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?

ANSWERS

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 van
der 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

 
 
period 4

(Berend van
der Zwaag)
period 3

(Berend van
der Zwaag)
period 14

(Berend van
der Zwaag)
period 24

(Berend van
der 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.