# Flea on infinite chessboard jumping with irrational vector eventually changes square color [closed]

Question from Engel's Problem Solving Strategies: An infinite chessboard consists of $$1 \times 1$$ squares. A flea starts on a white square and makes jumps by $$\alpha$$ to the right and $$\beta$$ upwards, where $$\alpha / \beta$$ is irrational. Prove that sooner or later it will reach a black square.

WLOG suppose that the flea starts at $$(0,0)$$. So the flea steps on the coordinates $$(k\alpha, k\beta)$$, for all $$k \in \mathbb{N}$$. I need to show that eventually $$\lfloor{k\alpha}\rfloor + \lfloor{k \beta}\rfloor$$ is an even number.

So I must show that $$\lfloor{k\alpha}\rfloor$$ and $$\lfloor{k \beta}\rfloor$$ must eventually have the same parity. Intuitively I know there must exist some $$k$$ such that $$\lfloor k\alpha \rfloor$$ and $$\lfloor (k+1) \alpha \rfloor$$ have same parity, while $$\lfloor k\beta \rfloor$$ and $$\lfloor (k+1)\beta \rfloor$$ have different parity.

How do I do this?

• 1. Do you mean an odd number rather than an even number? 2. I notice that the question doesn't say where in the white square the flea starts (in the middle? near a corner?) so the floor operation might not correspond accurately to the flea's position. – Rosie F May 11 '19 at 7:11
• I think WLOG we can assume that the square with bottom left coordinate being (0,0) is white – eatfood May 11 '19 at 8:16
• Doesn't seem all that textbook-y to me. VTO. – Bass May 11 '19 at 20:12