__The Question__1. Consider the L pentominoes, that is L and L’ together with their rotations.

a) For which n ϵ N is it possible to tile a 5 × n rectangle (to cover it without overlaps and without gaps)?

b) Find all m, n ϵ N for which one can tile an m × n rectangle.

c) In general, given positive integers m and n, what is the maximal number of L pentominoes that can be placed into an m × n rectangle along grid lines and without overlap?

2. Try the same problems for other types of pentominoes. If a pentomino has a chiral partner, then treat them together

The above question has been adapted from the International Tournament of Young Mathematicians 2013. The simpler parts of the original question have been covered here, here and here. It is not a quick question and, remember, that the ITYM questions may well have no known general solutions! How far you can get into it is the important thing.

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