Odd sums on a 3 × 3 grid

1. Step 1 · Only where the odd numbers go matters

   $\displaystyle 5 \text{ odd numbers } (1,\, 3,\, 5,\, 7,\, 9) \text{ and } 4 \text{ even}$

   A row’s sum is odd when it holds an odd count of odd numbers: 1 or 3. Evens don’t change that.

2. Step 2 · Patterns that work

   $\displaystyle \text{one full row of odds},\, \text{ and the other } 2 \text{ odds in the same column}\colon 3 \times 3 = 9$

   5 odds split into rows as $3 + 1 + 1$. The full row gives every column 1, so the 2 single odds must share a column to keep every column odd.

3. Step 3 · All ways to place the odds

   $\displaystyle \binom{9}{5} = 126$

   Every set of 5 squares for the odd numbers is equally likely.

4. Step 4 · Chance

   $\displaystyle 9 \div 126 = \frac{1}{14}$

   The good patterns out of all the equally likely ones.

Answer$\displaystyle \frac{1}{14} \approx 7.14\%$