I'm going to write an article in an educational magazine for middle school students, about the game Square It. The purpose of the game is to make lattice squares:
I want to introduce the game, and add some interesting facts and give some good problems about squares in a lattice (of course, they should be appropriate for middle school students). These are the only things I've thought of, by now:
- Pick's Theorem,
- Every square has integeral area.
- Pythagorean Theorem.
I'd be so thankful for any other idea.
Suggestions
1) What is the possible area of a lattice square? This ties in to Fermat's two square theorem.
2) Which player would you chose to be? The first or the second? Why?
3) Show that if the square (or rectangular) lattice has even length (e.g. a $ 4 \times 10 $ points), then the second player can do as well as the first player. (Do as well can be defined either as the number of squares, or even the sum of areas of squares.)
4) Show that if the square lattice has odd length, the first player can guarantee to do as well as the second player.