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.

Here’s a little bit of a surprising variant. Players alternate choosing a number from 1 to 9 (each number can only be used once in all). The first player to collect three numbers that sum to 15 wins. The game turns out to be identical to Tic-Tac-Toe, because 15 is the magic constant of a normal 3×3 magic square. So the strategy is exactly the same, and (in theory) the game should be equally boring — but it’s generally not, because

realizingthat the strategy is the same is nontrivial.Assuming the numbers in this variant correlate to spaces on a TIc-Tac-Toe board, wouldn’t choosing 1-2-3 win you Tic-Tac-Toe, but not the variant?

That’s true if you arrange the numbers in order, but in this variant, you’d actually arrange the number like this:

4 – 9 – 2

3 – 5 – 7

8 – 1 – 6

I discussed the Ultimate game with my colleague who teaches Computer Science. He proposed a rule variant where a player is not restricted to only playing X or only playing O. For example, I might play X to start, you might play O and then on my next turn I decide to piggy back on your move and I switch to O. The winner is whoever completes a three in a row first, regardless of whether it is X or O. I have not played through this to see if it is as interesting as it sounds.

Hmm, that’s an interesting variant. It has such a small space of possible moves that it should be solvable by hand. But now I’m curious to solve it…

Oh, you mean this as a variant on the Ultimate board! That’s even more interesting than I thought. (You can use the same variant on a small board, but it’s bound to end in a tie.)

How come it’d be a tie? In my eyes, as long as you are the first player you win. It should be different on an ultimate board, though.

Hmmm… I just played a quick game were player 2 actually won. Obviously I need to think a little more about this. Essentially, you’re trying to avoid creating any kind of two-in-a-row that could lead to a win, and you’re trying to trap your opponent into creating one. At first I thought that’d just lead to conservative play, resulting in a tie, but I don’t think that’s necessarily true anymore.

If I understand the variant correctly (play either X or O on your move) then the first player has a forced win on a 3×3 board by playing 1. X-Center.

If player 2 places an X anywhere on the board, player 1 completes the tic-tac-toe.

If player 2 places an O in a corner, player 1 plays an O in the opposite corner leaving player 2 with no safe moves.

If player 2 places an O on an edge, player 1 plays an O on the opposite edge. Player 2 must then play a 3rd O on one of the two remaining edges and Player 1 plays an X in the remaining edge leaving Player 2 with no safe moves.

Order and Chaos uses a rule like that on a 6×6 grid: https://en.m.wikipedia.org/wiki/Order_and_Chaos

My son and I usually play Ultimate TTT at restaurants when we’re bored. I like how it teaches the value of looking ahead several steps and not taking easy moves. We’ve always played to win the majority of the small boards instead of getting 3 in a row, we’ll have to try it that way next time and see if it makes a difference.

I think majority of boards and 3-boards-in-a-row both work well (and turn out to be pretty similar). One fairly different variant is to play until someone wins just one small board. It makes for a much faster game (though a slightly less interesting one, I think).

Also: a puzzle you play once, against the puzzlemaster who wrote it, but a game is social and on-going. There are chess and bridge puzzles, after all, and you could have students play a game of writing each other tricky math problems.

Well said. Puzzles are one-shot deals, whereas games regenerate.

*reads the text a second time to actually do the puzzles*

*considers how in the first picture, you should have drawn 500,000 people instead of ~40*

*moves mouse towards the picture at the same time*

OH SHIT YOU HAVE ALT TEXTS LIKE XKCD THAT ARE ALSO HILARIOUS I NEVER KNEW.

1. Olinguitos are top-to-bottom crazy awesome. The Orlinguito is henceforth my new online persona.

2. The alt-texts are the best-kept secret of this blog! That plus the haunted post of…. never mind. Anyway, to my knowledge you’re the first to comment on them.

1. If I continue at this pace (Ben Ohrlin, Orlinguito…), you will be able to change online personas like socks!

2a. > That plus the haunted post of…. never mind.

Clue, clue, clue! D:

2b. I’ve found another secret. You have a second blog! It can be reached by clicking your name. (PS: The domain expired 37 days ago.)

The rules for Tic-Tac-Grow, as I have learned them, state that you need to get *4* in a row. This makes it very fun to introduce the rules to students. I love showing them a 3×3 grid, telling them they need to get 4 in a row and listening to the cries of disbelief and dismay. Works like a charm!

Incidentally, glad you like the name–I came up with it–though I imagine I wasn’t the only one who has done so…

I also really enjoy the distinction you draw between puzzles and games. Though, I find the “steps” you talk about are what I really appreciate about puzzles. I love solving puzzles and applying what I discover about these steps to different puzzles of the same type. It’s true that once you’ve solved one puzzle you’re done–with that particular puzzle–but the beauty is that those rules apply to the next puzzle, and the next one too. I like to draw the analogy between puzzles and axiomatic systems–we build our understanding of the “world” of puzzles based on the conclusions we can draw from a limited number of first-principles, generally referred to as the “rules” of the puzzle. I find a huge amount of richness in this.

Ah, thanks for explaining that! I just tried a game of the 4-in-a-row version, and it seems to work quite well! I’m going to keep on thinking about it, and update this post.

That’s a nice analogy between puzzles and mathematically rigorous systems. I wonder what it is that makes puzzles so appealing to people, whereas a phrase like “axiomatic systems” turns so many people off? Maybe it’s that the puzzle has been crafted for enjoyment, whereas the mathematical system has been crafted for parsimony/elegance/abstract power/ease of proof?

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and this – https://www.academia.edu/1527773/New_Tic_Tac_Toe

And I love the form of Tic Tac Toe that involves 3 sizes of nesting pieces called Gobblet or Gobblers or Gobblet Gobblers where you can overlay a piece with a larger piece and pieces on the board can be moved. https://www.youtube.com/watch?v=Xpz7-8MbRvg

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