The Squares Problem is a purely mathematical problem. Although interesting, it has no known practical application.

Imagine that you have a square grid made out of cells, like a chessboard. Each cell has a probability p of being either black (associated with the number 1) or white (associated with the number 0). Every start of the game, the board is filled randomly with either white or black cells. It can form continuous regions of black cells, when cells share a black neighbor to either one of the directions up, down, left or right.   

A continuous black area with 5 cells has an area of 5. We are interesting in finding out what is the distribution of areas, how many of them are isolated (area 1), area 2, area 3, and so forth. 

How does the area distribution varies with p? 

Can you solve the problem analytically, and verify the results with the use of a computer?