Prove there exists $2\times 2$ checkerboard-colored square in a $100\times 100$ table colored black and white.

Hint: Stick $99\times 99$ needles on this grid, each at a place where four cells meet in a corner. For each pair of needles at distance one apart, connect them with a piece of string if the the two squares touching the edge between them have different colors.

Each needle with have either $2$ or $4$ pieces of string tied to it (why?). If a needle has four strings, then the four squares surrounding it are colored like a checkerboard. So, assume to the contrary that every needle only has two strings. What would the resulting picture look like? Why is that impossible?

Further hint:

If every needle only had two strings, then the needles would be partitioned into "loops," where each needle is connected to a the next and previous in a circular fashion. What are the possible sizes of a loop?

For example, you can have a loop of size four where the four needles are the vertices of a cell. Can you have a loop of size $5$?