Creating Squares

Pastels in a grid
In the game below, whoever completes a square first is the winner. You can play against the computer or a friend.
You can print off some dotty paper if you'd find it useful.
Once you've played the game a few times, you might like to think about the questions below.


Here are some questions you might like to consider:

  • How can you tell if four dots make a square?
  • Can you develop any winning strategies?
  • Can you suggest any mathematically interesting variations on the game?



In a 6 by 6 grid the blue ( or the starting player) will ALWAYS win!

Sergio, Interesting.

Do you mean blue will always win if they are both playing the best moves available to them? Why would that be so? And why do you say on the 6 by 6 grid?

No you can win as red.

I don't get what you mean Rajj, could you explain it a bit more, please?

it is fun trying to beat the computer but, I think there should be an easy mode.

I beat the computer! WHOO

Red can win. It is possible because sometimes blue thinks that red is never going to win, but when they don't pay attention, red wins. Red will have the mindset that they can't win because of the comment under the board. If they have the mindset that they can't win, then mostly they can't win.

One strategy is to always start at the center.
Then based on red's position always choose the opposite half (top or bottom). Choose the 2nd position from center.
If red comes adjacent to you. Move blue to diagonally opposite end of previous move
In the next move fill the middle (between center and the position described above) with blue.
In the next move try to create a T symbol with Blue.
Then in the next move red will try to block you from creating one of the squares, but you can always create the other. :-)

Took some time to figure this. But Blue can always win. One I have followed the above, blue has never lost.

That sounds like a good strategy! Does it work on all board sizes?

This is correct but it doesn't work for a 3x3 grid. Blue will win but you need to be careful. If you go on the same side as the computer, he can easily force a draw or win himself.

This mostly only works on a 5x5 grid

This is the only real strategy which works, but it only works on odd numbered grids

See, the thing is, I couldn't beat red until I thought about it:

You would need 3 blue dots adjacent to each other (like an ...)
Then, you place a dot below the middle dot.
Red will block one corner of your mini square, but you just use the other square and you win!

This will only work if Red does not block a corner before.

This game is really fun! The red is really good and I have only won this game once so far! I think there should be an easy, medium and hard mode for beginners!

What ever dots the blue team puts on ,the red team could block him by making a square. Even if the blue starts in the middle we can not be sure if blue will win. Please correct me if I am wrong .

I believe blue can usually - depending on the size of the board - force a win by developing a forking attack. Can you work out how?

A forking attack threatens to create 2 or more squares so red can only block one.

First I put my dot in the centre. Wherever the computer goes I put a dot 2 up or down with the aim of creating a pyramid by doing it between them so it is in the shape of a triangle. This threatens a win so they must stop it - they have no choice. Now I make a 'T' by doing a dot between my first 2 dots. Then I threaten 2 wins and I will win.

Thank you. I tried it and it worked. I can't remember which grid it was on though

I raged for so long against the computer but eventually I beat it.

Gets your brain thinking, so fun! I would play it all day if I could, I recommend it.

Start in the middle, then click on diagonal dots either side of the middle to make a diagonal line of 3.
Then click on a dot to the right, left, up, down or diagonal from the middle.
Create a T shape with your dots then you can usually win on a 5 x 5 grid.

Charlotte G & Sukie B

First place a dot in the middle, then try to make a t shape that has three dots up top and one at the bottom.

i had to play this game for a school project, once we'd played the game we had to write a comment to give our ideas on how we can make the game more complex. My idea is that there could be less squares and more people, this way it would make the game more of a challenge.

I played the game for a school project, and I didn't win; to make this game harder they could put more people in the game. This much of a challenging game is good because it gets people thinking about what the solution could be..

this game is more challenging than I thought

If you do it with this setting then the tactic is to use big tilted squares but keep an eye on the computer as it is very clever!

This game is very fun it took me a long time to learn the tactics but it is really worth it.

I agree with everyone. There really should be different levels: easy, medium and hard. On the one player game, it is really hard to win

It is difficult to win at first, but having a strategy may help. Several others have suggested trying to form a T-shape with your dots so you can threaten two squares at once.

the computers algorithms seems to rely on diamond shapes for its offence it likes to make a diamond or put you in a position where it has 2 individual options to win, this is also usually using diamonds .

What is it about diamond-shapes (or rather, tilted squares) that makes using them an effective strategy for the computer?

Is there some method you could use to help you spot any tilted squares the computer may be forming?

It seems that the computer prefers to beat the player with a diamond shape. It will always attempt to have to options available one diamond, one square. It will always defend as a last resort preferring to play more aggressively. But when it does defend it is to protect its diamond or to stop the player from winning.

So it would appear that the computer relies on trying to form a square that is hard to spot, while also blocking any square that you are about to complete.

If you are going first, is there any way you could adopt this strategy for yourself? Would going first give you an advantage if both players had the same strategy?

It is too hard playing vs the computer but I like the game!

Blue doesn't always win, red can also win. The winning strategy would be making two options for making a square so the opponent doesn't know which one to choose. Also to defend if the opponent goes first put the dot diagonally from it.

if you make a line of 3 not next to the side you can win by putting a dot above the middle one , you can then win by either side. What is good about this is that if you get blocked you can just do it again on the opposite side

Blue can win, but you just have to stay attached and keep your eyes on the game! You can win no matter what strategy you use, you just need to be focused on the game. If you can't do it the first time then keep trying, don't get frustrated just keep trying and trying until you find a strategy, because everyone has a different strategy!

If this is your first try of square it, don't look for a strategy, try and work it out on your own. If you've tried it lots of times and you can't find a strategy of your own then that's the time to find one. If you have found a strategy, well done and go and find a different strategy and try that one out. Thank you for reading this and well done everyone!

Wow that's amazing for the first time I had a draw with the computer!

Have any of you had a draw with the computer?

I managed to win but it took a great deal of thought. I had to map out the possibilities of every red dot.

Blue can win every single time. You just have to try and make a T shape and then red has no choice.