Square It
This game is part of a set of three. We recommend you play this version before having a go at Parallelogram It and then Rhombus It.
Square It 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 display the dots on a coordinate grid, 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.
Play with this interactivity until you can draw tilted squares confidently.
You could fix one side of a square and then work out where the other corners need to go.
Well done to everyone who developed a winning strategy. Olivia from McCauley Catholic High School in England dscribed an idea which most strategies are built on:
All I did was set up another square that I could go to if my first one didn't work.
Rishik had another general tip:
If given a choice a player should try to begin first as it not only increases the success rate by opening many possibilities but does not let the player became obstructed by disruptions. The first player should choose to place their dot on in the centre of their grid as this allows many chances for the player to win.
Tushar from NPS Rajajinagar in India, Finlay form Colyton Grammar School in England, Sydney from Acera, Aaron from the UK and Marco from International School of Lausanne in Switzerland, Nathan and Gabriel from The British School Al Kubairat in UAE and Rishik all described the same strategy. This is Tushar's work:
This is one of the winning strategies that I use. The main goal of this strategy is to get such a position that there are 2 possible squares that you can make. The other player can only stop you from making one square. After the opponent makes their move, you simply connect the other one. This strategy shows the quickest way to get that position. This strategy works if you are player 1 (triangle). The position that you are trying to get is this:
If you get this kind of position, you win. It doesn’t always have to be pointed upwards, it can be in any direction.
Rishik explained how to get to this point, and some other students from International School of Lausanne in Switzerland decided that the game was won before they had finished making the full shape. Vince wrote:
The blue triangles have already won at this point because the red squares see that the triangles are almost done with their square so red tries to block blue. Unfortunately for red it was too late because there are two little tricks that blue can do.
Trick 1 (right)
The first and most popular trick is to put a triangle in the middle of the three other triangles. This creates something where blue has two options to win. And when red blocks one of them, blue can use the other one to win.
Trick 2 (left)
The second but my personal favourite trick is to add a triangle in the close top right corner to make the same thing as before but just in a different format. In this case, the red player will have to pick a way to lose.
Can you see how closely related these tricks are? Charles, Mark and Yihang from Colyton Grammar and Issa M and William from The British School Al Kubairat described the technique in trick 2. Issa wrote:
First you must place a triangle in the middle of the grid. Then you must make a diagonal line starting from the top right to the bottom left of the middle triangle. Then place another, one space from the bottom left of the middle triangle. This will open up possibilities to make a diamond while the computer will still think that you will make a normal square and will block that.
Lottie from Colyton Grammar School explained how tricks 1 and 2 are the same by describing the strategy that links them:
Place three squares, one on top of the other, to make a straight line. This can be horizantally, diagonally or vertically. Then place another symbol adjacent to the square in the middle of the existing line, this can be on the left or right, it does not affect the method. If your opponent tries to block you by adding a square adjacent to the top square of the line, on the same side that you added a square on to the middle, you can still win in your next go as you have two possible squares you could make. You could place a square next to the bottom square, again on the same side that the other squares have been places. This is the same vise versa. This particular method ensures that you will make a square before your opponent and win.
Rishik explained that there are even more ways to create this shape - you don't actually need touching points - and also described another winning shape:
The player should always try to form two formations: the arrow and parallelogram.
Arrow: 
Although I have chosen to demonstrate these 2 formations with 3 by 2 grids, they can be formed with any size, by scaling these formations up or rotating or reflecting them in any way on the grid. There are many ways, but these two formations are the best way to secure victory as these create 2 possibilities for you to win and your opponent can only place one counter, thus letting you make a
square by completing the other.
Can you see why you have won if you create a "parallelogram"?
Dylan from Chesterton Community College in the UK had a strategy which involved responding to the computer, or opponent:
The method I found does not work on a 3 by 3 grid because the grid is too small. It works on any larger grid. To start you put a dot in the centre of an odd sized grid, or on one of the four centre points of an even sized grid. Then where ever the computer places its dot around your dot, your next dot should be adjacent to that dot on a horizontal or vertical line from your 1st dot. Next you place one above the computer’s next dot. Your 4th dot should be on a diagonal line with your 3rd dot and a horizontal or vertical line with your first dot. This will give you a winning position as you can make two different squares with your next dot.
Lewis from Colyton Grammar School and Millie from Thomas Gainsborough School in the UK used yet another strategy. This is Millie's work:
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.
Possible approach
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.
Key questions
- 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?
Possible support
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.
Possible extension
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.