What shape petit fours can I cut from a sheet cake s.t....
September 28, 2018 7:39 PM   Subscribe

(1) Each cut is a straight line across an entire piece of cake, but not necessarily across the entire pan. (2) Cake is wasted only at the edges of the pan. (3) All the resulting petit fours come out the same shape and size (except for the wasted bits at the edges). Equilateral triangles work, as do isosceles triangles. Rectangles work, either in a grid or staggered like brickwork. Hexagons do not work. I think I'm looking for tilings of the plane by one convex polygon, but not necessarily a regular, edge-to-edge, or even periodic tiling. Help?
posted by meaty shoe puppet to Science & Nature (11 answers total) 1 user marked this as a favorite
Diamonds, parallelograms, trapezoids all work.
posted by Eyebrows McGee at 8:02 PM on September 28, 2018 [1 favorite]

All triangles work. That's the only solution I can think of that hasn't been posted yet.

Can we use condition (1) to show that the average internal angle of each piece is 90° or less? If that's true, it would limit the possible polygons to triangles and quadrilaterals.
posted by aws17576 at 8:24 PM on September 28, 2018 [1 favorite]

I am interpreting your condition 1 to mean you only want convex polygons.

Can you say what your issue with regular hexagons is? They seem to fit your conditions, though they would be very annoying to actually cut.

There are also lots of pentagon tilings that might suit you.
posted by ktkt at 8:44 PM on September 28, 2018

You can get hexagons if you relax condition (3):


By relaxing (1) you could get L shapes:

posted by zompist at 8:55 PM on September 28, 2018

posted by oceano at 10:14 PM on September 28, 2018

Can you say what your issue with regular hexagons is?

I'm not the original poster, but I think these violate rule 1. If you're going for a honeycomb, you can't start with a straight cut all the way across the cake. My interpretation of rule 1 is that each cut must begin and end either at an edge of the pan or at an edge created by a previous cut.

Returning to the problem, I believe I can flesh out my previous comment about angles, though rule 2 poses a complication. I'll start by making the argument under the assumption that no cake can be discarded, then I'll comment on how to accommodate rule 2.

So: Assuming no cake gets discarded, I claim that the average of all interior angles of all pieces of cake must always be 90° or less. This is true at the beginning, when you have a single rectangle. Each subsequent cut creates new angles at its endpoints, which may land either at the corner of an existing piece or in the middle of an edge. In the former case, the total number of interior angles of all pieces increases by one, but the total of the angles does not, so the average angle decreases. In the latter case, two new angles are created which add up to 180°. The average of the new angles is 90°, and the average of the existing angles is 90° or less, so the overall average remains at or below 90°. This proves my claim.

As a result, if the pieces end up all being the same shape, then that shape must be a triangle or quadrilateral, since these are the only polygons whose average angle is 90° or less. (The sum of angles of an n-sided polygon is (n–2)*180°.)

Now, as for rule 2: I believe some clarification may be required, since I don't see anything in your question that technically rules out cutting away whatever you want at the edges to make a single petit four of any convex shape. But it's pretty clear that the intention is not to allow that. The mention of tilings of the plane suggests to me that you are looking for solutions with a fixed polygon that could be carried out on an arbitrarily large cake, producing an arbitrarily large number of petits fours. In that case, the boundary becomes relatively insignificant for large cakes. So if you have a large enough cake, you shouldn't be able to discard enough triangles from the edges to raise the average angle of the remaining pieces to 108° (which is what it would need to be to allow a solution with pentagons or higher).

If this reasoning holds up, then the only possible solutions are triangles and quadrilaterals, but it remains to determine which quadrilaterals work. Here's a solution with a quadrilateral that is not a parallelogram or trapezoid.
posted by aws17576 at 10:44 PM on September 28, 2018 [2 favorites]

If you use this, you could get hexagons: Ateco Stainless Steel Hexagon Cutter https://www.amazon.com/dp/B001BQXKQS/ref=cm_sw_r_cp_api_G1XRBbD5HM99C

Otherwise I think there’s more patterns that are createable if you cut, then remove pieces, then cut, remove pieces, etc.
posted by suedehead at 11:39 PM on September 28, 2018

I assumed the actual point was "shapes that are practical to cut" and not a Putnam exam problem. If that's the case, the most aesthetically pleasing solution is parallelograms (as mentioned previously). Evidence: traditional shape for cutting baklava.

I would suggest that cutting into various shapes is better, though, because there are always some deviant individuals that want "just a little piece" vs. the normal folks that want a huge one...
posted by Gilgamesh's Chauffeur at 6:34 PM on September 29, 2018

I assumed the actual point was "shapes that are practical to cut" and not a Putnam exam problem.

That's an entirely reasonable assumption to make. And while none of us have sat the Putnam, I got as far as the AIME, my wife did the AMC, and I would love to serve a selection of petit fours with proof that I had exhausted the space of petit four shapes satisfying some criteria.
posted by meaty shoe puppet at 10:04 PM on September 29, 2018

If you're not gonna hew to strict tiling but want to stick to long, consistent cuts across the board, zompist's hexagon slices with waste triangles suggestion would be my first thought (and you wouldn't have to do regular hexagons, if you wanted something a little more unconventional with the skew). Another take on that, still not tiling but with slightly less waste and a little more unusual zazzle: remove half of the cuts on one of the three axes of that hexagon cut and do irregular pentagons.
posted by cortex at 9:31 AM on September 30, 2018

In the spirit of Gilgamesh's Chauffeur's comment about people preferring different sizes, here's a pattern that obeys rules 1 and 2 and produces pieces of the same shape but three different sizes. Or, infinitely many sizes.
posted by aws17576 at 4:40 PM on September 30, 2018 [1 favorite]

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