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Rotating Triangle

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

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A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

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Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

A Long Time at the Till

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?
Follow up task 
If you were now to be given related problems with £7.11 replaced by £7.12 or £7.13 or £7.14 how would you now choose to proceed? Can you assess in advance which of these problems will probably be harder or easier? Can you efficiently solve any of these problems with the benefit of hindsight?

 This task  was inspired by a problem from "Sums for Smart Kids" by Laurie Buxton, published by BEAM Education