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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.

Iffy Logic

Stage: 4 and 5 Challenge Level: Challenge Level:1

Mathematical logic and thinking are grounded in a clear understanding of how the truths of various mathematical statements are linked together.

For example, for any number $x$ the expressions $x> 1$ and $x^2> 1$ are both mathematical statements which might be true or might be false. However, we always know that $x^2> 1$ IF $x> 1$, whereas it is not always the case that $x> 1$ IF $x^2> 1$ (consider $x=-2$, for example). Thus:

It is correct to write $\quad\quad x^2> 1$ IF $x> 1$

It is incorrect to write $\quad\quad x> 1$ IF $x^2> 1$

Test out your logical thinking with these statements where n and m are positive integers (full screen version ).

This text is usually replaced by the Flash movie.

You can view and print out the cards here.

Are there multiple solutions? If not, how do you know?

How would the logic change if $n$ and $m$ were not necessarily positive or not necessarily integers?

Extension: Note that this activity does not prove that the statements are true. How might you go about proving that certain combinations are correct? How might you go about proving that certain combinations are incorrect?