Watermelon - The Idea

The trick is that you need $w > 2$ AND $w$ must be even. Why both conditions? Let me break it down for you. If $w$ is odd, any split gives one even and one odd part. For example, splitting 7 gives either 1+6, 2+5, or 3+4.

In each case, one number is odd. That fails the requirement since both parts must be even. If $w = 2$, the only split is $1 + 1$, but 1 is odd. So $w$ must be even AND greater than 2.

When both conditions hold, you can always split as $2 + (w-2)$, and both parts are even positive integers.