Did you solve it? Cheese cube nibbles

Earlier today I set you the following four puzzles:

1. You have a cube of cheese that measures 3 x 3 x 3 inches, and you want to slice it into 27 smaller 1 x 1 x 1 inch cubes, as shown below. If you have a straight knife, what’s the minimum number of slices you need to do it? You are allowed to rearrange the pieces after each slice.

Does it make any difference if you have a really, really elastic cheese?

Solution

With a rigid cheese, the minimum number of slices is 6, which are the slices along the dotted lines in the picture. That’s two through each set of parallel faces. Consider it this way: if you are going to cut the cube into 27 smaller ‘subcubes’, these subcubes will be the 9 on the bottom horizontal slice, the 9 on the middle horizontal slice and the 9 on the top horizontal slice. The subcube in the centre of the middle slice will require six slices, because it has six faces, and each of these faces will need to be cut from a different slice. (All the other subcubes have once face that is part of an external face of the original cube, so these subcubes require less slices. But the middle one requires six).

In the case of a really, really elastic cheese, if it were possible to fold the cube over, so that one slice could cut two faces (These faces would be parallel once the cheese was unfolded), then theoretically you only need three slices: one slice for each set of parallel faces. You would need to fold the cheese over in three different (perpendicular) ways for each of these slices. Messy!

2. A cube of cheese is divided into 27 identical smaller ‘subcubes’, as above. A mouse starts at one of the corner subcubes and eats his way through the cheese (without ever passing through the same subcube more than once.) Whatever subcube the mouse is in, it can only move into a subcube that is horizontally or vertically adjacent to it.

Show a path through the cheese that passes through every subcube and ends in the centre subcube, or prove that such a path is impossible.

Solution

It’s impossible. Imagine that the 27 subcubes are coloured as in a 3-d chequerboard, so every white cube is vertically and horizontally adjacent only to black cubes, and vice versa. Let the corner sub cubes all be black. The cube thus contains 14 black sub cubes, and 13 white sub cubes. The centre subcube will be white.

If the mouse can only travel horizontally or vertically, the subcubes it travels through must alternate black and white. So if the first cube visited is black, every odd-numbered cube it visits will be black, and every even-numbered cube will be white. It is therefore impossible for the 27th cube to be white, so the (white) centre cube cannot be the final cube visited.

3. A cheese cube is sitting on a horizontal table. If you slice it in half by making a vertical cut, midway between, and parallel to, two of the sides, the cross section of the cut will be a square.

How do you slice a cube into two equal parts, such that the cross-section of the slice is a hexagon?

Solution

The fact you can get a hexagon out of a cube is a lovely surprise. You cut it like this:

Reader Rachel Baker made one out of butter (she had ran out of cheese):

4. How do you slice a cube of cheese into six pieces of identical volume with only three cuts?

Solution

The most pleasing symmetrical method is to cut down the diagonals, as Rachel Baker demonstrates here with a cube of parmigiano reggiano. Slice 1 is vertical from the diagonal on the top face to the diagonal on the bottom face.

Slices 2 and three are between the diagonals on other pairs of faces. Each of the three diagonals needs to meet at one of the vertices of the cube.

The six pieces are all tetrahedrons – four-sided objects where each side is a triangle, in this case isoceles triangles. The pieces have identical volume but are not identical, coming in two orientations. Rachel’s impressively neat slicing earns her today’s prize of one of my books.

Rachel’s is not the only solution. A simpler method is demonstrated here by Ed Southall. Easy cheesy!

In fact, Ed came up with many creative solutions. Just as there is more than one way to skin a cat, there is more than one way to slice a cheese cube.