Prepare to square off

Consider an 8x8 chess board as shown in the adjacent figure. How many squares can you count without repetition?





Now consider the infinite chess board shown in the next figure. What is the inverse of the sum of the squares of all the distinct areas of squares that can formed from this board? What is the inverse of the sum of squares of all the square areas that can be formed from this board?

Playing with dominoes - II

This one is often attributed to Martin Gardner.

How many dominoes would you require to fill up a 8x8 chess board?

Now, let us give vent to the suppressed child in us, and mangle the chess board by removing the opposite corner squares (see picture). How many dominoes would you require now?

Playing with dominoes

A domino is a 2x1 piece of rectangle (see picture). Now consider a rectangular area of size nx2. The question is simple: given a large number of identical dominoes, how many different ways of arrangements of the 2x1 dominoes into the nx2 area are possible? Assume all the dominoes are identical. Note that as always, key ideas are more important in this blog than complete solutions.