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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.

Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

You do not need to be able to play chess to solve this problem.

The standard move for a knight on a chess board is $2$ steps in one direction and one step in the other direction. A knight's tour is a sequence of moves in which the knight visits every square on the board once and only once and a circuit is a tour in which the knight returns to the starting point.

Prove that a knight cannot make a tour on a $2$ by $n$ board for any value of $n$.

How many different tours can you find on a $3$ by $4$ rectangular board?

4x3 table with letters a to l

Prove that a knight cannot make a circuit on a $3$ by $4$ rectangular board.