The classic completion of this question is:

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

The correct probability is:

118\frac{1}{18}181​

How that answer comes about (short version)

Think of an 8×8 chessboard:

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

Now count how many pairs of squares actually share a common side:

  • Interior squares (not on any edge): 6×6 = 36 squares.
    Each touches 4 squares by a side.

  • Edge (non‑corner) squares : each row/column edge has 6 such squares, 4 edges → 24 squares.
    Each touches 3 squares by a side.

  • Corner squares : 4 of them.
    Each touches 2 squares by a side.

If you pick the first square at random and then pick an adjacent one:

P(pair shares a side)=3664⋅463+2464⋅363+464⋅263=118.P(\text{pair shares a side}) =\frac{36}{64}\cdot\frac{4}{63} +\frac{24}{64}\cdot\frac{3}{63} +\frac{4}{64}\cdot\frac{2}{63} =\frac{1}{18}.P(pair shares a side)=6436​⋅634​+6424​⋅633​+644​⋅632​=181​.

So, the probability that two randomly chosen squares on a chessboard share a side is 1/18.

If you meant a different condition

Sometimes textbooks ask variations like:

  • “What is the probability they have no side in common?”
  • “What is the probability they are on the same diagonal?”
  • “What is the probability they are of the same color?”

If you were aiming for one of these other endings (“…that they are of the same color”, “…that they lie on the same diagonal”, etc.), tell me the exact condition and I’ll compute that probability as well.