### 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?

# A Long Time at the Till

##### Stage: 4 and 5 Challenge Level:

This problem involves considering, comparing and assessing different ways to solve a very difficult 'background' problem. There are three different parts to this unusual and thought-provoking task.

Background problem
Note: solving the background problem is very involved and not the main focus of this task!
A mathematician goes into a supermarket and buys four items. It has been a while since she has used a calculator and she multiplies the cost (in pounds, using the decimal point for the pence) instead of adding them. At the checkout she says, "So that's £7.11" and the checkout man, correctly adding the items, agrees.

The mathematician very, very slowly puts the items into her bag whilst thinking and tapping away on her calculator. She eventually says "I believe that the prices of four items with this property is unique".

Spend a few minutes trying this problem yourself to get a feel for its mathematical structure.

Please note: Although it involves only the basic properties of numbers, the background problem is very difficult and time-consuming to solve directly. Now there's a challenge ....

Main problem
Read carefully the two solutions provided in the hints tab. How do your attempts at the first part compare to, or differ from, these two solutions? Which of the two solutions do you prefer? Why?