The math inside nested tic-tac-toe
Play →In Ben Orlin's Math Games with Bad Drawings this game is called ultimate tic-tac-toe. The name the author likes best is fractal tic-tac-toe. Why would tic-tac-toe have anything to do with fractals?
Secret 1: a small board hides inside the big one
Zoom in on one cell of the big board and you find a whole tic-tac-toe board. The whole and a piece of it have the same shape. Mathematicians call that self-similarity.
Self-similar things are everywhere. A big branch splits into smaller branches, and each of those splits again, and every piece looks like a smaller copy of the tree. A river splits into streams, and streams into creeks. A jagged coastline looks much the same from far away and from close up. In your lungs the airway splits in two, again and again, about 23 times, until it becomes tiny air sacs.
The mathematician Benoit Mandelbrot gave these shapes a name: fractals. Some people say a shape needs at least 3 levels of self-similarity before it counts as a fractal. Ultimate tic-tac-toe has exactly 3: one cell, one small board, one big board.
Think about this: a branch splits into 2 smaller branches each time, and it splits 8 times. How many twigs are at the end?
Worked it out? Check the answer
After 1 split there are 2. After 2 splits, 2 × 2 = 4. After 3, 8. Each split doubles the count. After 8 splits: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 twigs.
One simple rule, repeated, grows a shape this tangled. That is what makes fractals interesting.
Secret 2: what if you nest it once more
Ordinary tic-tac-toe has 9 cells. In the ultimate game each of those cells holds 9 more, so 9 × 9 = 81 cells.
The book's author says that if that is still not enough, you can join 9 ultimate boards into one still bigger board: 81 × 9 = 729 cells. The rules have to change with it. The move before last chooses the middle board, and the last move chooses the small board inside it. The story is that nobody has dared to try.
| Levels | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cells | 9 | 81 | 729 | ? |
Each extra level multiplies the number of cells by 9. Growth that keeps multiplying by the same number is called exponential growth. It looks mild at first, then it runs away.
Think about this: how many cells are in 4 levels? How many levels until there are more than ten thousand?
Worked it out? Check the answer
4 levels is 729 × 9 = 6,561 cells, still under ten thousand. 5 levels is 6,561 × 9 = 59,049, well past fifty thousand. So you need 5 levels.
From 4 levels to 5, you add one level and more than fifty thousand cells. That is exponential growth.
Secret 3: a good small move can be a bad big one
In ordinary tic-tac-toe everyone wants the center, because the most lines pass through it. In the ultimate game, playing in the center of a small board sends your opponent to the center board of the big one. If that is exactly where they wanted to go, your "good move" just helped them.
So every move has two questions. Is this cell good on the small board? And the small board it sends them to: who does that help on the big board? The book calls this "think globally, act locally".
One more thing to watch. Fill a small board and it becomes shared. Both players can use it in a line. The move that fills it might complete a line for your opponent.
Think about this: strong players rarely send an opponent into a board that is already closed. Why?
Got an idea? Here is how a strong player sees it
Sent into a closed board, your opponent may play in any open board they like. You used to choose where they go. Now you have handed them the choice, and they will pick the place that helps them most.
In this game, leaving your opponent with no choice is an advantage by itself.