# Seeing Squares

*This game is part of a set of three. We recommend you play this version before having a go at Seeing Parallelograms and then Seeing Rhombuses.*

*Seeing Squares printable sheet**Printable dotted grid*

This game can be played against a friend or against the computer.

Players take it in turns to click on a dot on the grid - the first player will place blue triangles and the second player will place pink squares.

The winner is the first to have chosen four dots that can be joined to form a square.

Squares can be anywhere and any size.

Clicking on the purple settings cog allows you to select the size of the grid, who the players are, and who goes first.

Once you've played a few times against a friend, you might like to discuss your strategies, and then test them by playing against the computer.

**Can you find a winning strategy?***If you are not using the interactive game, you may like to print off some dotty paper.*

*You may be interested in the other problems in our Strategy Games Feature.*

Play with this interactivity until you can draw squares confidently.

Can you draw tilted squares, when the sides are not horizontal and vertical?

You could fix one side of a square and then work out where the other corners need to go.

Thank you to everybody who shared their ideas about this game with us. Amélie from St Nicholas CE Primary in the UK said:

I try to make it so there are two possible squares I can make.

Good idea, Amélie - often when the computer wins, it wins because it has two possibilities for making a square and you can't stop both of them!

Luca from Village School in the USA said:

Try to make a T-shape so you can fork the other person. Also try to start in the middle.

That 'T-shape' idea came up a lot in these solutions. If you choose to watch two computer players, you might notice one of them making that 'T-shape' a couple of moves before they win. Dhruv from The Glasgow Academy in the UK drew this shape in their solution, and agreed that the middle was a good place to start:

Good ideas, Dhruv! From your pictures, I wonder if you are imagining the diamond as being a tilted square or a rhombus? We talk about 'diamond' shapes a lot but this isn't actually a mathematical shape name - some 'diamonds' are squares and some are rhombuses or kites.

Lachlan gave some step-by-step instructions for how to win:

Step 1 = Place a triangle one above the centre dot.

Step 2 = If the computer doesn't place the square 2 dots below, you place it there.

Step 3 = Place a triangle 2 dots along from the first triangle you placed.

Step 4 = Place a triangle on the other side of the first triangle.

Step 5 = After that you can make a square.

These are very clear instructions, Lachlan, and I can see the same 'T-shape' in your strategy. I wonder what you would do differently in Step 2 if the computer does choose the dot two below your dot?

Om described a few different possible strategies, some with the same 'T-shape':

Good ideas, Om! I wonder if the first player can always win using one of these strategies?

Will from Modbury School explained how to make the 'T-shape':

Make a triangle with a hollow middle, then fill the middle dot in the top line and wait for the computer to make its decision.

I think you mean like this:

Good idea, Will - you can make this 'T-shape' easily from here. I think even if the middle dot has already been taken by the computer, you can win in two moves from here. Can you see how this can become the same as Om's first or second strategy, depending on which other dots are still available to take?

Why play this game?

This game will help pupils deepen their understanding of the properties of squares. It is also a useful context in which to challenge pupils who have got used to calling tilted squares 'diamonds'. As they play the game, learners will have plenty of opportunities to visualise squares and to develop winning strategies for beating an opponent.

### Possible approach

*This game featured in an NRICH Primary webinar in November 2021, although the interactivity has been updated since. This game also featured in the NRICH Primary and Secondary webinar in April 2023.*

Introduce the game on the interactive whiteboard, either with you playing against a pair of pupils, or the whole class playing against the computer. Play the game a few times and when the winning square is 'tilted', use this opportunity to talk about what makes a square a square, and encourage learners to explain how they know that the shape drawn is indeed a square.

*The Settings menu (purple cog) offers the chance to have different sized grids, and coordinate axes if you prefer.*

Once everyone has understood how to play, encourage learners to play in pairs against another pair, ideally on tablets or computers using the interactivity. If this is not possible, learners could use dotty paper, or sets of differently coloured counters with a printed (and laminated) copy of this grid.

Allow time for learners to play several games and as they play, listen out for those pairs who are thinking strategically. They may, for example, be looking at where the best place to start could be. They may be thinking ahead and considering consequences of their possible moves. They may be focusing on what their opponents are doing too, trying to block them where possible. Can learners set up a situation in which they can create two different squares on the next turn and ensure a win?

*If they are using the interactivity, learners might like to use the 'Game report' to help them look back on the game and analyse possible alternative moves.*

You could encourage learners to share their various strategies in a mini-plenary, then encourage everyone to try to make use of what they have heard as they play more games.

The final plenary might involve the whole class playing against the computer, putting into practice the strategies that have been discussed. If a game is lost, you could use the 'Game report' to help the class analyse what went wrong and what they could have done differently.

Key questions

Where is a good place to start? Why?

How are you deciding where to go?

How do you know this is a square?

How could you set up a situation in which you can create two different squares on your next turn?

Can you beat the computer if the computer goes first?

### Possible support

It may be worthwhile looking at Complete the Square before playing this game.

The game can be built up gradually from a 25 dot board and a 36 dot board to the 49 dot board in the question.

Some learners might find 'believing' in the tilted squares difficult. On paper they could use the corner of a piece of paper, for example, to convince themselves that the angles in a shape are $90^\circ$. Alternatively, they could be encouraged to cut the shapes out and move them around to see if the cut-out really looks square.

Possible extension

These three activities can be useful follow-up to this game: Square Corners, Seeing Parallelograms and Seeing Rhombuses.