### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

# Why 24?

##### Stage: 4 Challenge Level:

This problem is in two parts. The first part provides some building blocks which will help you to solve the final challenge. These can be attempted in any order. Of course, you are welcome to go straight to the Final Challenge without looking at the building blocks!

Click on any of the questions below to get started.

 Question A Choose any whole number. What happens when you multiply the numbers either side of it? For example, if you choose $7$, work out $6 \times 8$. Repeat several times. Notice anything interesting? Convince yourself it always happens. Question B Write down three consecutive numbers, none of which is a multiple of $3$. If you can't, explain why. Question C Choose two factors of $120$ which are coprime (they have a highest common factor of $1$). Multiply them together and record the result. Repeat several times. Notice anything about your results? Start with numbers other than $120$. Does the same thing always happen? Convince yourself. Question D Choose any two consecutive even numbers. Multiply them together and record the result. Repeat several times. Notice anything interesting? Convince yourself it always happens. FINAL CHALLENGE Take any prime number greater than $3$, square it and subtract one. Repeat several times. Notice anything interesting? Convince yourself it always happens.