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Age 14 to 16
Challenge Level

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.