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small-pearlOne of my students recently sent me an email with a link to the game Shinju. (Thanks Chris Z!) He included the note, “I thought you would enjoy this game I came across. It deals with Maxi geometry!”

The game is played on an 8×8 board that is partially covered with shells, one of which has a golden pearl inside. You click on a shell, and it either reveals the hidden pearl (You Win!) or it gives a number indicating how many steps away the pearl is, where each step can be horizontal, vertical, or diagonal. It’s fun to play the game, and I found it surprisingly addicting.

For example, in the screen shot below I had clicked on a shell and it indicated that the pearl was 8 steps away:

pearl-game-1

Now here’s where the neat geometry part comes in. Moving in horizontal, vertical, and/or diagonal steps, the number of squares that are 8 units away lie in a giant square around the chosen shell.This isn’t immediately obvious — it took me a while of playing around before I noticed that. And so in the picture below I drew a yellow line through all the unit squares that were exactly 8 steps away. If the board were bigger, or I had’t picked a shell in the corner, then the yellow line would form a square completely around the chosen shelll.

pearl-game-2

For my next try, I just picked a shell that was along that yellow square. It wasn’t the right one, and indeed was 6 steps away from the right one. In the picture below I’ve drawn an orange line through all of the unit squares that are 6 vertical, horizontal, and/or diagonal steps away from the chosen shell. Again, if my board were bigger the orange lines would form a square completely around that second shell.

pearl-game-3

Then I looked at the intersection. It turned out that there was only one space that worked, and fortunately it had a shell in it. That’s what I picked.

pearl-game-4

This method worked in general, although sometimes there was more than one shell that was the appropriate number of steps from both of my first two choices, and I didn’t find it until clicking on the fourth (and final) shell.

It was only after playing this for a while that I realized why Chris had made the reference to Maxi Geometry, which is a geometry that uses Chebyshev (or Tchebyshev) distance. Maxi Geometry is a geometry in which the distance between two points is the maximum of (the difference of the x-values) and (the difference of the y-values). The Chebyshev between the point (-1,3) and the point (2,10) is the maximum of 3 (the difference of the x-values) and 7 (the difference of the y-values), so those points are Chebyshev distance 7 apart. You can define a circle to be the set of points that are a fixed distance from a given point, and it turns out that the circles look like squares. That is, the yellow lines form a Chebychev “circle” around the first shell, and the orange lines form a Chebyshev “circle” around the second shell.

All of which is to say that playing this game could easily count as prepping for teaching Geometry again in the spring.

Tags: Chebyshev, geometry, Shell game

This entry was posted on November 13, 2008 at 7:22 pm and is filed under Miscellaneous. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.