Problem 2. Find the number of all squares, which can be found in a chess-board.
Solution: The classic chess-board is a 8 8 u square, but we consider the general situation of a n n u board.
Take the cases n 1 , n 2 , n 3 and n 4 (Fig. 7). In each case the first step is to count the initial square. At the
second step consider all possible equal squares with side length, which is smaller by one unit square. At the third
step consider all possible equal squares with side length, which is smaller by two unit squares, and s. o. Note that at
each step the equal squares are grouped in a square and it is enough to consider the dimensions of this square. Thus,
the counting of the cases under consideration gives: