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# Square It

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Age 11 to 16

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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: Parallelogram:

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:

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?