Question 2

We can break down a square grid into layers of odd numbers of dots.

Step 1 — The largest layer

Let us take a $3 \times 3$ square grid. It has 9 dots in total. We can remove the outermost 'L' shape. This 'L' shape has 5 dots. We are left with a $2 \times 2$ square.

$9 - 5$

$= 4$

$\boxed{4 \text{ dots left}}$

Step 2 — The next layer

Now we have a $2 \times 2$ square. It has 4 dots. We can remove its outermost 'L' shape. This 'L' shape has 3 dots. We are left with a $1 \times 1$ square.

$4 - 3$

$= 1$

$\boxed{1 \text{ dot left}}$

Step 3 — The final layer

Now we have a $1 \times 1$ square. It has 1 dot. This is the last odd number. We have successfully partitioned the $3 \times 3$ grid. The parts are 5, 3, and 1 dots.

$1 \text{ dot}$

$\boxed{1 \text{ dot}}$

Step 4 — Generalizing the pattern

This method works for any square grid. For a $4 \times 4$ grid, we would first remove 7 dots. Then we remove 5 dots. Then we remove 3 dots. Finally, we are left with 1 dot. The parts are 7, 5, 3, and 1. This partitions the $4 \times 4$ grid into odd numbers.