The probability that two randomly chosen squares on a standard 8×8 chessboard share a common side is 118\frac{1}{18}181​.

Explanation:

• Total number of squares on the chessboard = 64.
• Total ways to choose 2 squares from 64 = (642)=2016\binom{64}{2}=2016(264​)=2016.

To find the number of pairs of squares that share a side (are adjacent):

• Count horizontal adjacent pairs: Each of the 8 rows has 7 adjacent pairs → 8×7=568\times 7=568×7=56.
• Count vertical adjacent pairs: Each of the 8 columns has 7 adjacent pairs → 8×7=568\times 7=568×7=56.
• Total adjacent pairs = 56+56=11256+56=11256+56=112.

Thus, the probability that two chosen squares share a side is:

1122016=118\frac{112}{2016}=\frac{1}{18}2016112​=181​

This result is confirmed by counting adjacency types or by direct combinatorial reasoning