Problem 9 · 3×3 boards of zeros and ones
Equal rows and columns: all zeros, all ones, and permutation matrices.
Integer answer, at most 4 digitsIn how many ways can a $3 \times 3$ board be filled with the digits 0 and 1 so that every row and every column has the same sum?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: if every row sums to $s$, the total is $3s$; and if every column also sums to $s$, we get the same total. We analyse $s = 0, 1, 2, 3$.
$s = 0$: all zeros, $1$ way. $s = 3$: all ones, $1$ way.
$s = 1$: exactly one $1$ in each row and column — these are the $3 \times 3$ permutation matrices: $3! = 6$ ways.
$s = 2$: exactly one $0$ in each row and column — the complements of the previous ones: $6$ more ways.
$$1 + 6 + 6 + 1 = 14.$$
Answer: 14