MathDB

Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle.
Determine who wins each of the following games:
(a) The starting rectangles are

1 \times 2020

and

2 \times 4040

. (b) The starting rectangles are

100 \times 100

and

100 \times 500

Problem Statement