two squares are chosen at random on a chessboard. what is the probability that they have a side in common?

The probability is 19\displaystyle \frac{1}{9}91​.

Step-by-step reasoning

A standard chessboard has 8×8=648\times 8=648×8=64 squares.

1. Total ways to choose 2 squares

Any 2 distinct squares can be chosen in

(642)=64×632=2016\binom{64}{2}=\frac{64\times 63}{2}=2016(264​)=264×63​=2016

ways.

2. Count pairs of squares sharing a side

Classify squares by how many neighbors (sharing a side) they have:

• Corner squares: 4 corners, each has 2 neighbors → 4×2=84\times 2=84×2=8 neighbor-relations.
• Edge (non-corner) squares:
  • Each edge has 6 such squares, and there are 4 edges → 4×6=244\times 6=244×6=24 squares.
  • Each has 3 neighbors → 24×3=7224\times 3=7224×3=72 neighbor-relations.
• Interior squares:
  • Remaining squares: 64−4−24=3664-4-24=3664−4−24=36.
  • Each has 4 neighbors → 36×4=14436\times 4=14436×4=144 neighbor-relations.

Total “square–neighbor” relations:

8+72+144=224.8+72+144=224.8+72+144=224.

But each pair of adjacent squares is counted twice in that total (once from each square’s perspective), so the actual number of unordered adjacent pairs is:

2242=112.\frac{224}{2}=112.2224​=112.

3. Compute the probability

P(share a side)=favorable pairstotal pairs=1122016=118×2=19.P(\text{share a side})=\frac{\text{favorable pairs}}{\text{total pairs}} =\frac{112}{2016}=\frac{1}{18}\times 2=\frac{1}{9}.P(share a side)=total pairsfavorable pairs​=2016112​=181​×2=91​.

So, when two squares are chosen at random on a chessboard, the probability that they share a common side is 19\boxed{\frac{1}{9}}91​​.