Math circle 10.14.2014(October 14, 2014) “I’d like to show you my creatures,”  I announced as everyone arrived.  I already had everyone’s rapt attention – the word “creatures” will do that.  I opened a box that seemingly contained a Go board and Go stones.

“That’s a Go board,” protested J, as I unfolded the board.

“No, this is the habitat of my creatures.  It does resemble a Go board, but it’s not one.  I’d like to tell you about how this species of creatures exists over time.”  I placed a few black counters on the board.

“They can live if they have 2 or 3 neighbors.”
“What if they only have one?” asked A.
“Then they die.”

“Of loneliness,” posited K.

“What if they have 4 neighbors?” asked someone else.

“They die.”  I went on like this for a few minutes, letting the kids’ questions elicit the rules of death and survival.1  “In real life,” I added, “something else happens besides survival versus death.”

After a couple of conjectures, H posited excitedly “new ones are born!”  Then I explained the rule about births – they only occur in a cell/location with exactly 3 neighbors.

“Is this something Martin Gardner invented?” asked both L and K early on.

“No, but… “ I started to say, and before I got the sentence out, the kids said “Martin Gardner made it famous.”  They were right.  I showed them his article about it.

“Are you going to ask us if we can fill the whole board with creatures before they die out?” asked L.

“That’s a really interesting question to explore.  Today, though, I wasn’t going to ask you any questions.  My plan is just to show you the creatures and they rules they live by, and have you all come up with questions.”

“Is it a board game?” asked J.  I suggested they try it out and see what they think is the answer to that.  The kids started to play.

Other questions and conjectures quickly arose:

  • “It matters how you start the game”
  • “How do you win?”
  • “Will they all die?”

The kids played more and saw that some initial positions resulted in immediate or quick death to all, some seemed to keep going, and some ended up in a stable position, which they identified as a “Steady Square.”  Are there outcomes, they wondered, other than a steady square or extinction?

During this experimentation, I talked to the kids about how this game was one of the early ways people used mathematical modeling to predict outcomes for populations of species.  Someone brought up Ebola, made some logical leaps, and got worried that people with the wrong number of neighbors might die of it.  (“Is that how it works in real life,” they asked.)   I had to do some reassuring that this model is really just a game, and it’s purpose mainly is to get you thinking.  It is not a literal model of human lives and deaths.  Remember, many factors influence our survival.  All this game specifically examines is population density.2

Almost an hour later, we were still at it, and possibly ready for a change of pace.  The students asked about the Pentominoes that were on another table.  They couldn’t figure out why they’re called that – what pent means.  (They knew pentagon, tricycle/triathlon/trionimoes.)  No matter what I asked them, they didn’t think of counting up how many unit squares comprised each piece.  I was wiggling in my seat with the self-control it took to not blurt it out.  So it’s still a mystery.

The kids wanted free exploration with pentominoes for last 10 minutes, so we did just that, and went home.


1 I read the rules as Martin Gardner described them in his SA article “The fantastic combinations of John Conway’s new solitaire game “life.”  This article describes the rules very well.  It also mentions a $50 prize for anyone who can answer a question that Gardner posited that our kids posited too.

2 Paul Callahan has a very nice write-up on the applications of this game, as well as detailed contemporary approaches.  (Garner brought Life to the public’s attention in 1970.) I’d beg you to please, pretty please, though, at this point don’t have your kids  use his applet – it does too much too soon if we’re hoping to develop mathematical thinking in these young kids.  I must admit that the applet is really cool and fun and provides immediate insight.  I’m just saying let’s not rush with the kids.

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