Cutting the Cheese

A puzzle from Henry Ernest Dudeney:

Here is a simple question that will require just a few moments' throught to get an exact answer. I have a piece of cheese in the shape of a cube. How am I to cut it in two pieces with one straight cut of the knife so that the two new surfaces produced by the cut shall each be a perfect hexagon? Of course, if cut in the direction of the dotted line the surfaces would be squares.

http://farm4.static.flickr.com/3201/235 ... 82ed_m.jpg

Now produce hexagons.

I started by cutting it corner to corner and then realized one wouldn't have enough surfaces - however if one comes in a little from the corner and cuts down through the block at the same diagonal angle...voila.

I'm sure that was about as clear as mud.

I think the key is to not require the pieces to be the same volume, just to have the hexagons to have the same area. Hey, Bob, does it need to be a regular hexagon?

I can't believe that you haven't solved this yet, Stuart. Maybe you should try it with an actual cube of cheese.

I did that, but the ricotta didn't hold together very well.

BEWARE: The spoiler is below.

This page has a pretty neat 3-D illustration three images down that shows how - and you can rotate it with your mouse:

http://mathworld.wolfram.com/Hexagon.html

I think the key is to not require the pieces to be the same volume, just to have the hexagons to have the same area. Hey, Bob, does it need to be a regular hexagon?

Yes.

Spoiler alert -- not as fancy as Fred's 3D solution, but it does have the virtue of coming from Dudeney himself:


Mark the mid-points in BC, CH, HE, EF, FG, and GB. Then insert the knife at the top and follow the direction indicated by the dotted plane. Then the two surfaces will each be a perfect hexagon, and the piece on the right will, in perspective, resemble Figure 2.

http://farm3.static.flickr.com/2382/235 ... f6d335.jpg

In practice, the effect is subtle, and some folks do notice the shape.

Mark the mid-points in BC, CH, HE, EF, FG, and GB. Then insert the knife at the top and follow the direction indicated by the dotted plane. Then the two surfaces will each be a perfect hexagon, and the piece on the right will, in perspective, resemble Figure 2.

http://farm3.static.flickr.com/2382/235 ... f6d335.jpg

In practice, the effect is subtle, and some folks do notice the shape.

For anyone versed in the language of symmetry, one could say that you make the cut along a plane perpendicular any of the the C3 axes that run through the cube. For Stuart and any other chemically inclined people, the question is: are the two pieces produced chiral or achiral? I'm sure that there's a crystallographic parallel here for anyone motivated to look for it.

Mark Lipton