Noughts and Crosses: a Defence of Unusual Openings

Noughts and Crosses: a Defence of Unusual Openings

When was the last time you won a game of Noughts and Crosses? I'm going to guess it was in early high school. By playing in the centre as your first move you will almost certainly draw, since I think most people know to play in the corner as a response because they've seen that position in almost every game.

Then, at some point someone will show you this trap:

 Common Trap

Common Trap

They open in the corner and you quite naturally take the centre. They go in the opposite corner and now there is potential for you to go astray. If you now go in either corner then they have a forced win. It is surprising when you first see it, but once again you won't be fooled more than once.

Well here is my solution for getting a more interesting game: try starting at the edge. No one plays it, but the worst that can happen is you get a draw (which was probably going to happen if you went for the centre anyway), but there is a possibility of actually winning this inherently drawn game because you are taking your opponent away from what they have seen before.

In fact, from the 8 possible responses that your opponent can choose from after your first move is on the edge, 4 of them lead to forced wins for you and non of them particularly obvious. Go and try to work them out now. These will be the first new thoughts you've had on this game you thought you had mastered a decade ago.

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... Go on,  do some thinking. It's good for you.

 

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Ok here's my analysis. I'm going to start with a cross in the top middle square, but the whole thing is symmetrical. Here's a picture of which responses lead to wins for you:

 The cross at the top is your first move. The other spaces are what you will be able to force if your opponent responds in that place.

The cross at the top is your first move. The other spaces are what you will be able to force if your opponent responds in that place.

So the middle unfortunately leads to a draw unless someone does something stupid: so even though you may be itching to try out your new trap lines when you play the edge opening, realise that most of the time you will be just faced with this.

I'm going to start by analysing what happens if they respond middle left. Our only second move which guarantees a win is in the top left corner like this:

 I drew the arrow too long. There are two threatened lines, just not where I pointed. Sorry.

I drew the arrow too long. There are two threatened lines, just not where I pointed. Sorry.

You'll see that after that move in the top left everything was forced until it gets to the point where we threaten two separate lines.

Now lets try analysing the best move if their first response was in the bottom left corner. Our only winning move is (once again) the top left corner. Let's see it play out:

Once again we have a long line of forced moves until we get to the point where we threaten two lines. By symmetry we also have our answers for the other two losing first responses from our opponents. The other first responses that draw can play out in several different, dull ways. Common sense is enough to proceed in them to not lose.

Now that you have seen this line of thought, maybe you should think about what you should respond to someone playing not the centre or the edge openings, but the corner opening instead?

13 Months in a Year

Buckminster Fuller

Buckminster Fuller