Copyright © University of Cambridge. All rights reserved.
Andre's (Tudor Vianu School) started of
by giving his thoughts on the problem. Many thanks for this Andre -
I think it is an excellent attempt to unpick what is
happening.
For a 2 x 2
Cube
First, I analysed a 2x2x2 cube. After dipping it into the pot
of yellow paint, each cube has three faces painted yellow and three
faces painted red (the original colour). So, the other three faces
of the cube remain red. Reversing each cube (so that each exterior
vertex goes into the centre of the big cube) all the red faces
remain at the exterior. After dipping the cube into the green pot,
the rest of the face that remained red are now coloured green. So,
it is sufficient to use two colours.
For a 3 x 3 Cube
Now, to analyse further, for increasing numbers of small cubes, I
will create a table, with columns indicating: the side of the big
cube, the total numbers of small cubes, the total number of faces,
the number of faces at the exterior, and the number of colours that
should be used (in principle, obtained as a division of the total
number of faces by the number of exterior faces):
| Side of cube |
Small cubes |
Faces (total) |
Faces (exterior) |
Colours |
| 2 |
8 |
6 x 8 = 48 |
6 x 4 = 24 |
2 |
| 3 |
27 |
6x 27 = 162 |
6 x 9 = 54 |
3 |
| 4 |
64 |
6 x 64 = 384 |
6 x 16 = 96 |
4 |
|
|
|
|
|
| n |
n x n x n |
6 x n x n x n |
6 x n x n |
n |
Now, I analysed the situation with 3x3x3 cubes. I created 27
cubes from paper in order to analyse the situation experimentally.
I see that in order to make possible to colour all the faces using
3 colours, each face must be coloured only once. After trying
several times I observed that this is impossible, because the
number of cubes with different numbers of coloured faces is very
different:
After the first dipping:
- there are 8 cubes with three coloured faces (vertices)
- there are 12 cubes with 2 coloured faces (centres of
edges)
- there are 6 cubes with 1 coloured 1 face (centres of
faces)
- there is 1 cube with no coloured faces (centre of big
cube)
I used in my solutions the following considerations:-
- the cube from the centre must go into one vertex, to maximise
its coloured faces in the following step
- one cube from one vertex must go into the centre of the big
cube
- the other cubes from the vertices must go into the centres of
the faces (6), and one in a centre of the edges
- the cubes from the centres of the faces must go into the
vertices
The third of these assumptions was
incorrect. Someone else then completed the solution:
For the 3x3x3 cube, 2 of the corner slots and the middle slot can
be used to swap 3 cubes around and colour then in the three goes.
The rest of the 6 corner pieces are moved to 6 of the edge pieces,
the 6 edges pieces are moved to the 6 centre slots and the 6 centre
pieces are moved to the corners. Doing this after the second
dipping as well will cover all 18 of these pieces in the three
goes. Finally, the remaining 6 edge pieces can be left where they
are, and simply rotated each time so that after the three dippings
they are all covered. Thus a 3x3x3 cube can be totally
coated.
For an n x n x n cube (if n is 6 or greater, or is 4), can be
coloured as follows:
- We start of with 8 corner pieces, 12(n-2) edge pieces, 6(n-2)^2
centres, and (n-3)^3 middles
- 4n blocks are looped around spending 2 dips in the corners, and
(n-2) dips in the middles. This leaves 12(n-2) edges, 6(n-2)^2
centres and n(n-2)(n-4) middles.
- 6n(n-2) gaps are kept for blocks spending 2 times at the edge,
2 times at the centre, and (n-4) times in the middle
If n=4 then all the blocks are covered. If n is greater than or
equal to 6, then there are still some edge and middle pieces we
haven't describe. Blocks loop around these n(n-2)(n-4) slots,
spending 6 times at the centres, and (n-6) times at the edges.
The 'missing' case is n=5. Here instead of the second point we
would use 52 of the centre slots and 13 of the middle slots to move
around a total of 65 cubes in to cover them in the five dips. We
would then use 2 of the edge slots, two of the centre slots, and
one middle to colour 5 more blocks in the 5 dips. Finally we would
use the 21 edge slots and 14 middle slots to cover a total of 35
blocks using a total of 5 dips (so each block spends two 3 times at
an edge and 2 times in the middle).
So an n*n*n block can be covered in n dips of paint as
expected.