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

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!]

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.

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.