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A coin puzzle

A coin problem
by Gary Antonik, Numberplay blog, New York Times, 17 March 2014

The post begins with this simple problem, posed by Daniel Finkel:

Consider this simple game: flip a fair coin twice. You win if you get two heads, and lose otherwise. It’s not hard to calculate that the chances of winning are 1/4… . Your challenge is to design a game, using only a fair coin, that you have a 1/3 chance of winning.

Continues "And here is my recipe for getting the most out of this problem: if you can solve it, do not stop with one answer. Rather, see how many answers you can come up with. I’ve posed this problem to many people, and I continue to hear novel solutions."

Here are three ways that suggested themselves (I notice they also had soon showed up in readers' comments to the NYT!):

• Toss the coin until the first head appears. You win if this takes an even number of tosses
• Toss the coin twice. You win on HH and lose on HT or TH. If TT appears, ignore the result and make another two tosses.
• Toss the coin until the first appearance of HTT or HHT on consecutive tosses. You win HTT.

The third is an instance of the game Penney-Ante, invented by William Penney. It is a famous example of non-transitivity: whatever triple you choose, I can choose one that has a better than even chance of coming up first. What I did not know until I searched for this online description, was that there is a known variation with cards, called the Humble-Nishiyama Randomness Game:

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Submitted by Bill Peterson