# Fair share of a square watermelon?

One hot day, Stan, Kyle, and Kenny were sitting outside with a square watermelon (actually it was a cube like the picture below).

Stan says "Let's cut the watermelon into 3 equal slices (like the right-side diagram below)."

Kyle says "But then the end slices have more rind."

Can you show them how to cut this square watermelon into 3 equal (congruent) pieces, so that each piece has an equal amount of rind? The correct answer should include a sketch or a picture and uses just a few straight cuts (no tricks).

• Cut the whole rind and then divide it the way Stan has said. – Spikatrix Feb 1 '15 at 10:26
• @CoolGuy That's 6+2 cuts. Too many. You can do it with less. – Somnium Feb 1 '15 at 12:02
• I wouldn't bother even trying. Cartman would just take the lot anyway. – Mac Cooper Feb 1 '15 at 19:36
• Who cares about getting equal amounts of rind? Really they should be trying to get equal amounts of pink stuff, which is trivially easy. :) – fluffy Feb 1 '15 at 19:39
• @Somnium You'd need to cut the extra rind off only on the ends, so it'd be 2+2 cuts, not 6+2 cuts, although perhaps that's still too many for you. – jamesdlin Feb 2 '15 at 10:33

solves the problem - but Stan or Kyle or Kenny would have trouble cutting their cubical watermelon this way since the point of the knife needs to follow the long diagonal of the cube while the edge of the blade follows a face diagonal.

Each individual pyramid has a square base and one vertical edge. It's the most interesting piece in the geoblocks available in many K-2 classrooms - the central one in this image from http://catalog.mathlearningcenter.org/store/product-924015.htm

This dissection is really much more than the answer to a puzzle. It's the essence of the theorem that says that the volume of a pyramid is 1/3 of the height times the area of the base, and of the fact that the integral of $x^2$ is $x^3/3$. The analogue in the plane is cutting a square in half along the diagonal. In four dimensions you can cut a tesseract into four congruent pyramids each of which has a cube for a base.

• They agreed that the cuts would require some skill. So they asked Chef because he had done this before in a previous life. – Len Feb 1 '15 at 19:54
• I may be being pedantic, but "cutting a triangle along the diagonal" makes no sense - a triangle has no diagonals ;) – Niet the Dark Absol Feb 2 '15 at 15:05
• @NiettheDarkAbsol Not pedantic, correct. I changed "triangle" to "square". Thanks. – Ethan Bolker Feb 2 '15 at 15:09

Look at the cube down a space diagonal. It should appear as a hexagon, which can be divided into three rhombuses by line segments from the center to every other corner. If these three cuts are made, they will divide the cube into three pieces, which must be congruent by symmetry.

• A picture will be fine! – Somnium Feb 1 '15 at 17:21
• @user9291 +1 for a good answer. A picture would have helped. – Len Feb 1 '15 at 19:56
• This is actually the same as Ethan Bolker's answer. – Joe K Feb 5 '15 at 21:38

Assume that the cube/watermelon has edge length $1$, and let $M$ denote the point in the very middle of the cube.

• For each of the six faces of the cube, we define a corresponding pyramid that has the cube-face as its ground-square and that has point $M$ as its top vertex.
• The ground-square of such a pyramid has area $1$, and its height (distance from ground-square to $M$) equals $1/2$. Hence the volume of the pyramid is $1/6$ of the total cube volume.

Now we can define three congruent pieces that each have equal amounts of cube surface and cube volume:

• The first piece consists of the two pyramids associated with the front face and the right face.
• The second piece consists of the two pyramids associated with the bottom face and the back face.
• The third piece consists of the two pyramids associated with the top face and the left face.

Added by Len - Diagrams show the difference between the two answers:

• The question asks for three pieces specifically while your answer gives six pieces. Is it possible to cut it so that it's only three pieces, each consisting of two pyramids? (And can you show a pic of the cuts?) – EFrog Feb 1 '15 at 15:11
• However, I think it will be a bit difficut to cut watermelon in such way. – Somnium Feb 1 '15 at 15:42
• @EFrog: In my solution there are six pyramids, but only three pieces. Every piece consists of two pyramids that share a common face, and hence together form a polyhedron. – Gamow Feb 1 '15 at 17:31

You can cut it in six pyramids (each with the base one face of the cube, and the top in the middle of the cube). To cut in three parts, you take an edge and cut away pyramids formed by the two squares that define that edge (let's say top and left). After that, you do the same for front and right, and end with back and bottom as the third piece. This way, you also solve the problem of sharing the sweetness (the center of a watermelon is sweeter than the margins).

Just cut the rind off 2 opposing sides then cut the remains into 3 equal parts.

• -1 Your method cuts the melon into five non-congruent pieces; the question clearly specifies that it should be cut into three congruent ones. – David Richerby Feb 2 '15 at 17:43
• @DavidRicherby The question never said exactly 3, so I assumed I could have extra pieces. – Abraham Zhang Feb 3 '15 at 6:07
• @EpicGuy In this community, we generally dislike answers that go against the spirit of the constraints in their puzzles. (We even don't like puzzles that require those types of answers in order to be solvable at all!) – Kevin Feb 3 '15 at 8:14
• @Kevin Ok. I guess I could've even done it in 2 cuts in the shape of a cross, resulting in 4 congruent pieces. However, there would be less pink stuff for everyone :) – Abraham Zhang Feb 3 '15 at 9:38

Cut the watermelon in a cross.

From corner to corner. Like an X!

Then, the remaining slice, you cut in 3 parts.

This is the best option since it also avoid wastes in case one of them doesn't want that piece. Thus, easily dividing the remaining slice in only 2.

Also, the slices will have the same ammount of 'shell' and a very ergonomic shape, that helps you to bite it or cut it.

As pointed by @Len, there is one problem...

Now, we have to cut the slice in 3.

The solution is to grab a piece of string with the length of the slice.

Then you make a circle with it and mark 1 line at the center.

Try to make the "peace" symbol (You can check it here: http://en.wikipedia.org/wiki/Peace_symbols).

Cut it along one of the lines.

The string will have 1/3 of the size!

Now cut the slice with the same width as the string.

• Hmmm. So you are creating 4 wedges. But the 4th wedge cannot be cut into 3 identical pieces? – Len Feb 1 '15 at 19:25
• @Len It can. But you will need a piece of string and math. Eventually, one of them will give up and just say "keep my share from that slice". And done! Slice it in half! But I will add the mathmatical part. – Ismael Miguel Feb 1 '15 at 19:29
• @Len Actually, it was harder than I though, and no math envolved! – Ismael Miguel Feb 1 '15 at 19:34