A Quick Cake Conundrum - Numberphile
video description
you know that the area of the right angle triangle up thrre equals half the area of the 1/4 square under it.
eyball an imaginary line through the bottom 1/4squares half points on the y axis and that leaves you with a trap made up of 3 equal area shapes.
one, triangle on top
one, rectangle in the middle
one, rectangle at the bottom
place your cut through the middle rectangles mid points on the y axus.
cut along the x axis.
do the same for the other trapezoid.
you now have 4 equal peices.
Date: 2022-04-08
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Comments and reviews: 9
anty
I thought about this way of doing it before he showed his way:
- horizontally cut off the bottom half of the bottom -square- (assuming this is the same perspective of the cake as in the video)
- insert that piece into the corner of the cake
- you have now converted the L shape into a rectangle
- you can now easily cut this in four, with 3 people getting a piece that is 3/16th of the total original cake
- however, one person must have two slices; one that is 1/8th of the cake, and one that is 1/16th of the cake
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I thought about this way of doing it before he showed his way:
- horizontally cut off the bottom half of the bottom -square- (assuming this is the same perspective of the cake as in the video)
- insert that piece into the corner of the cake
- you have now converted the L shape into a rectangle
- you can now easily cut this in four, with 3 people getting a piece that is 3/16th of the total original cake
- however, one person must have two slices; one that is 1/8th of the cake, and one that is 1/16th of the cake
reply
Marcus
Could you connect this to fibonacci numbers arising (as absolutes) along a diagonal of, absolutes of; complex numbers? Also crazily enough, I'd ask if fibonacci numbers with multiplikative, additive and subtractive(ordinary squareroots and squares of imaginary -units-=numbers) operations does span the field of integrals? (we have two followers along the diagonal -possibly- creating a third follower on the diagonal!
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Could you connect this to fibonacci numbers arising (as absolutes) along a diagonal of, absolutes of; complex numbers? Also crazily enough, I'd ask if fibonacci numbers with multiplikative, additive and subtractive(ordinary squareroots and squares of imaginary -units-=numbers) operations does span the field of integrals? (we have two followers along the diagonal -possibly- creating a third follower on the diagonal!
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John
Aha! Since you've solved that one, try this (a practical problem from my childhood):
Square cake again, this time out of the bowl, covered top and sides with butter icing (frosting. Frosting is regarded highly by the child consumers of said cake. How to divide the cake fairly so each child gets a fair share of both cake and icing.
Easy when there are 4 children. But there were 5 of us.
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Aha! Since you've solved that one, try this (a practical problem from my childhood):
Square cake again, this time out of the bowl, covered top and sides with butter icing (frosting. Frosting is regarded highly by the child consumers of said cake. How to divide the cake fairly so each child gets a fair share of both cake and icing.
Easy when there are 4 children. But there were 5 of us.
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Kevin
Four cuts. Center of top left square to center of bottom right square. Center of top left square to top center of pan. Middle of pan to right top corner of pan. Center of bottom right square to right middle of pan. Basically cutting into 12 equal triangles and each person gets 3 triangles. I read this as each person gets a one piece, so cutting it this way will satisfy that requirement.
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Four cuts. Center of top left square to center of bottom right square. Center of top left square to top center of pan. Middle of pan to right top corner of pan. Center of bottom right square to right middle of pan. Basically cutting into 12 equal triangles and each person gets 3 triangles. I read this as each person gets a one piece, so cutting it this way will satisfy that requirement.
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education
Option B: Cut into two congruent trapezoids by making a 45 degree cut from the center to the corner. Then leverage the 3-dimensionality of the cake by making a second cut horizontally through the remaining cake. Voila, four equal pieces. (This solution hinges on the uniformity of the cake from top to bottom. Would not work well with frosting)
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Option B: Cut into two congruent trapezoids by making a 45 degree cut from the center to the corner. Then leverage the 3-dimensionality of the cake by making a second cut horizontally through the remaining cake. Voila, four equal pieces. (This solution hinges on the uniformity of the cake from top to bottom. Would not work well with frosting)
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Adam
I'm at 1: 44. My solution: cut the remaining L shape into 3 even squares. Then cut each square twice along the diagonals. You now have 12 triangular shaped pieces. Each of the four family members gets 3 triangular shape bits. Seems easy enough to me. Now I watch the rest of the video for the solution. Hope my answer is right. :)
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I'm at 1: 44. My solution: cut the remaining L shape into 3 even squares. Then cut each square twice along the diagonals. You now have 12 triangular shaped pieces. Each of the four family members gets 3 triangular shape bits. Seems easy enough to me. Now I watch the rest of the video for the solution. Hope my answer is right. :)
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Kyle
I like the fractal, recursive nature of Cliff's solution, but a more simple solution came to mind at first. The L is made of three squares. Take one that isn't the middle, corner square and cut it in half to make two rectangles. Put them flat against the remaining two squares. Now cut that rectangle into four equal slices.
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I like the fractal, recursive nature of Cliff's solution, but a more simple solution came to mind at first. The L is made of three squares. Take one that isn't the middle, corner square and cut it in half to make two rectangles. Put them flat against the remaining two squares. Now cut that rectangle into four equal slices.
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Cactuscobbler
So what is this technique that he used called?
Also, How about taking the area of the remaining cake, and dividing that number by four? That will yield 4 equal numbers. These numbers will be the amount of area you get. As long as you get this much area, it doesn't matter what your cake looks like
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So what is this technique that he used called?
Also, How about taking the area of the remaining cake, and dividing that number by four? That will yield 4 equal numbers. These numbers will be the amount of area you get. As long as you get this much area, it doesn't matter what your cake looks like
reply
njack1994
How to divide it and make it still look like a slice of cake? Well you want one piece one third bigger then the rest and 4 pieces of equal size for a total of five pieces. So imagine a circle instead of a square one piece makes up 96 degrees of the circle and the others make up 66 degrees of the circle.
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How to divide it and make it still look like a slice of cake? Well you want one piece one third bigger then the rest and 4 pieces of equal size for a total of five pieces. So imagine a circle instead of a square one piece makes up 96 degrees of the circle and the others make up 66 degrees of the circle.
reply
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