Feb 12 2023
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Dec 27 2018
STATUS
approved