Can you cut a cake fairly to solve this middle school math problem?

[Whateverist]

Goddammit, Whateverist, this is one of those problems that I know has to be fairly simple to solve, but I'm sitting here racking my brain over it and loathing myself for not being able to figure it out.

My stepson stole my opportunity to enjoy what you're experiencing right now by telling me his solution before I had a chance to consider the problem.  

[Whateverist]

You a funny girl!  Me like.

[Anomalocaris]

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5 cuts. Each running from the center to the outside edge. In the case of a 20 x 20 x 5 cake each cut needs to be 16 linear centimeters along the outside edge. That would give everyone 160 square centimeters of icing and 400 cubic centimeters of cake.

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Mathematically this is a easy and clear solution, but I rejected it because I missed the part about the cake being a square, and thought the original question indicated the dimension, and presumably the aspect ratio, of the cake didn't matter.  This implies the solution had to be elegant enough to not require careful measurements of the aspect ratio of the cake to come up with angles of the cuts.

[brewer]

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5 cuts. Each running from the center to the outside edge. In the case of a 20 x 20 x 5 cake each cut needs to be 16 linear centimeters along the outside edge. That would give everyone 160 square centimeters of icing and 400 cubic centimeters of cake.

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Exactamundo.  But to elaborate

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Of course every one of the triangles have the same height regardless of where their base may be.  For those which straddle a corner, each can be dissected into two triangles with a common height whose bases sum to the same as the other.  So for a square with side length = 20, the base of each triangle must be 4(20)/5 = 16  and the combined sides of a piece which includes a corner, like your red one, where 12 + 4 = 16.  Since the height of both triangles remains 8, the combined area of the two triangles would be 12(8)/2 + 4(8)/2 which can be rewritten as (12 + 8)(8)/2.

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If that's what Losty has to do, none of you are getting equal portions of cake. Apologies Losty.

[Edwardo Piet]

BTW, Losty isn't really dressed for kitchen success either . . . .

I'm sure Losty doesn't need to dress at all for kitchen success. She got zeh kitchen skillz regardless of clothing or lack thereof.

[popeyespappy]

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5 cuts. Each running from the center to the outside edge. In the case of a 20 x 20 x 5 cake each cut needs to be 16 linear centimeters along the outside edge. That would give everyone 160 square centimeters of icing and 400 cubic centimeters of cake.

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Mathematically this is a easy and clear solution, but I rejected it because I missed the part about the cake being a square, and thought the original question indicated the dimension, and presumably the aspect ratio, of the cake didn't matter.  This implies the solution had to be elegant enough to not require careful measurements of the aspect ratio of the cake to come up with angles of the cuts.

Correct. This simple solution only works for a square (or round) cake like the one Whateverist listed as an example. I'm not even sure there is a solution that yields equal shares of surface area (icing) and volume (cake) for an oblong shape because if you solve for surface area the volume is unequal. If you solve for volume the surface area is unequal.

[Pat Mustard]

When Poca, Alex, Rob and Whatshisname came for a visit, Losty baked a cake.  It was a square cake which she frosted on the sides and on top with yummy cream cheese frosting.  "I want a big piece of cake" said Poca.  "Just make sure my piece has a maximally quantum-big portion of frosting" said Alex.  "Be fair!" exclaimed Rob.  "Whatever" said Whatshisname.

How can Losty cut the cake so as to give all five of them the largest piece possible with each containing an equal share of both cake and frosting?  Oh, and to keep it interesting, lets do it with the fewest possible straight cuts with a knife (so no, you can't dump it all in a blender and measure out equal shares of sludge).  Surprisingly (hint, hint) the dimensions of the cake are irrelevant but if you really, really want to know, the size of the actual cake is given below.

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The cake was 20 cm on each of its four sides and 5 cm tall.  Good luck .. and be fair!

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*Edited to clear up the fact that the entire cake is to be used up making these five equal servings.  Thank you: 

Anomalocaris 

The definitive answer:

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While Losty was busy greeting her guests Tazzycorn snuck in the back window and ate all the cake. Therefore all five of them got nothing, the maximal amount each of what was left of the cake.

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[Losty]

Why can't Losty make 5 cakes ??


She lazy or sumthin' ??

Wow I didn't even know about this thread and you're in here calling me lazy. Rood

[Losty]

Also...whatevs....you better get someone else to cut the cake I can't cut in a straight line!

[Crossless2.0]

Why can't Losty make 5 cakes ??


She lazy or sumthin' ??

Wow I didn't even know about this thread and you're in here calling me lazy. Rood

Don't you have some baking to do? Get your ass back in the kitchen, woman.

<smacks Losty on backside>