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

[Whateverist]

Now I'm jealous of your ability to illustrate what you have in mind on a computer screen. I think I see what you have in mind but I can't quite see whether it makes a difference. But I think I see a way to investigate it. Let me mess with it a bit and I'll see if I can find a way to show it.

[popeyespappy]

Green, blue and purple gets a whole corner each. Red and yellow have to share one.

[The Valkyrie]

Losty gets to tell the fussy bastards to go fuck themselves and keep the cake for herself (and the kids).

[Whateverist]

Just as the area of the square representing the base of the cake alone must be divided into five equal pieces so long as the cuts are made from the center to five equally spaced points along the perimeter, so must the larger square consisting of the cake plus frosting.  Since the portions contained inside the square representing the unfrosted cake's base are equal, if you subtract them from each piece of the larger square representing the cake plus frosting, the remaining area (representing the base area of the frosted region alone) must also be equal.  Therefore the frosting is also divided equally in this solution, including the fours corner regions.

To see this I made a drawing to show directly that the base area of the frosted regions would still be equal when the frosting is applied to a thickness half the side length of the cake alone.   A larger scale version of the photo and an explanation of the drawing are below.



Larger scale photo:

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Explanation of the drawing:

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Okay so I drew a square cake 40 units on a side.  (This grid paper is ruled with a darker line every 8th line, or 8 lines to the inch.)  So the perimeter of the cake is 160 units which divides by 5 to make 32 units of perimeter per piece of cake.

I then drew very dimensional frosting, 20 units deep all the way around the cake.  The resulting cake has an 80 units by 80 square base. 

Next I drew cuts from the center to five points along the perimeter of the cake, each one 32 units from its neighbor all the way around the cake.  The cuts continue all the way through the cake and frosting.

The total area of the portion of the drawing representing the frosting, has a total area of 4800 sq. units, consisting of 4 squares at the corners of 20 by 20 and four rectangles between them of 20 by 40.  If the frosting is to be divided evenly each piece must contain 4800/5 = 960 sq. units of this base area.  

To find the base area of each of the five pieces I partitioned each one into rectangles and triangles and summed all its parts.  The resulting area worked out to be the 960 sq units required.

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Green, blue and purple gets a whole corner each. Red and yellow have to share one.

Notice that though the yellow and red pieces get only half a corner of frosting, each one gets portion of frosting that extends into the area directly above its neighbor.

[Rev. Rye]

I think the answer lies in the perimeters. a 20 cm square cake will have a perimeter of 80 cm. Find a corner and measure (or guess) a length of 16 cm. Repeat until you arrive back at the original corner.

Coordinates of each vertex: (0,0), (0,16), (20,12), (12,20), (16,0).

Then find the centre and cut from each vertex to the centre.



Full disclosure: I looked up the problem on Google, and found that Alex Bellos of The Guardian actually did a whole write-up on the problem last year. It's where I got the diagram from.

[Jackalope]

As long as Losty used a square pan to bake her square cake and spread the frosting evenly there isn't a problem.

Y'all aren't accusing Losty of sloping the icing on unevenly, are you? I sure hope not because those kinds of accusations have consequences.

I think Chthulthuluuu's point was that if we cut it into five equal rectangular strips lengthwise then besides the issue of the frosting on the end pieces, there would also be less cake on those two end pieces.

Assuming our cutting into five equally thick rectangular strips is based on the external dimensions of the cake as measured on the frosted cake, then the thickness of the frosting itself will be contributing to the thickness of the slice for the end pieces but  not for the three middle pieces.  So, however thick the frosting is applied, that is how much cake the end pieces are not getting.  (Naturally -if I wasn't counting calories- I'd gladly trade that thin slice of cake for extra frosting.)

^^^ This frood knows where his towel is ^^^

[popeyespappy]

Notice that though the yellow and red pieces get only half a corner of frosting, each one gets portion of frosting that extends into the area directly above its neighbor.

OK so I'm making it harder than it needs to be...

I 3d modeled the damn thing and let the computer tell me the volume of all the different parts. A 20x20 cm cake covered with 0.5 cm of icing on the top and sides has a volume of 2425.53 cm. That's 20003 cm of cake and 425.53 cm of icing. If you cut it like we did way back on page 3 everyone gets 4003 cm of cake and 85.13 cm of icing. The thickness of the icing doesn't make a difference as long as it is spread evenly.

Oh, and everyone gets diabetes too from eating that much cake.

[Rev. Rye]

Notice that though the yellow and red pieces get only half a corner of frosting, each one gets portion of frosting that extends into the area directly above its neighbor.

OK so I'm making it harder than it needs to be...

I 3d modeled the damn thing and let the computer tell me the volume of all the different parts. A 20x20 cm cake covered with 0.5 cm of icing on the top and sides has a volume of 2425.53 cm. That's 20003 cm of cake and 425.53 cm of icing. If you cut it like we did way back on page 3 everyone gets 4003 cm of cake and 85.13 cm of icing. The thickness of the icing doesn't make a difference as long as it is spread evenly.

Oh, and everyone gets diabetes too from eating that much cake.

Which is why nobody ever slices their cakes the way the problem suggests. That and it being needlessly counter-intuitive.

[Joods]

While everyone stands around a big white board, drawing schematics and diagrams of how to properly divide the cake Losty made, Nymphadora and her epic stealth skills quickly and cunningly removes said cake from the counter, grabs two forks and two bottles of wine, along with Losty and we sneak out the back door to the deck, where we each grab a seat at the table, facing the house. We each take up a fork and a bottle of wine and laugh heartily at the fools inside who are still bickering about how to cut the cake, and we consume said cake - all without a feeling of guilt because we are women with cake and wine and we are invincible!

And a lot smarter than the men still inside, arguing about the now eaten cake.

[vorlon13]

I still think Losty just needs to make a cake for everyone.

Get cooking, sweetie !!