We had some great ideas sent in for this task, so thank you to everybody who sent us their solutions.

Lots of children noticed that if there are three counters left at the beginning of your go, you can't win. Josh, Arthur and Mia from Kirkby on Bain in the UK said:

We have noticed that when there are three counters left no matter what the next player takes, the other player will win. This is because there will be one left if two are taken, and two left if one is taken. So the other player will win!

Josh, Arthur and Mia then explained how the first player could win:

Player 1 should take one counter as their first go. This means they will get to leave three counters after their second go and then win. We are going to try with 9 counters!

Thank you for sending in these ideas. I wonder what the best strategy for winning is with nine counters?

We also had lots of ideas sent in from the children at St Charles Primary School Ryde in Australia. Samuel suggested that if there are seven counters left, you should take one counter so that there are six left. Can you definitely win if you leave six counters for the other player? Why?

The children from Olga Primary School in the UK sent in lots of videos explaining their ideas. Francis and Caitlin explained why leaving three counters for the other player is a winning strategy:

Ben and Ayaan explained why going first means you can always win:

Thank you all for sending in your thoughts about this game!

Thank you as well to Eloan and Ella from Clifton College Prep School in England, Joshua from Spring Hill Primary School in Australia, the children from Waverley Primary School in the UK, and Dhruv from Pict in India, who all had similar strategies.

Students at Clifton Hill Primary in Australia worked hard to beat the computer at Got It with a target of 23, and with numbers 1, 2, 3 and 4 to choose from.

J and C said:

You can win the game with a target number of 23 and range numbers of 1-4 by trying as hard as possible to land on the number 18. By landing on 18, you make your opponent off by 1 of getting the end number. If you get the number 18, your opponent will be stuck and you have a guaranteed win.

Danica and Eve took this a little bit further:

Can you find a winning strategy?

Yes, you can by getting to 13 then the other player has to say their number (1-4) you can get to 18.

After that, no matter what number they say there will always be a way for you to get to 23.

Leon and Jack built on this even further:

There are three key numbers:

8

13

18

Firstly, you have to be the one that gets the total to eight. You can do this by starting with 3 and the opponent has to let you get to 8. Once you have accomplished that, it is their turn with a total of 8. They then can't stop you from getting to 13. After you get 13 they can't stop you from getting to 18. After this, they can't stop you from reaching 23.

So, in fact rather than there being three ‘key numbers’, it sounds like you’re saying there are four key numbers: 3, 8, 13, 18.

Golden Eagles Class at Ditchingham Primary Academy described their strategy in a similar way:

We found that it was easier to win once we had found the stepping stones for the target number.  This was a key number that if we reached we could win any game.

As we could add up to 4 each time the difference between each number was 5. The stepping stones were 3, 8, 13, 18 for 23.

For example, if we started with 3, our opponent could only make it up to 7, we could then add 1 to make 8. The most they could then make is 12 but with all variation of numbers we could then make 13 and so on up to 23.

When we played the computer using this strategy and we went first we always won, however if the computer started we always lost. It was possible to beat an opponent if they went first, but only if they hadn't worked the stepping stones out. Once they had we always drew.

We tried using 17 as a target number and found stepping stones at 2, 7, 12 using the numbers 1-4. This was again as the difference between these numbers was 5.

We would like to try dropping the 4, using numbers 1-3 and seeing if the stepping stones reduced to a difference of 4.

I like the fact that you are making a conjecture about how you could apply your strategy to a version of the game using only numbers 1, 2 and 3, Golden Eagles Class.  This is exactly what mathematicians do!

This idea of ‘key numbers’ of ‘stepping stones’ is crucial in this game. Mateusz and Joseph from St Ann’s Primary agreed with this approach, so did Amelie and Sam from St Francis of Assisi Primary School; Gabriel from Yavneh College; Alaric from Haydon Wick Primary; Harry from St Michael’s Preparatory School; Ruby, Alice, Meiyla, Lilly, Olivia and Sophie from Anston Greenland’s Primary; Joy, Akosua, Asuo and Kumi from Torrens Valley Christian School in Australia; as well as three groups of students from Ganit Kreeda, Vicharvatika, India.

Anirudh from Hamilton Primary School shared this video, in which he explains how he worked backwards to find the stepping stones:

And in this video, Kaana from Ganit Kreeda explains the importance of multiples of 5:

Can you see how Anirudh’s and Kaana’s solutions are connected?

Joseph from Jesmond Park Academy sent a very comprehensive solution:

I initially approached this problem by just playing some games with the computer and tried to recognise some patterns. I started by playing first but then swapped to the computer playing first. It was here that I noticed that it would always start with the same first number depending on the target number.

I noted this down and also noted that it would try to reach certain values consistently. I called these 'landing values' and wrote them down.

From here, I noticed that there was a fixed distance between each landing value that was 1 greater than the largest number I could choose. This revelation led to further investigations that led to the process below.

For anyone who wanted to consider solving this problem, or ones like it, I think the best first step is to just play around and consider different values. After you've considered different values, you can assess whether there is a pattern or not. This pattern is the real starting off point for finding a general solution.

Thank you, Joseph, what a lovely description of the process you went through. We agree that ‘playing around’ with a new problem is a great way to start. The exploration then leads to some noticing, as Joseph said, and then some more systematic working, some generalising and even proof.

Here is Joseph’s general strategy for winning at Got It:

The numbers that are chosen to be added up range from 1 -> n, where n is a positive integer and the spaces between 1 and "n" are integers. (In fact I wonder whether they have to be consecutive numbers, Joseph...)

For example, "1, 2, 3, 4" where n is 4.

The player will have a target number, which we will call "m". To begin, player 1 will write down the target and subtract (n+1) away from the number "m" until there is a number that they are able to choose, we will call this "b".

For example, consider the target is 22 (so m=22) and the player can add up to 4 (so n=4), they will subtract 5 (n+1) until they reach the number 2 (so b=2).

Player 1 will choose this number and begin the game. From this point forward, it is Player 1's objective to reach the next multiple of (n+1) that starts from "b". In our example, these "Landing Values" would be 7, 12, 17 and finally, 22. Player 1 will always be able to force a Landing Value on their turn assuming they started from "b" and their opponent plays perfectly.

If player 1 aims for these numbers, Player 2's choice will not matter as they will end up unable to reach the target number.

The exception to this trend comes if the target number is a multiple of (n+1). This is because you can't keep subtracting (n+1) to reach the initial number "b" as you will eventually reach (n+1) and the next number would be 0.

For example, if we could choose to add up numbers between 1 and 4, if the target was 10, we could subtract 5 (n+1) once.

Following this, subtracting 5 again would reach 0, which isn't an option with our initial conditions.

Thus, the first player would have to choose any number between 1 and 4.

Assuming that there is a perfect opponent, they would reach the multiple of (n+1) instead, in this example it would be 5. The game would continue as stated previously, resulting in player 2 winning instead.

Thus, you can reach the conclusion that it is only sensible to be Player 1 if the target number is not a multiple of (n+1). If it is, then you should choose to be Player 2.

Abishai expressed a general solution in a very similar way, so did Xavier, and so did students from Ganit Kreeda, India. Thank you to you all. How fantastic that we have used our mathematics to find a winning strategy that will ALWAYS work when playing Got It.

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:

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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:

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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

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.

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Trick 2 

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.

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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:   

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                    Parallelogram:   

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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:

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