### Why do this problem?

One way to encourage students to be more curious about the mathematical world is to present them with challenges or puzzles to be solved, or intriguing contexts that require explanation.

The challenges in this problem offer a starting point to explore properties of numbers, and provides a good introduction to the ideas developed in the

Factors and Multiples Puzzle and in

Charlie's Delightful Machine.

### Possible approach

You could begin the lesson by dividing the board into two columns, one headed with a tick and the other headed with a cross.

Ask students to suggest numbers, and write each suggestion in the appropriate column according to a rule of your own choice. Possible rules could be prime numbers, odd numbers, multiples of 4...

Once a few students think they know what your rule might be, invite them to make guesses that will help the rest of the class guess the rule too.

"Some friends played a game like this with two-digit numbers, and they used these four rules:

- Multiples of five
- Triangular numbers
- Even, but not multiples of four
- Multiples of three but not multiples of nine

Choose any two of the rules, and see if you can find some two-digit numbers that satisfy both of them."

Give students a little bit of time to find some examples, and then extend the problem...

"Now choose one of the other rules. Do any of your numbers satisfy that one too?"

Again, give students some thinking time. Then set the final challenge:

"Can you find a two-digit number that satisfies all four of the statements?"

Again, give students some time to work on this, then finish by bringing the class together to discuss how they know it's not possible to satisfy all four rules using two-digit numbers.

You could follow this up by asking students to consider how they would describe the numbers that satisfy pairs of statements:

"How could you describe the pattern of the numbers that satisfy both Alison's and Sam's statements?

How about the numbers that satisfy both Alison's and Matt's statements?

Can you describe patterns for other pairs of statements?"

### Key questions

Which rules have lots of numbers that satisfy them?

Which rules don't have very many numbers that satisfy them?

Where is it best to start if you're looking for numbers that satisfy more than one rule?

### Possible extension

Invite students to relax the two-digit number restriction - can they satisfy all four rules now?

Charlie's Delightful Machine would follow on nicely from this task.

### Possible support

One of Thirty-six could be a good way to get students thinking about systematic methods of recording. You could print off some number grids from our

Printable Resources Page.