The probability that two squares chosen at random on a chessboard have a side in common is 118\frac{1}{18}181. Explanation:
- Total number of squares on a chessboard: 64 (8 rows × 8 columns).
- Total ways to choose any 2 squares: (642)=64×632=2016\binom{64}{2}=\frac{64\times 63}{2}=2016(264)=264×63=2016.
- Number of pairs of squares that share a common side (adjacent squares):
- Horizontal adjacent pairs: Each of the 8 rows has 7 adjacent pairs → 8×7=568\times 7=568×7=56.
- 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.
- Probability = Number of adjacent pairs / Total pairs = 1122016=118\frac{112}{2016}=\frac{1}{18}2016112=181.
This result is confirmed by multiple sources and is a standard probability for adjacency on a chessboard
