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Cubist Cuts

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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:

diagrams showing 6 cuts

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 $2^k$ $k \times3$