homeschooling,  mathematics

Endless Patterning Game(s)

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My oldest son and I stumbled onto a fun game the other day. We played the game with math tiles, which are little colored plastic squares we have, but the game could just as easily be played with lego, blocks, probably even coins, anything where you have lots and lots of four types of pieces.
To begin the game, lay out the first pattern. We started with one of each color. We also laid out a key, which was a set of rules for replacing the colored blocks in the pattern. Our key looked something like this:

The top row tells which color is going to be replaced, and each column shows what colors to replace the colored square with. Since we were including the top row in the column of what was being replaced we were in effect simply adding the second two tiles to the first one, but I’m going to stick to saying we replaced the red with one red and two yellows, the yellow with a yellow, blue, yellow, etc, etc.

We rolled a die. If it was one, we replaced all instances of the first color in our key (in this example, red) tiles with one red and two yellow. If it was a two, we used the second column as our guide, and so on for three and four. If it was five we removed all of the blue and one red tile, and if it was six we removed every third tile.

So our pattern started out with four blocks, and on the second roll it grew to six. Our second roll was three, so we ended up with the above.  If we rolled a 1 – 4, the pattern grew. If we rolled a 5 or 6, the pattern shrunk. We tried this a couple of times changing the key to find different results.

Then we installed a new rule. We decided that anytime there were two identical patterns of six or more tiles within our long chain, we could remove those twelve (or more) tiles and put them to the side. Now we had a goal. We had to watch as patterns grew and changed and pull out repeating parts. Rolling a five became particularly fun because it involved choosing which red tile to remove.

There are lots of potential variations to this. The fun is in changing the rules. Draw up a new key! Start with a different pattern! I tended to keep it so that all colors were “called” by at least one other color, so if all of one color disappears from the pattern for a while, it would come back but that doesn’t have to be so. One challenge would be to try to remove one color completely (by finding matching patterns in the parts with the tiles, not by having rolling the dice remove all of that color).

We played cooperatively with one big pattern between us, but two players could play competitively. Maybe both players set up the starting pattern for each other, and then they take turns rolling the die. At the beginning choose whether both players have to make the transformations called by every roll of the die (which would be different if they are starting with a different starting pattern), or whether each player changes his or her pattern only for his or her roll of the dice. See who can clear out one color first, or who can create the biggest collection of “matched patterns.”

You could take the luck out of the game by choosing which transformation to do instead of rolling the die. You could make it harder for yourself by creating rules limiting your choices. For example, a meta-rule could be that you can only do a transformation twice in a row before you have to choose a different transformation to do. Or a rule that every third transformation is to remove every fourth tile. The possibilities are endless. Just write down your rules, try them out, and see what types of patterns you can create.

Or set up a simple key, and each player sets up a starting pattern and makes four transformations of his or her choice. Trade seats, stare at each other’s patterns and try to figure out what transformations were made. The following started off as: red, yellow, green, blue, and four transformations were used on it according to the key at the top. What transformations were they?

The possibilities are endless! And the best part is, this is actually an introduction to formal systems theory and it works similar to the L-Systems coding I wrote about a couple of months ago.

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