thanks for a terrific site! I've got a group of little kids- ages 8 through 17- that I meet with twice a month for what I call "Math Maniacs League". We do puzzles and projects,as well as art, related to math. The common theme is MATH IS FUN! Sooo, I'm planning a little exploration with squares for our next time. The kids will manipulate squares to find out how many possible ways they can arrange two,three,four and finally five squares where at least one edge of each square lines up with one edge of another square. For two squares, there is one possible solution;for three squares there is two possible solutions; for four squares there arefive possible solutions. A big jump occurs next where five squares can be arranged in twelve different shapes that satisfy the requirements, not counting flips and turns that are
really the same shape.

My question for you is what is the math, or the theory behind this? By the way, im as far from a mathmetician as anyone can get so if you could explain this to me like you would to a group of elementary school aged kids I think I'll get it!

Thanks, in advance for your time-I appreciate it!!
Bonnie Lake

thank you for the kind words.

To answer your question, I am pretty confident that no formula has been found for an arbitrary number of squares. It's just too difficult and inconsequential. It's known that for 6 squares the number is 35 and for 7 it's 108.

I think what you do with the kids is quite wonderful. If I may suggest a problem that may be enjoyed by kids of all age groups. It's known as breaking chocolate bars. I have an applet at my site to do that online. However one breaks an NxM bar into the constituent squares it always takes exactly the same number of breaks to separate the bar into single squares. (Pieces must be broken individually: no stacking is permitted.) When the solution is understood a natural question to be answered (without experimenting) would be "If there were an NxMxK chocolate brick, how many breaks would it take to split it into single cubes?"

All the best,
Alexander Bogomolny