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Three coins (Posted on 2015-08-20) Difficulty: 3 of 5
Three coins are lying on a table: a quarter, a half dollar, and a silver dollar. You claim one coin, I’ll claim the other two, and then we’ll toss all three.
A coin that lands tails counts zero, and a coin that lands heads wins its value
(in cents, 25, 50, or 100) for its owner.
Whichever of us has the larger score wins all three coins. If all three coins land tails then we call it a draw and toss again.

Which coin should you claim to make the game fair — that is, so that each of us has an expected win of zero?

Source: Martin Gardner, “Charles Addams’ Skier and Other Problems,” in Wheels, Life and Other Mathematical Amusements, 1983.

See The Solution Submitted by Ady TZIDON    
Rating: 2.0000 (1 votes)

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re(6): question Comment 10 of 10 |
(In reply to re(5): question by Ady TZIDON)

I'll quote from the original book.  Note the key word is "own", in that the owner stands to lose the given amount--he is the supplier of the coin:

Three coins ar on the table; a quarter, a half-dollar and a silver dollar.  Smith owns one coin and Jones owns the other two.  All three coins are tossed simultaneously.

It is agreed that any coin falling tails counts zero for its owner.  Any coin falling heads counts its value in cents.  The tosser who gets the larger score wins all three coins.  If all three come up tails, no one wins and the toss is repeated.

What coin should Smith own so that the game is fair, that is, so that the expected monetary win for each player is zero?

The payoff matrix is verbatim from Gardner's book except for the added titles "You Choose ..." and is found at

The solution page does have a caveat:

UPDATE: The mathematical principle here is sound, but the presentation muddies it a bit. The expected outcomes are equal after a coin has been chosen, but properly speaking choosing a coin is part of the game. If both players start with nothing, as described, then choosing the silver dollar gives the best chance of winning. (Thanks, Seth and Tim.)

But of course the payoff matrix makes it clear that the loser actually loses the amount of the owned (called chosen, even here in the caveat) coins. With only winners and no losers of actual value there can't be a balance between wins and losses.

Edited on August 23, 2015, 3:10 pm
  Posted by Charlie on 2015-08-23 15:08:46

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