32 cookies are there to start.

Oldest gets 1/2 of the 32 cookies and 1/2 of the 16 remainders for a total of 24 (16 + 8).

So 8 cookies are left.

Middle gets 1/2 of the 8 cookies and 1/2 of the 4 remainders for a total of 6 (4 + 2).

So 2 cookies are left.

Youngest gets 1/2 of the 2 cookies and 1/2 of the remainder for a total of 2 (1 + 1).

By definition, “plus a half” will always leave either a partial cookie or some number of cookies thus prohibiting Homer from giving them ALL away.

It’s like I tell you to step halfway towards the door. Now step halfway from that new point to the door. And repeat. Mathematically, you’ll never reach the door.

So I suspect that the problem is not worded correctly.

The middle one gets half of what was left from the oldest, plus a half, leaving one cookie for the youngest, so there were three before he got his share.

Likewise there were seven cookies before the oldest got his share, because he got three and a half cookies, plus a half, leaving three for the middle brother.