I’m still stunned by the response to my post on Ultimate Tic-Tac-Toe, which spawned a whole fleet of mobile apps, was translated into Spanish by the Argentine Department of Education, and has drawn more than half a million visitors.
I take no credit. I didn’t invent this game, just drew some silly pictures explaining it.
In response, commenters suggested lots of other variants on Tic-Tac-Toe. They ranged from well-known to obscure, from simple to complex, from fun to “I guess somebody must find this fun.” I’ll post someday about the variants that make good games. But this is a post about the ones that make good puzzles, and why “puzzle” isn’t the same as “game.”
Puzzle #1: Tic-Tac-Toe with No Starting Grid
Suggested by several folks, this fairly self-explanatory variant is easiest to follow if you picture, instead of no grid, a never-ending grid. Eventually, we’ll narrow this endless grid down to a standard 3×3 board. That narrowing down will happen gradually, in a manner determined by the moves we make.
On my first move, I can go anywhere I want. Let’s suppose I pick here.
Next, you can go anywhere you want, so long as it could conceivably share a board with my move. (This means you can’t play too far away, or our moves could never occupy the same 3×3 board.) So you’re limited to these spaces:
So let’s suppose you go here:
For my next move, I’m limited to spaces that could share a 3×3 board with the two moves already made:
For example, I might go here:
By now, you get the idea. You’re limited to these spaces:
You need to block, so let’s suppose you go here:
Now, finally, we’ve narrowed ourselves down to a final board, and the game becomes a regular round of tic-tac-toe. So I’ll go here:
Now, if you’re familiar with tic-tac-toe, you can recognize that X will win on its next move. (And if you’re not familiar with tic-tac-toe, then whoa, how did you miss that growing up?)
With a little free time, it’s possible to solve this game—that is, to figure out exactly what X’s and O’s best moves are at each step. I recommend giving it a shot—it’s a fun problem!
Hint #1 (highlight to read): X’s first move doesn’t really matter, since it doesn’t help define the board at all. Then, for O’s first move, there are five distinct options. Start by identifying those five options.
Hint #2: 4 of the 5 moves O has will lead to defeat! Figure out which ones they are, and then see what happens with the fifth possibility.
Puzzle #2: Pay-to-Play Tic-Tac-Toe
Ethan Bradford describes “Pay-to-Play Tic-Tac-Toe.” The rules are a little complicated, but in essence, you need to “buy” squares to go in them. The center is most expensive, and edges are cheapest, with corners falling in between.
In this game, X starts with slightly less “money”—presumably to counterbalance its first-mover advantage. But is that penalty too severe? Or perhaps not severe enough?
So here’s your puzzle. Assuming you’re not allowed to pass on your turn, who wins this game? Follow-up: How does allowing players to pass change the outcome of the game?
Puzzle #3: Tic-Tac-Grow
One proposal was the adorably named “Tic-Tac-Grow,” in which each time you mark a square, you add another square to the board. So after my first move, I might do this:
Then you might do this:
Then I might do this:
Uh-oh! Your next move can only block one of my two threats for victory. So I’ve won.
In fact (spoilers!), X always wins. Just a single “grow” allows X to win every game in precisely three moves, no matter where on the board it starts. (If you’re curious, go figure out how!) “Tic-Tac-Grow” isn’t really a game, or even a puzzle; it’s a textbook sidebar on “first-mover advantage.” The “grow” move is a cute rule change, but the game seems to flop.
Then Breedeen Murray, who suggested the game, pointed out my mistake. You need to get four in a row! This actually turns Tic-Tac-Grow from a lame non-puzzle into a pretty interesting game, which leads to two interesting questions.
What makes Ultimate Tic-Tac-Toe (or anything) a good game?
My take? It’s a welcoming playground for strategic thinking.
First, every move offers a small handful of options, ranging from 2 to 9. That’s a perfect balance between childish games with no strategic element, and sophisticated games with dozens or hundreds of moves to consider.
Second, there’s a clear final goal (win 3 boards in a row), with straightforward minor goals along the way (win small boards), and familiar stepping stones to achieving those minor goals (get two in a row, or claim key spaces like the center and the corners). This makes it easy to decipher the strategic implications of every move—not always true in other games.
As a result, it feels like the right move is always just within reach. The game is complex, but not mysterious or intimidating.
What’s the difference between a puzzle and a game?
Puzzles are made to be solved. And once they’re solved, they’re done. That’s how it is with the three games I’ve outlined above, and with classic tic-tac-toe as well. A few minutes of playing out possibilities, and you know exactly how every scenario will end. If you’re challenged to “play again,” you won’t even need to turn your brain back on, just to follow the steps you’ve already discovered.
Games are different. Games pack surprises, even for experienced players. When you play a game, you’re not just executing an algorithm. You’re thinking, strategizing, discovering.
As a math teacher, I mostly traffic in puzzles. I try to engage my students with tricky ideas. I encourage them to work out the possibilities, to master the scenarios, to boil their understanding down to steps that can later be applied automatically.
But when it comes to math itself, I hope my students see it as more than a collection of puzzles. I hope they see it as a game, full of promise and possibility. I hope math never exhausts its ability to surprise, to stump, and to delight them.