Pigeonhole principle with two groups of balls labeled $1$ through $50$

Question

Say you have $50$ red balls, labeled $1$ through $50$, and $50$ blue balls, also labeled $1$ through $50$. You pick $10$ red balls and $10$ blue balls at random. Prove that you can make the sum of $1$ red and blue ball exactly match the sum of a different red ball and different blue ball.

My thought process: The max combined number of one red and one blue is $100$. The lowest combined combined number is $2$. So there are $99$ possible sums. However, there are $100$ possible combinations of balls. There is one more combination than possible sum, so two combinations must have the same sum.

Question: Am I missing something? I feel like I'm missing something. I'm sure there must be a trick and I'm not seeing it.

Answer 1

I'd argue that the "trick" to pigeonhole problems is figuring out these things:

1. What is the pigeon?
2. How many pigeons are there?
3. What is the hole?
4. How many holes are there?

The pigeons are the sums of the twenty balls that you picked, and there are $100$ of them. The holes are the possible different sums, and there are $99$ of them.

That's it!