# Got It

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

## Problem

Got It is an adding game for two players. You can play against the computer or with a friend. It is a version of a well-known game called Nim.

Start with the Got It target 23.

The first player chooses a whole number from 1 to 4.

Players take turns to add a whole number from 1 to 4 to the running total.

The player who hits the target of 23 wins the game.

Play the game several times.

Can you find a winning strategy?

Can you always win?

Does your strategy depend on whether or not you go first?

To change the game, choose a new Got It target or a new range of numbers to add on.

Test out the strategy you found earlier. Does it need adapting?

Can you work out a winning strategy for any target?

Can you work out a winning strategy for any range of numbers?

Is it best to start the game? Always?

Away from the computer, challenge your friends:

One of you names the target and range and lets the other player start.**Extensions:**

Can you play without writing anything down?

Consider playing the game where a player CANNOT add the same number as that used previously by the opponent.

## Getting Started

Try to work out the 'stepping stones' that you must 'land on' on your way to the target?

Alter the settings on the game to have a lower target and a shorter range of numbers (e.g. a target of $10$ using the numbers $1$ and $2$). As you play, note down the running totals to refer back to later.

## Student Solutions

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.

## Teachers' Resources

**Why play this game?**

Got It is a motivating context in which learners can apply simple addition and subtraction. However, the real challenge here is to find a winning strategy that always works, and this involves working systematically, conjecturing, refining ideas, generalising, and using knowledge of factors and multiples.

### Possible approach

*This problem featured in an **NRICH Secondary webinar** in June 2021.*

Introduce the game to the class by inviting a volunteer to play against the computer. Do this a couple of times, giving them the option of going first or second each time (you can use the "Change settings" button to do this).

Ask the students to play the game in pairs, either at computers or on paper. Challenge them to find a strategy for beating the computer. As they play, circulate around the classroom and ask them what they think is important so far. Some might suggest that in order to win, they must be on 18. Others may have thought further back and have ideas about how they can make sure they get to 18, and therefore 23.

After a suitable length of time bring the whole class together and invite one pair to demonstrate their strategy, explaining their decisions as they go along. Use other ideas to refine the strategy.

Demonstrate how you can vary the game by choosing different targets and different ranges of numbers. Ask the students to play the game in pairs, either at computers or on paper, using settings of their own choice. Challenge them to find a winning strategy that will ensure they will always win, whatever the setting.