Journal of Integer Sequences, Vol. 24 (2021), Article 21.6.2

A Golden Penney Game

Issa Dababneh
University of Bridgeport
Bridgeport, CT 06604

Mark Elmer
SUNY Oswego
Oswego, NY 13126

Ryan McCulloch
Elmira College
Elmira, NY 14901


Penney's game (also known as Penney ante) is a counter-intuitive coin flip game that has attracted much attention due to Gardner's Scientific American column. We concern ourselves with just one case of Penney's game: player I choosing HHH vs. player II choosing HTH. If a trick golden penny is minted to have the probability of heads equal to 1/ϕ, where ϕ is the golden ratio, then neither player has an advantage in this game.

We discover that counting the number of winning player I sequences in this game that have exactly n number of tails and k number of heads appearing before the final HHH is equivalent to counting the n-tilings of a board using exactly k fences. We derive combinatorial identities related to this counting formula, all of which are fascinating and many of which appear to be new. Some of the sequences that we encounter along the way are Pascal's triangle and a related Pascal-like triangle, the Fibonacci sequence, the Jacobsthal sequence, the golden rectangle numbers, the squared Fibonacci numbers, and more.

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(Concerned with sequences A000045 A000079 A001045 A001654 A006498 A007318 A007598 A059259.)

Received December 18 2020; revised version received May 15 2021; May 16 2021. Published in Journal of Integer Sequences, May 17 2021.

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