Q1.
Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product?
Answer
Yes. Just look at the parity of the two dimensions.
| Grid m × n | Number of small squares | Example |
|---|---|---|
| both m and n odd | odd | 3 × 3 = 9 |
| one of them even | even | 3 × 4 = 12 |
| both even | even | 4 × 6 = 24 |
The count of small squares is m × n.
odd × odd = odd
even × any number = even
odd × odd = odd
even × any number = even
Why it happens: if one side is even, the whole grid can be cut into complete pairs of columns (or rows), so the squares pair up perfectly. If both sides are odd, each row has one square left over, and there is an odd number of rows, so one square is finally stranded.