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# Got It

## Got It

**Why play this game?**

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Age 7 to 14

Challenge Level

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

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.

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.

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

All the notes that follow assume that the game's default setting is a target of 23 using the numbers 1 to 4.

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.

How can I work out the 'stepping stones' that I must 'hit' on my way to the target?

Is there an efficient way of finding the first 'stepping stone'?

When is it better to go first and when is it better to let the computer go first?

If the computer says 1, I say...?

If the computer says 2, I say...?

If the computer says 3, I say...?

...

A more demanding game, requiring similar strategic thinking, is Last Biscuit.

You could demonstrate the game a few more times at the start. Alter the settings on the game to have a lower target and a shorter range of numbers (for example a target of 10 using the numbers 1 and 2). As you play, note down the running totals to refer back to later.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?