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Thirds

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"How do you split a rectangle into thirds?"

"Well, you could cut from the center of the rectangle to the middle of one of the long sides, then from the center out at an angle that isn't really 120˚ from the previous cut..."

"What would that angle be?"

"Um... You could cut it in half and redefine equal."

"..."

"Or, cut it into quarters, we each take a piece, and wrestle for the last one."

"Let's go with that."

Comments

Because I'm That Kind of nerd, I decided to work this out.

Given a rectangle that's X by Y units, you'll have two trapezoids on the top and bottom with area (X+K)*Y/4, where K is the point along the top/bottom edge where the angular cut needs to be. Solve for (X+K)*Y/4 = X*Y/3:

(K+X)*Y*3=X*Y*4
K+X=X*Y*4/(Y*3)
K+X=X*4/3
K=X/3

So, it turns out that you make your fan cuts go from the center of the pastry to the 1/3 point opposite the center cut.

If you care about an even distribution of edge-crust (which I assume you do since you didn't just cut across in thirds), this one gives you Y/2+2X/3 edge for the two trapezoids and Y+2X/3 for the wedge. So the person with the wedge does get more edge, regardless of aspect ratio, unfortunately (but at least it's closer than the cross-cut approach which gives the middle piece 2X/3 and the end pieces 2X/3+Y).

Figuring out the equal share of stuffing volume requires levels of calculus I don't feel like dealing with at the moment.

This is just so so so awesome! <3

Oops, I just checked my work and I hecked it up. K was meant to be the size of the wedge's slice, not the trapezoid's slice, and when I work it out correctly, it gives me an impossible fan.

But there's a much simpler solution! Cut across 1/3 of the way through, then cut the remaining 2/3 in half. This gives the person who receives the first cut way more edge, though.

Anyway I really want to add some diagrams to more complicated solutions so I think I'll be writing a blog entry of my own. :)

Okay I spent *way* too long on this, but: http://beesbuzz.biz/blog/5206-Thirds

<3