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Platonic Planet

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?


Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Cubic Conundrum

Which of the following cubes can be made from these nets?

Chopped Dice

Age 14 to 16
Challenge Level

Alison and Steve wish to make a new sort of die which can land on its faces and its corners. 
They plan to start with a cube and make planar cuts across the corners to create a solid which, when rolled, has a good chance of landing on the 'corners' and the 'faces'
Consider designing such a die such that it is symmetrical -- i.e. each corner is to be cut off in the same way.
How many faces would the die have and what shape would they each be?
Draw a net of a cube and indicate accurately the lines along which the cuts are to be made.
Where would you align the cuts such that each of the new faces was of the same area?
Collaboration/cross curricular activity: Suppose that we wish to make a physical die of this sort such that there are equal probabilities of landing on each  of the 'faces' and 'corners'. Discuss with DT the possible construction of such a die and plan a way of producing the die.