Representation - October 2008, Stage 3&4

Problems

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Crossed Ends

Age 11 to 14 Challenge Level:

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

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Legs Eleven

Age 11 to 14 Challenge Level:

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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Domino Square

Age 7 to 16 Challenge Level:

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

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Differences

Age 11 to 14 Challenge Level:

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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Tet-trouble

Age 14 to 16 Challenge Level:

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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Hexy-metry

Age 14 to 16 Challenge Level:

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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Napoleon's Theorem

Age 14 to 18 Challenge Level:

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?