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Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

### Counting Factors

Is there an efficient way to work out how many factors a large number has?

### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

# Have You Got It?

### Why play this game?

Have You Got It? offers an engaging context which requires students to use simple addition and subtraction to find a winning strategy. The challenge involves working systematically and strategically, conjecturing, refining ideas, generalising, and using knowledge of factors and multiples.

### Possible approach

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

All the notes that follow assume that the game's default setting has a target of $23$ and uses 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 "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.

### Key questions

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...?
...

### Possible support

You can 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.

### Possible extension

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