Ways in which a wave can hit an n x n array of points.

1, 1, 2, 3, 4, 1, 3, 2, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 3, 5, 7, 6, 8, 9, 1, 4, 2, 7, 5, 3, 8, 6, 9, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 5, 4, 6, 7, 9, 8, 10, 11, 13, 12, 14, 15, 16, 1, 2, 5, 3, 6, 4, 9, 7, 10, 8, 13, 11, 14, 12, 15, 16

1

We actually given only 1/4 of the number of ways because this problem has symmetry. The number of n x n vectors is given by A206350(n).

T. D. Noe, Plot of 7 arrays

T. D. Noe, Table of 7 arrays

Stan Wagon, Problem of the week 1230: Water Wave

(Mma) Table[b =.; tb = Flatten[Table[i - j*b, {i, n}, {j, n}]]; Union[Table[Ordering[tb], {b, -n, 0, (1.0/n^2)/Sqrt[2]}]], {n, 7}]

Cf. A206350.

nonn,tabf,nice

T. D. Noe, Dec 02 2016