# Cut up a cheese cube into $6$ equal parts with $5$ slices

Bob has a cheese cube and wants to cut it up into exactly $6$ equal pieces for his guests. These guests are professional cheese connoisseurs who will complain if they get less than anyone else.

Bob doesn't have the tools necessary to make arbitrary and precise cuts. He can be precise enough if he slices from a corner to corner or bisecting along an edge, but not randomly through an edge.

Notes:

The guests only care about volume, the shape of the pieces doesn't matter.

Is it possible for Bob to please his guests?

Edit: Cleaned up unnecessary restrictions.

• After we make a cut, can we then remove one of the cut pieces and cut it without cutting the rest of the cube? – 2012rcampion Jul 4 '15 at 2:37
• @2012rcampion2 yeah that should be fine. I'm going to have to make an edit though. – Quark Jul 4 '15 at 3:08
• Related (title has spoilers). – xnor Jul 4 '15 at 9:09

Here's how I propose to solve this conundrum.

We need to cut out a tetrahedron from the cube. This can be done by cutting along three corners each time as shown in the image. As a result, we have one tetrahedron, whose volume is 1/3rd of the original cube.

We are left with four pieces, which are equal in volume, whose three faces are half the triangle from the cube face. The volume of those pieces cannot be calculated very easily, but we know that the sum of their volume is 2/3rd of the original cube (since 1/3rd of the volume went into the tetrahedron). This means each of them are 2/3 $\div$ 4, i.e 1/6th of the original cube.

Since we know how to bisect a piece, we can cut the tetrahedron into two equal pieces and the volume of each piece is 1/3 $\div$ 2, i.e 1/6th of the original cube.

Hence, we have our 6 pieces whose size is 1/6th of the original cube.

Image for reference:

• Nice job, you solved it much faster than I expected. – Quark Jul 4 '15 at 5:05

I think he can do it with 3 cuts!

Cut along the planes marked in red, green and blue. Each plane divides the cube in half, and all three intersect along a space diagonal. The middle image shows what these cuts look like when the cube is viewed in the direction of that space diagonal. The right image shows what each piece looks like.

• Can't believe I forgot this was a solution. Originally I had it limited to 5 cuts because I remembered this, but I edited it out after mistakenly thinking it wasn't needed anymore. I wish I could accept two answers but most I can do is give an upvote for finding the only other solution that will fit the criteria. – Quark Jul 4 '15 at 7:03

I think that this can be achieved by cutting the cube into 6 pyramids like this:

• Although this is a pretty cool idea, Bob wouldn't be able to do this with a knife. – Quark Jul 4 '15 at 5:04
• @Quark Sure he could, just could it along from one edge to the opposing one and so on... I guess – Beta Decay Jul 4 '15 at 22:37
• @BetaDecay But these cuts would also cut sections of from each pyramid, thus you will end up with more then six pieces. – fibonatic Jul 7 '15 at 10:37
• I agree with Quark. But on a more positive note, I think this is an awesome proof that volume of a pyramid is $$A=\frac{1}{3}Area(Base)\times Height$$ :) – John Joy Jan 26 '16 at 15:40

Alternate solution :

1-Cut the cube through the centre of each side. You now have 8 identical cubes.
2-Quietly leave 2 pieces in the fridge. Serve the remaining pieces to your friends.
3-They are happy to have the same amount of cheese, while you are happy to have some more for later (just don't tell them).

Cut the cube into six spheres in six cuts.

1. Pick three adjacent sides of the cube.
2. Lets call those sides A, B, and C.
3. Rotate the cube so that side A is facing up.
4. Chose a pair of opposing corners on the top face of the cube.
5. Align the knife with the corners and cut downwards completely through the cube.
6. Repeat step 5 for the other pair of corners.
7. The cheese should be sufficiently tacky to hold together for the remainder of this process.
8. Repeat steps 3-6 for sides B and C.
9. Separate the pieces into six pyramids.
10. Roll each pyramid into a ball and serve.