You toss a circular coin randomly onto the Adversary’s† infinite checkerboard. That is, it is as...

Question

You toss a circular coin randomly onto the Adversary’s† infinite checkerboard. That is, it is as likely to turn up in one location as any other location. The coin’s diameter is exactly half the length of the sides of the squares of the checkerboard, and the coin does not come to rest on its edge. If the coin touches a side of one or more of the squares, the Adversary keeps it; if it does not, the Adversary returns the coin and gives you three more coins. Is this a fair game? Explain why or why not.

David · Accepted Answer

Infinite checker
Consider the square with side say 1 unit.  Then the circle inside will have diameter ½ unit.   
Area of the circle = pi/16, and area of square = 1. 
If the coin has to touch one of the sides, then it has to settle at a distance less than ½ unit from any side 
of the square. 
i.e.

You toss a circular coin randomly onto the Adversary’s† infinite checkerboard. That is, it is as likely to turn up in one location as any other location. The coin’s diameter is exactly half the length...

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