Two squares are chosen at random on a chessboard (see figure). The...

Total ways of choosing 2 squares $= C2 64 = 32 \times 63$

Now, each corners has two neighbours, and each $24$ squares across the outer edge of the chess board has $3$ neighbours, and remaining $36$ squares have four neighbours

Favorable ways $= \dfrac{1}{2}\left[4 ( 2 ) + 24 ( 3 ) + 36 ( 4 )\right] = \dfrac{1}{2}\left[224\right] = 112 .$Since, every pair counted twice.

Required probability $= \dfrac{112}{32 \times 63} = \dfrac{1}{18}$