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A good game for visualization

December 23, 2006 am31 2:41 am

Today was the last day before break. I had four classes in the morning. We did little bits of math, but mostly we played. Today’s game was 3-D tic-tac-toe.

Now, I need to add, I introduced this in each class without naming the game. “In the simpler version of the game we are going to learn, there are two players who, each using a different marker, tries to create a row of maximum possible length while preventing his opponent from doing the same.” You could see the gears in the little brains spinning, hard. Of course each class eventually guessed at tic-tac-toe.

Then I played a game of regular tic-tac-toe in each class, and won each time. (Started on the edge, confused them). And then I drew a cube, we discussed why it was hard to play on the cube, how the 4-in-a-rows could be hidden, etc, and then “sliced” the cube to create 4 four by four squares, found a few four in a rows, and let them play.

       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       

In combinatorics, I offered a prize to the first one to count all the ways of winning. In Algebra II we did some real work first (20 minutes of extending compound inequalities to absolute value inequalities). In two classes I also showed them how to manage the four-in-a-rows without reference to the cube*.

But mostly they played. And loved it.

more, and links to java gamelets –>

Here is a link to a 3-D tic-tac-toe game. Chris Malumphy wrote some nice java (I think) to create a tough but beatable computer opponent.

This version is more like what I handed out.

*Now, how can you find 4-in-a-rows without worrying about geometry? Label each square by its level, row, and column. For example, A(2,3,4) is on the 2nd level, in the 3rd row, all the way on the right. The lower front left corner is B(4, 4,1). Using this notation, look at the level numbers, the row numbers, and the column numbers. If each set is all the same, or 1,2,3,4 in that order or 4,3,2,1 in that order, we have 4-in-a-row.

  D    
       
C C C C
E      
  D    
       
E     A
       
  D    
E      
       
       
E D    
       
       
B      

A few examples: across the next to front row of the top: C(1,3,1) (1,3,2) (1,3,3) (1,3,4). Back middle-left, straight down: D(1,1,2), (2,1,2), (3,1,2), (4,1,2). Diagonally from the upper left front to the lower left back: E(1,4,1) (2,3,1) (3,2,1) (4,1,1)

A few kids did better once they had the numbers.

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