### All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

### Painting Cubes

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?

### Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

# Cubist Cuts

##### Stage: 3 Challenge Level:

Jake, Ryo and Charlie from Moorfield Junior School have explained how to cut up a 3x3x3 cube using 6 cuts:

"We got a 3 by 3 cube and then we cut it 2 times to make 3 lots of 9 cubes. Then we piled all the cubes on top of each other. Then we took another 2 cuts to leave 9 towers of 3 cubes. Next we layed them next to each other. After that we took another 2 cuts to leave the 27 unit cubes."

Chris B, Elliot and Joseph, also from Moorfield Juniors, sent us a diagram to show where these cuts should be:

Juliette noticed that it wouldn't be possible with fewer than 6 cuts:

"We need at least 6 cuts because we need one cut for each face of the small cube in the middle of the $3\times 3 \times 3$ cube."

Anthony noticed that, with a $4 \times 4 \times 4$ cube, we can use 6 cuts if we rearrange the cubes:

"First cut the cube in half down the middle, then stack the halves on top of each other (in an $8 \times 2 \times 4$ arrangement) and cut down the middle, to make four $4 \times 4$ slices each 1 unit thick. Then rearrange the cubes into the original arrangement and repeat the process in the other two directions. This will cut the cube into $1 \times 1 \times 1$ cubes. It cannot be done with fewer than 6 cuts because the cubes in the middle will each need at least one cut for each face"

The $n \times n \times n$ cube is a bit trickier. Try a few yourself before looking at this explanation.

First of all let's see how many cuts are needed to cut the cube into slices 1 unit deep. We can then do this in each of the three directions to cut the $n \times n \times n$ cube into unit cubes, and can multiply by three to find out how many cuts are needed in total.

For a cube with side length 3 or 4 units, we need 2 cuts, as Juliette and Anthony explained. For 5 units, we'll need an extra cut in each direction. To cut as efficiently as possible, we should use a method similar to Anthony's: first cut in half (or as close to in half as possible), then stack up the "halves" and repeat until we are left with "slices" 1 unit thick. We can then put the cube back together and repeat for the other two directions.

The general pattern is: for each doubling of $n$, we need an extra 3 cuts to cut an $n \times n \times n$ cube into $1 \times 1 \times 1$ cubes. This is shown in the table below.

 $n$ Number of cuts $1$ to $2$ $1 \times3$ $3$ to $4$ $2 \times3$ $5$ to $8$ $3 \times3$ $9$ to $16$ $4 \times3$ $(2^{k-1} + 1)$ to $2^k$ $k \times3$