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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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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!]

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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.

Iffy Logic

Stage: 4 Short Challenge Level: Challenge Level:1

Why do this problem ?

Learning to write mathematical statements clearly and simply is one of the most important skills that a mathematician needs to learn. In order to do this, a mathematician needs to have a clear understanding of the logical flow 'IF something is true, THEN something else is true'. This problem gives practice in this process. It could be accessed at any point from advanced stage 3 to stage 5 and will really help to sharpen up the students' written mathematics.

Possible approach

The interactivity lends itself to a group approach. It is important that students are given the opportunity to talk about their logical statements and to try to explain their answers verbally. To do this, you might ask someone to suggest an answer to a line in the card-sorter and then ask for feedback on whether others agree or disagree. There are multiple possible ways in which certain cards can be matched.

Key questions

  • Which cards have a good chance of fitting with other cards?
  • Can you explain why your logic holds for each correct answers?

Possible extension

The cards have been chosen so that the completed statements are 'obviously' true. Good students might be encouraged to think more about WHY the statements are true. Can they give a strong argument or proof?

Extension work of a similar type is provided in the question Contrary Logic. Activities extending the 'proof' theme are given in Direct Logic.

Possible support

Rather than attempt to fill in all of the boxes, students could try to make just a selection of true statements.

You might start by discussing these statements:

If my team wins the world cup tomorrow then I'll be happy tomorrow.

If I am happy tomorrow then my team will win the world cup tomorrow.

If I am not happy tomorrow then my team will not win the world cup tomorrow.

If my team does not win the world cup tomorrow then I will not be happy tomorrow.



If this is maize then it grew from a seed


If this grew from a seed then it is maize


If this did not grow from a seed then it is not maize


If this is not maize then it did not grow from a seed


If Rover is a dog then Rover is an animal

If Rover is not an animal then Rover is not a dog

If Rover is not a dog then Rover is an animal

If Rover is an animal then Rover is a dog