Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in
every possible order to give 5 digit numbers. Find the sum of the
Not a popular question - was it harder than I thought maybe?
Pen Areecharoenlert, Suzanne Abbott and Rachel Walker of The
Mount School, York were close to the first part, showing there are
only two essentially different "standard dice" with alternative
faces adding to 7. To quote them,
". . . if you look at the corner surrounded by 1,2 & 3 you
either count clockwise or anticlockwise . . .".
They used diagrams of the net of a cube to help them.
My solution is as follows;
so there are 2 and only 2 standard dice.
Suppose we relax the condition that opposite faces sum to 7. How
many different dice can we make now? Pen, Suzanne and Rachel
mention a school colleague in Year 12 who "wished to remain
nameless" but who told them there were 6! = 6x5x4x3x2x1 = 720
different dice. They thought it ". . . seemed a lot . . .". Quite
right, there are in fact only 30 possible dice.
I give two proofs for you to choose from;
so the number of choices is 5 x3 x 2 = 30
Write [(1,6), (2,5), (3,4)] to represent the "pairing" for the
so there are 30 dice altogether.
I stole the idea for the second proof from Chiara Colli's
solution to the Russian Cubes problem of last month. Chiara is from
the Liceo Cairoli in Vigevano, Italy.