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!