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### Number and algebra

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### Advanced mathematics

# Seeing Squares

## Seeing Squares

Why play this game?

### Possible approach

Key questions

### Possible support

Possible extension

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Age 5 to 11

Challenge Level

- Problem
- Getting Started
- Teachers' Resources

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.*

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.

*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?

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.