# Problem of the Month(May 2009)

Take a collection of integer-sided blocks inside a square box. Give each block a unit initial velocity in a horizontal or vertical direction, with the understanding that if a block touches another block or the edge of the box, it turns left. We insist that the blocks be positioned so that these left turns are always possible, and that blocks only touch at integer times.

What sort of periodic behaviors are possible? What if we allow rectangular blocks as well?

Boxes containing one block are well understood. They have a period that is some multiple of 4 (or 2 for rectangular blocks). And the centers of the blocks trace out a square (or rectangle for rectangular blocks) within the box.

Here are some small block configurations with more than one block, their periods, and the paths that the centers of the blocks take. You can click on a configurations to see an animation.

4×4 Boxes
 period 4 period 12(Berend vander Zwaag) period 2 period 2

5×5 Boxes
 period 48 period 16 period 16 period 16 period 8
 period 32(Berend van der Zwaag) period 48(Berend van der Zwaag) period 16(Berend van der Zwaag) period 16(Berend van der Zwaag)

6×6 Boxes
 period 100 period 20 period 20 period 10 period 20

 period 60 period 60 period 60 period 60 period 60

 period 4 period 4 period 20(Berend van der Zwaag) period 20(Berend van der Zwaag)

Here are some configurations using rectangular blocks.

Rectangular Blocks
 period 36 period 12 period 6 period 4
Rectangular Blocks
 period 4(Berend vander Zwaag) period 3(Berend vander Zwaag) period 14(Berend vander Zwaag) period 24(Berend vander Zwaag)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/22/09.