__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.

**Feel free to comment, ask questions and even check your answer in the comments box below powered by**~~Disqus~~ Google+.

**If you enjoy using this website then please consider making a donation - every little helps :-)**

**You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.**

**Don’t forget to follow Gifted Mathematics on Google+, Facebook or Twitter. You may add your own interesting questions on our Google+ Community and Facebook..**

You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.You can also subscribe to our Bookmarks on StumbleUpon and Pinterest. Many resources never make it onto the pages of Gifted Mathematics but are stored in these bookmarking websites to share with you.