Challenge Level

Why play this game?

This game offers an excellent opportunity to practise visualising squares and angles on grids, and encourages students to develop winning strategies for beating an opponent. Describing strategies to others is always a good way to focus and clarify mathematical thought.

Working with tilted squares provides an opportunity to examine the properties of gradients of parallel and perpendicular lines. This can lead on to Square Coordinates and Opposite Vertices, and Tilted Squares for students who are going on to work on Pythagoras'
Theorem.

*This game featured in the NRICH Primary and Secondary webinar in April 2023.*

Start with a demonstration playing against a student rather than against the computer. Students are often surprised when the winning square isn't aligned with the grid. This leads to discussions about what makes a square a square.

After a demonstration of the game, students could be left to play for a while in pairs, ideally on tablets or computers using the interactivity. If this is not possible, students could use this paper grid.

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

Bring the class together for a discussion of their thoughts on the game. Did anyone consistently win or lose? Can anyone think of any good strategies which might help them win? Are they able to ensure a win by setting up a situation in which they can create two different squares on the next turn?

Once ideas have been shared the group can return to playing in pairs, ideally against the computer. Encourage each student to explain the reasoning behind the moves they suggest.

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

One aspect of developing a winning strategy that could be considered is the number of distinctly different starting points ($6$ on a $5 \times 5$ board) and the number of different squares that can be drawn that include each of those points. This can help students decide on a good place to start.

The final plenary might involve the whole class playing against the computer, putting into practice the strategies that have been discussed.

- Why did you make that move?
- Why do you think the computer made that move?
- How do you know a square 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?

Students could experiment making different squares using the interactivity in Square Coordinates and then have a go at Square Corners.

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. Students could be asked to draw examples of all the different possible squares on their specific board size, and to compare notes to check for wrong or omitted solutions.

Some students might find 'believing' in the tilted squares difficult. On paper they could use the corner of a piece of paper or a set square, 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.

Suitable follow up problems involving generalising about squares are Square Coordinates and Tilted Squares. Alternatively, students could move on to other quadrilaterals by working on Parallelogram It, Rhombus It and Opposite Vertices.