Review

For those who have come across this post by accident, this is Part 2 of a 2-part post on a tic-tac-toe-like game called Notakto. Part 1 can be found here, and will give you a more complete introduction to the game, as well as lessons on how to play and win on 1 or 2 boards against someone else.

Everything from this point on assumes that you've practiced all the strategies and can win 1- and 2-board Notakto games every time.

You should know and understand the importance of terms like sacrifice, boot trap, and 2X trap. You should also be very familiar with the few rules we've introduced so far:

• When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.

• Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).

• When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

Starting from this point, I'll show you how to generalize what you already know to play and win Notakto on 3 boards, 4 boards, and beyond!

In the next section, you'll start by learning how to play 3 boards.

3 Boards

Let's start with the simplest example of a 3-board Notakto game.

Since there are an odd number of boards, you start the game, and mark an X in the center of any board. In this example, the other player responds by marking an X on the same board, and your reply is to sacrifice that board.


At that point, the game reduces to a 2-board Notakto game with the other player effectively going first. Even before they place their first X on the remaining boards, you should already know that you're going to win.

This is the basis of another general rule you should keep in mind:

Your basic strategy is to sacrifice boards, until you get down to the final remaining 2 boards. These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.

Marking a Different Board

In this next example, you play first once again, but your opponent marks an X on the edge (it could just as easily be a corner or the center) of a different board. You recall this rule, and respond accordingly:

When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

With 3 boards, all 3 now have an X on them. From there, you play to eliminate the first board you can, and play the rest as a standard 2-board Notakto game.


There are a few more lessons we can learn from this game. First, if your opponent doesn't mark a previously-empty board in the center, the defenses you've learned will usually prevent them from placing an X in the center of that same board until they're forced to. Of course, if they do mark a previously-empty board with an X in the center, you can easily sacrifice it or set up a boot trap, as needed.

Because of the rule that tells you when to be player 1 or player 2, combined with the rule that tells you when to place an X in the center of a previously-empty board, you can always guarantee yourself a minimum number of boards with Xs in the center.

With an even number, such as 4, going second and only marking previously-empty boards after your opponent does the same thing, it's easy to see a pattern of theirs-yours-theirs-yours guarantees you a minimum of 2 boards with Xs in the center (it's also possible ALL of them could have Xs in the center). Extending this to 6, you should easily see that the minimum number of boards with Xs in the center will be 3. Given any even number n, the minimum number of Xs in the center you'll have is n/2.

What about odd numbers? With an odd number of boards, you always start, effectively guaranteeing yourself an extra board with an X in the center. So, for any odd number of boards n, the minimum number of boards with Xs in the center works out to be (n + 1)/2. For 3 boards, that is (3+1)/2 = 4/2 = 2 boards minimum with Xs in the center. For 5 boards, you'll have a minimum of 3 boards with Xs in the center, and so on.

Why are the Xs in the center so important? When learning how to play 2-board Notakto, your defense depended on at least one of the two boards having an X in the center. Even if you didn't end the game on this particular board, it was the board with the X in the center which allowed you to keep control of both boards and win with either one.

Because of this, we're going to amend the rule taught from the simple 3-game demonstration above. The revised rule is below:

Your basic strategy is to sacrifice boards, until you get down to a set of boards, only 1 of which has an X in the center. On the sole remaining board with an X in the center, you will build your boot trap. If there are two or more boards without Xs in the center, play to sacrifice them until you get down to 2 boards (one of which is the board with an X in the center). These remaining 2 boards will be played just as you would play any standard 2-board Notakto game.

It's time once again to practice. Practice 3-board Notakto either online or with the iPad Notakto app until you can win every time.

Once you feel confident playing and winning 3-board Notakto every time, you're ready to apply what you know to win on 4 boards, 5 boards, and beyond! Just make sure you can win every time on a given number of boards, before moving on. You'll probably notice that it takes you less and less time to play perfect games on each level.

Once you can win every time on any level, it's time to stop playing against computers and learn how to present this as a game to real people.

How To Present Notakto

Now that you know how to play and win Notakto every time, it's time to learn how to present this to a real person. My first bit of advice comes from Bob Farmer, who would often include the following warning in his Flim-Flam! column:

Caveat Scamtor: Ethical Hustlers warn the Mark the game is fixed. Money lost is an educational investment. Gambling may be illegal where you live. Information in this column may be wrong, so don't bet the farm until you've verified it's right.

Since Notakto is so similar to tic-tac-toe, that's often the best way to introduce it. Here are a few key points that help when introducing the game:

• “Ever play tic-tac-toe? It's fun until you get to the point where both players are good enough that it's always a tie.” - This brings up the topic, and establishes familiarity.

• “My friends and I liked the fact you could play it almost anywhere, but hated those tie games, too.” - This takes the familiarity, and gives a reason for the non-standard rules you're about to introduce.

• “To prevent the ties, we decided that both players should play as X. After all, if you only mark Xs on the board, somebody has to get 3 in a row sooner or later, right?” - This introduces the all-Xs rule in a sensible way.

• “After some testing, we found it makes for a longer and more interesting game when the person who makes 3 Xs in a row is the loser instead of the winner.” - This brings up the other major rule change. Note that both new rules are introduces with the real reasons for their existence.

• “We've found it's even more fun when played on more than one board at the same time! When we do that, any board with 3 Xs in a row is out of play, and the last person to make 3 Xs in a row on the last available board is the loser.” - This quickly establishes the idea and rules of multiple board play. It is important for later than you do not mention a specific number of boards at this point.

Who Plays First?

At this point, you've already mentioned that the game will be played on multiple boards, but avoided mentioning exactly how many. This is about to give you the advantage in the game.

Ask the other player whether they'd like to go first.

If they decide that they're going first, draw 2 boards (or 4, if you prefer), and invite them to place their X anywhere. If they decide you're going first, draw 3 boards (5 is often too intimidating for a first game), and mark your X in the center square of any board, as usual.

You've subtly guaranteed yourself a win by having their choice of who goes first determine the number of boards!

If you want to state the number of boards before asking whether they go first, mention specifically that you'll be playing on 3 boards (or any odd number), and ask whether they'd like to go first. If they let you play first, play and win the game as normal.

If they decide they're going to play first instead, mention that, since the game is new to them, you should start with a practice game, so they can get the idea. Play the practice game, preferably playing to lose to build their confidence. Remind them once or twice through this game that it's a practice game. Once this “practice game” is over, play a legitimate game, mentioning that, because they went first last time, you get to go first this time.

You can find more tips on presenting Notakto in the next section.

Tips

• One more suggestion for the “practice game” technique: You could always have your practice game take place on a single board, explaining that this is to help them get the idea. Then, for the real game, you can move on to 3 boards as promised, in which you go first in that game, since they went first in the practice game.

• After you make each move during the game, you'll notice that usually all available boards will have an odd number (1, 3, or 5) of Xs. Naturally, this means that before you make your move, there will be only one board with an even number of Xs. This can act as a strong signal to tell you which board you need to play.

There are two exceptions to this pattern:

1) At the beginning of the game, when your opponent has just marked an X on a previously-empty board. In this situation, you're going to mark a new board with an X in the center square, of course. That makes this one case where each board played starts with an odd number of Xs before you play, while the same is true after you play (since you're marking a new board).

2) The other exception happens at the end of the game, the one described in the Attacked! section of this tutorial. If you're down to two boards, one of which has an X at the center and the other one doesn't, and your opponent sacrifices the board with the X at the center, then each move you make will leave an even number of Xs on the remaining board, instead.

• If you want to understand more about Notakto, here are a few helpful resources:

- Another puzzle (bgonline.org discussion, where the idea was first posted to the internet)
- Neutral Tic-Tac-Toe (MathOverflow discussion)
- Impartial Tic-Tac-Toe Presentation (PDF)
- The Secrets of Notakto: Winning at X-only Tic-Tac-Toe (PDF)


• This winning strategies don't need to be top secret. If someone genuinely shows an interest in learning how to play Notakto and win every time, feel free to teach them and/or send them to this tutorial. Above all, enjoy it and have fun!

Introduction

Math professor and Backgammon expert Bob Koca was playing tic-tac-toe with his 5-year-old nephew, when the nephew whimsically suggested that they should both play X. After mathematically analyzing such a game, Professor Koca realized that this was a deceptive new version of the classic game of Nim.

Thane Plambeck later dubbed this game Notakto (pronounced No-tac-toe). In this tutorial, you'll learn how to play this game so that you can win every time!

Notakto is played on a standard 3-by-3 tic-tac-toe board, and the rules are as follows:

• Both players alternate making an X on the board.

• Players may mark on X on an available space (again, any space not already occupied by an X) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the board is the loser.

If you'd like to try out this game for yourself, you can play the first level online here (click reset if you pass the first level to stay). If you have an iPad, you can download and play the Notakto app here.

When referring to general types of squares, I'll use the terms center, corner, and edge to apply to the various squares as follows:



When I need to refer to a square in more specific terms, I'll refer to the various squares with these terms:



I'll also frequently use the terms rotations and reflections.

If you rotate a given pattern of Xs through quarter turn (90°) increments to match another pattern of Xs, then those two patterns are rotations of each other. Two given patterns of Xs that are mirror images of each other, with left and right switched, and/or top and bottom switched, are reflections of each other.

These are important concepts, since patterns that are rotations or reflections will share the same strategy.

Now that you've got a basic understanding of the concepts involved, it's time to learn how to win 1-board Notakto!

Winning 1-board Notakto

Mathematician Timothy Chow originally likened the winning moves to that of a chess knight. If you're not familiar with chess, chess knights move either 1 square vertically and 2 squares horizontally or 1 square horizontally and 2 squares vertically, resulting in an unusual L-shaped move.

If we think of a chess knight on a Notakto board, you'll note that the knight in the corner square below (on your left) can only move into either of the edge squares marked with a dot. The knight located in the edge square (on your right) can only move into either of the corner squares marked with a dot.



Because of the unusual way the knight moves, a knight in the center doesn't have any possible moves.

Here's how to use the knowledge of a knight's move to win at Notakto.

To guarantee yourself a win on 1-board Notakto, you must play first, and start by placing your X in the center.

After each time the other player marks an X in a given square, you'll place your X a knight's move away from that square.

As seen in the knight's move graphic above, there will often be two open squares that qualify. Before placing your X, make sure that you're not inadvertently making 3 in a horizontal, vertical, or diagonal line of 3 Xs, thus losing the game.

Played properly, the resulting game should look something like this:

Why This Works

By starting with your X in the center, that limits every following move to be played on centers and edges, which are squares where you can use the knight's move strategy.

If the other player places an X in a corner at this point, the knight's move will place your X on an edge square. Conversely, If the other player places an X in the edge at this point, the knight's move will place your X on a corner square. This results in a boot-shaped arrangement of Xs at this point:


I refer to this arrangement (and any rotations and reflections of this arrangement) as the boot trap. Take a close look at it. The other player in the boot trap above won't place their X in the lower edge or the upper right corner, because they would instantly lose the game.

That leaves 4 remaining squares that seem harmless enough. However, when they place their X on any one of those squares, and you place your X a knight's move away (again, making sure you don't accidentally complete a horizontal, vertical, or diagonal line of 3 Xs), the board then looks something like this:


Notice that placing ANY X on this board must result in a losing play! From the boot trap, you can always force this situation, which means you'll always be able to win the game.

Practice 1-board Notakto online or with the iPad Notakto app until you can always win.

Once you feel comfortable enough with the 1-board strategy, it's time to learn to win 2-board Notakto!

Playing 2-board Notakto

For playing Notakto with 2 or more boards simultaneously, the rules are similar, but there are a few changes:

• Both players alternate making an X on the board.

• Once an individual board has a horizontal, vertical, or diagonal line of 3 Xs on it, that board becomes unavailable (no more moves may be made on it).

• Players may mark on X on an available space (any space not already occupied by an X on any available board) any available board during their turn.

• The person who makes a horizontal, vertical, or diagonal line of 3 Xs on the last available board is the loser.

While you'll still use the boot trap and the knight's move strategy, there are some new adjustments and new strategies to learn in order to win multi-board Notakto every time.

To guarantee yourself a win in 2-board Notakto, the other player must go first. In fact, there is a simple pattern that will help guarantee you a win with any number of boards:

When playing on an odd number (1, 3, 5, etc.) of Notakto boards, you can guarantee yourself a win by being Player 1 - the odd-numbered player. When playing on an even number (2, 4, 6, etc.) of Notakto boards, you can guarantee yourself a win by being Player 2 - the even-numbered player.

There's also a simple rule for remembering where to mark your first X:

Your first move will always be placing your X in the center square of any empty board (empty board refers to any Notakto board with no Xs already on it).

When you go first, all the boards are empty, so of course you can make your first move this way. If you're Player 2, there will always be at least one other empty board after the other player's first move, so you can be assured of still having an empty board on which to mark the center square.

These are great strategies for knowing how to start the game in your favor, but as I mentioned earlier, you'll need to learn some more strategies to assure yourself a win. The first new strategy is learning how to sacrifice a board.

Sacrificing

Since only the person who makes a horizontal, vertical, or diagonal line of 3 Xs on the final available board loses, taking other boards out of play by purposely completing a line of 3 Xs can be very helpful. Taking a board out of play in this way is called sacrificing, just like the same concept in chess.

Let's start with the simplest possible example of sacrificing a board. In the example below, the other player is Player 1, and you are Player 2 (because there's an even number of boards, remember?).

Their first move is the center square of one of the boards, and your reply is the center square of the remaining empty board. From this arrangement, no matter where they put their next X, there will always be 2 Xs in a line. Your response is to complete that line with a 3rd X, and sacrifice that board:


Once that sacrifice is made, the game effectively reduces to a 1-board Notakto in which the other person is the second player. As shown in the animation above, you set up your boot trap, and proceed to win the game just as before!

While the above game can and does happen, not every game happens in such a simple and straightforward manner.

Our next example will still show the concept of sacrificing a board, but it will happen later, and after a more complex series of moves.

This game starts, again, with the other player going first. This time, they're going to start in a corner (though it could just as easily be an edge), and you respond just as you should, marking an X in the center square of the remaining board.

This time, however, the other player marks an X on the same board on which you just played your center X. What do you do in this case?

Simple! You respond just as you would in 1-board Notakto, and set up your boot trap.

From here, we'll assume the other player marks an X in line with the X they played first. Naturally, you complete the row of 3 and sacrifice that board. The game then returns to the board on which you've set up your boot trap, and you win in the usual way:


This latter, more complex game actually shows a number of concepts that will be important through the rest of this tutorial:

1) Any strategy you learn in this tutorial can be delayed and still remain effective.

2) This is a another good lesson in determining which board to play:

When the other player marks their X on a board with pre-existing Xs, your next move will be made on that same board. When the other player marks their X on a previously-empty board, your next X will be placed in the center of another empty board.

In the previous tab, we discussed a similar rule that helps determine where you start. This rule, on the other hand, determines on which board you will continue play.

Try practicing 2-board Notakto online or with the iPad Notakto app using this approach, but don't be discouraged if you don't win.

Playing this way will guarantee a win if the computer responds as we've assumed above, but you'll notice that there are some situations that haven't been covered yet. The most common is when they start on a corner or an edge, and the next time they make a move on that board, there's no way to make 3 Xs and sacrifice the board!

In the next section, you'll learn how to use that kind of play to set up another trap for the other player!

The 2X Trap

When the other player marks their first X in a corner or an edge, it's quite possible that their next move could prevent you from sacrificing the board on your next move. It's not uncommon to see a situation such as this (the X in the right board is yours, the 2 on the left belong to the other player):


Another arrangement that could prevent an immediate sacrifice happens when two Xs are marked on edges that aren't directly across from each other (Again, the X in the right board is yours, the 2 on the left belong to the other player):


In both of the above cases, the next move is yours. What is the best move to make?

With both arrangements, the answer is exactly the same! You should place an X to make the pattern shown on the left board:


Why? The next time your opponent places an X anywhere on a left board with that arrangement, they'll wind up with 2 Xs in a line. When they do, you can place the 3rd X and sacrifice that board! Yet again, you can prepare your boot trap on the right board, and win the game as usual.

Because this design forces your opponent to place 2 Xs in a line, I refer to this as the 2X trap. It's easy to visualize, as the 2X trap always has 1 X in a corner, and 2 edge Xs, both of which are a knight's move away from the corner X.

Below is a full game animated, in which the 2X trap is used. This particular game involves a rotation of the 2X trap depicted above, so you can get used to seeing it in another arrangment. As with all 2-board Notakto games, the other player goes first.


As with the earlier example games, the 2X trap could be played even if you had developed the boot trap further on the left board. Delayed use of tactics, as I've said before, can still be effective.

Go practice 2-board Notakto either online or with the iPad Notakto app with your knowledge of the 2X trap, and you should find that you're winning more 2-board games than before.

However, your 2-board game still won't be perfect. Sometimes, your opponent can be sneaky and sacrifice your carefully-prepared boot trap! You'll learn how to handle that situation in the next section.

When They Sacrifice Your Boot Trap

Up to this point, every strategy discussed has involved sacrificing all but the last board (if any), and then winning the final board via the boot trap and the knight's move strategy that has been taught.

It isn't difficult to conceive of sacrificing a board, however, and your opponent can do it just as easily as you can. Below is a snapshot of a 2-board Notakto game where the following has occurred:

1) The other player moved first, and placed their X on an edge square (left board). You responded by placing your X in the center of the other (right) board.

2) They marked their next x on an edge square on the right board, and you responded preparing your boot trap.

3) They then marked an X on an edge square on the left board in such a way that you couldn't immediately sacrifice the board. You respond by setting up the 2X trap.

4) They then surprise you by sacrificing the right board, ruining your boot trap.

It's now your move. What do you do?


In this situation, there's only one effective response. You must place your X in the corner directly opposite the other corner X in the 2X trap:


Why does that work? While there are 5 remaining spaces open, the other player cannot place their X in the center or either of the open corner squares, because they'd lose the game. Their only possible response is to place their next X in one of the remaining edge squares, and your response will be to place your X in the other remaining edge square.

After they mark their edge square, and you mark your edge square, it's their move again, and the board now looks like this:


Anywhere they place their X on the left board, they must lose (Remember, the right board is out of play).

When They Sacrifice Early

Sometimes, when the other player sacrifices your boot trap early, the other board only has a single X on a corner or an edge (If the other board had a single X in the center, then you built your boot trap too early and on the wrong board), like this:


The response for this situation is easy, but different enough that it warrants its own section.

The strategy for this situation is simple. Your next move is to mirror the placement of the other player's Xs. What do I mean by mirroring the placement?

If they placed an X on the upper edge, you place your X on the lower edge (and vice-versa). If they placed an X on the right edge, you place your X on the left edge (and vice-versa).

Similarly with the corners, if they placed an X on the upper left corner, you place your X on the lower right corner (and vice-versa). If they placed an X on the upper right corner, you place your X on the lower left corner (and vice-versa).

In the case shown above, there an X in the lower left corner of the right board, and it's your move. The mirroring strategy tells you to place your X in the upper right corner, and keep mirroring their plays as discussed above.

If they mark a mixture of corner and edge squares, and you mirror those appropriately the boards will eventually look like this after your last move:


Does the pattern on the right board look familiar? It's the pattern of 6 Xs with an open diagonal (albeit a rotation of that earlier pattern) that ended the game discussed above.

Alternatively, if the other player keeps marking Xs only in corners, and you keep mirroring the appropriate corners in response, you'll come to this pattern much more quickly after your last move:


Even though there's only 4 Xs in this pattern, the other player must lose. It's their turn, and marking an X in the center or any edge means completing a horizontal, vertical, or diagonal row of 3 Xs, thus costing them the game.

With everything you've learned here, you're now equipped with enough knowledge to win every 1-board and 2-board Notakto game everytime.

To make sure you know how to use this knowledge, it's time again to practice 1-board and 2-board Notakto either online or with the iPad Notakto app.

Keep practicing until you can win the 1-board and 2-board games every time! Try and play without looking back at these strategies, but don't be afraid to look back at them when you need to do so.

Not only have you learned all the strategies you need to win every 1- and 2-board Notakto game, you've also learned all the strategies you need to play on any number of boards! In the next post, I'll take what you've already learned and show you how to generalize those strategies so you can apply them to any number of boards.