### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Give Me Four Clues

## Give Me Four Clues

I am thinking of a number between 1 and 100. What's my number?

Look at the eight clues below.
Can you sort out the four clues that enable you to deduce my number?
Which four clues are not necessary, even though they are true?

Clues
1. The number is greater than $9$.
2. The number is not a multiple of $10$.
3. The number is a multiple of $7$.
4. The number is odd.
5. The number is not a multiple of $11$.
6. The number is less than $200$.
7. Its ones digit is larger than its tens digit.
8. Its tens digit is odd.

Challenge:
Choose your own secret number between 1 and 100.
Write exactly four clues so that your friend needs each of them to guess your number.
Try it out on your friend.

Remember, avoid clues that double-up and therefore are not needed such as both 'it's even' and 'it's in the 4 times table'. Your four clues must lead to just one number between 1 and 100.

### Why do this problem?

This problem is an accessible context in which pupils can apply their knowledge of number properties. It provides a great opportunity for learners to reason logically and to communicate their reasoning with others.

### Possible approach

Introduce the first part of the challenge to the whole group and give them time to work individually, then in pairs without saying too much.  (It might be useful to print out copies of it from this sheet.)   You could bring them together for a mini plenary after a short time, asking whether they can suggest some clues that are not needed and how they know that they are redundant.

Suggest that pairs continue to work on the problem, recording whatever and however they find useful. Let them know that you will be asking them to explain their reasoning, as opposed to simply focusing on the answer.

As you go round the room, listen out for children who are using logical reasoning to eliminate the redundant clues and to find the number.  They might well use vocabulary such as 'because' and/or 'if ... then ...'.  You could warn a few pairs that you'd like them to share what they have been saying with the whole group in due course.

Bring everyone together again to share their solution but in particular to share examples of logical reasoning that led to it.  You can then set the group off on the follow-up challenge where they could work in pairs to create a similar task for another pair with exactly four clues, none of which are superfluous.

To end the lesson, place learners in groups of four so two pairs can try out each other's new challenges and report back.

### Key questions

Which clues do you have to have to find the number?