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Card Trick 2

Can you explain how this card trick works?

Cube Paths

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

How Many Dice?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?

Painting Cubes

Age 11 to 14
Challenge Level

30 different painted cubes.

Let the six faces be painted a, b, c, d, e, and f.

With face a opposite face b there are six arrangements for the other four colours around the cube: cdef, cdfe, cedf, cefd, cfde and cfed.

Likewise for the face a opposite face c; face a opposite face d; face a opposite face e; and face a opposite face f. All have six arrangements for the remaining four colours.

Hence the total is 5 x 6 = 30 arrangements.