Length@Reduce[And
@@ Table[Sum[a[i, j], {j, 8}] == 1, {i, 8}] &&
And @@ Table[Sum[a[i, j], {i, 8}] == 1, {j, 8}] &&
And @@ Table[Sum[a[i, j - i + 1], {i, j}] <= 1, {j, 2, 8}] &&
And @@ Table[Sum[a[9 - i, i - j + 8], {i, j}] <= 1, {j, 2, 7}] &&
And @@ Table[Sum[a[i, i + 8 - j], {i, j}] <= 1, {j, 2, 8}] &&
And @@ Table[Sum[a[9 - i, 1 - i + j], {i, j}] <= 1, {j, 2, 7}] &&
And @@ Flatten@Table[0 <= a[i, j] <= 1, {i, 8}, {j, 8}], Integers]