### Brain Teasers

# Coin Dropped on a Chessboard

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

A table has a chessboard integrated into the center of the table. A round coin fits exactly inside of a square on a chessboard. The coin is dropped on the table and lands face-down, with at least part of the coin on the chessboard.

What is the probability that the coin covers the corner of four chessboard squares?

What is the probability that the coin covers the corner of four chessboard squares?

### Answer

49Pi/(320+Pi) ~ .47638The diameter of the coin is equal to the length of the side of a square on the chessboard. For part of the coin to land on the chessboard, the center of the coin must be no more than the radius of the coin from the chessboard. If the length of a side of a square on the chessboard is one, then this describes a 9 x 9 square with rounded corners. The area missing from the rounded corners is equal to the size of a chessboard square minus the size of an inscribed circle (the coin). So the size of the area that the center of the coin must land in is:

9 x 9 - (1 - Pi/4) = 80 + Pi/4

For part of the coin to cover a corner, the center of the coin must be within a circle centered over the corner that has a radius equal to the radius of the coin. The area of this circle is

(1/2)^2 x Pi = Pi/4

There are 7 x 7 = 49 corners that are shared by four squares, so the total area that the center of the coin can land in to cover the corner of four squares is:

49 * Pi/4

The probability is then:

(49Pi/4) / (80 + Pi/4)

= 49Pi / (4 x (80 + Pi/4))

= 49Pi / (320 - Pi) ~ .47638

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