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# Have You Got It?

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Adhvik from Inventure Academy in India (buried under other stuff), Daniel from Wordsley in the UK, Miss Kelly's 7a3 students from Acklam Grange School in the UK, Aayan from Robert Gordon's College in the UK all noticed a number that helps you win. Aayan wrote:

First player to get 18 has the chance to win.

Daniel said this in a different way:

In order to win you would have to get the person to get to 19

Shaunak from Ganit Manthan, VicharVatika in India explained why:

I played several times with the computer and observed that it was defeating me when it achieved $18.$ I realised this would always work because $23 - 5 = 18,$ and if I play any number from $1$ to $4$ called $x$, the computer subtracts $x$ from $5.$

Dylan from Forest School in the UK, Ci Hui from Queensland Academy of Science Mathematics and Technology (with diagram), Miss Kelly's 7a3 students and Shaunak extended this to a more complete strategy. Miss Kelly and her students wrote:

You always need to get the number 18 before your opponent.

Look at opponents last score.

Work backwards.

If both opponents know to get 18, if they get to 13 before you then they

have won, as no matter what numbers you use, they can still make 18.

For example, 3, 4 (7), 3 (10), 3 (13)..... 1(14), 4 (18), 1 (19), 4 (23).

Dylan wrote:

I realised that I needed to get the AI to play on 18, 13, and 8 so I realised that if I started at 3 then then AI would to play something to get me to 8 and then I coukd win from there.

Ci Hui used a diagram to explain the idea (click on the image to open a larger version):

Adhvik, Ci Hui, Miss Kelly's 13 year old students from Acklam Grange School and Shaunak explained how their strategies could be adapted for other sets of numbers. Click here to see Shaunak's full video solution on YouTube.

This is Adhvik's work (click on the image to open a larger version):

Miss Kelly and her 13 year old students wrote something similar, but their strategy starts from the very first turn:

We can always win, if you start first or if your opponent does not know the trick.

If we add the biggest and smallest number (from the list of numbers in which we can add), we subtract this from the target and continue subtracting until we get to the smallest number.

For example, if we can add the numbers 1, 2, 3, 4, 5 and the target is 29. The first step is 1 + 5 = 6, we then subtract 6 from 29 to get 23, and continue until we get to the lowest we possibly can...... 29, 23, 17, 11, 5. We need to get to all of these numbers (which are our milestones), for us to definitely win.

This works for any set of numbers and targets.

Ci Hui showed some more examples and used another diamgram to explain why this method works for finding winning strategies (click on the image to open a larger version):