Determinant of a matrix of "ones" and "minus ones"?

Cute problem. You want to pull out a factor of two so add any row (other than the top row) to the top row. The determinant stays the same, but now you have a factor of two that you can pull out (since every value on top will now be 2, 0, or -2), and then expand upon that row. All the submatrices you'll compute the determinant of still have only 1s and -1s, so you can apply induction.