Chessboard Combinatorics!!!

There are 3 'types' of starting squares that can be selected:

1) The first square is a Corner square (4 such squares)

2) The first square is an edge square (and not a corner square) (24 such squares)

3) The first square is an interior square (36 such squares)

For case (1), each corner square has a choice of exactly 2 adjacent squares (squares that share a side).

For case (2), each edge square has exactly 3 adjacent squares.

For case (3), each interior square has exactly 4 adjacent squares. .

Thus the probability of finding a pair of squares adjacent to each other can simply be expressed as a sum of the aforementioned mutually exclusive events:

P(corner-adjacent-pair)= (4/64)*(2/63)

P(edge-adjacent-pair)=(24/64)*(3/63)

P(interior-adjacent-pair)=(36/64)*(4/63)

The sum of these 3 events adds up to 1/18 (a/b). Thus a+b=19.

There is a one-to-one correspondence between the 112 interior "barrier lines" on the chessboard and pairs of adjacent squares. Since there are C(64,2) = 2016 possible pairs of squares in general, we end up with the a probability of 112 / 2016 = 1 / 18.

This gives us a = 1, b = 18 and thus a + b = $\boxed{19}$ .

no. of favourable choices:112 (general formula for n*n square board is 2n(n-1)) Total choices:64C2 Probability:1/18