Brain Teasers
Easy As Cake
In our business, my two partners and I cut a lot of cake.
We work exclusively with square cakes. The cakes come in sizes that are meant to be divided into a square number of square pieces. Each portion comes decorated with an "X" stretching to its corners, which makes it easy to cut the usual square pieces and also offer half and quarter portions.
These markings also help on Wednesday when we savor the previous week's success by dividing a cake equally amongst ourselves. Even when a good week deserves a 64-portion cake, we can each ceremonially make a single standard cut (straight and perpendicular to the plate), and we each go home with a single piece of cake.
Using no other cuts or devices, how do we evenly divide the 8-by-8 cake? (EXTRA - How could we get 7 (and only 7) equal-size pieces from the same size cake?)
We work exclusively with square cakes. The cakes come in sizes that are meant to be divided into a square number of square pieces. Each portion comes decorated with an "X" stretching to its corners, which makes it easy to cut the usual square pieces and also offer half and quarter portions.
These markings also help on Wednesday when we savor the previous week's success by dividing a cake equally amongst ourselves. Even when a good week deserves a 64-portion cake, we can each ceremonially make a single standard cut (straight and perpendicular to the plate), and we each go home with a single piece of cake.
Using no other cuts or devices, how do we evenly divide the 8-by-8 cake? (EXTRA - How could we get 7 (and only 7) equal-size pieces from the same size cake?)
Hint
Think smaller.Answer
To get three twenty-one-and-one-third (64/3) portion-sized pieces, we momentarily ignore the 28 X's/portions/square-units which form the 32 unit perimeter of the cake. There is more than one way to divide the 6-by-6 (36 X) cake inside into three equal pieces. Using one based on its 24 unit perimeter, this cake could be divided into 24 wedges of equal size. Each triangular piece would have as its base one unit of the perimeter; the other two sides would be formed by cuts to the center of the cake. Every triangle would have the same altitude (3 units) giving the same area (A = ½ * b * h = ½ * 1 * 3 = 3/2 square units). 8 adjacent triangles, even though they include a corner or two, will be 12 square units or 1/3 of the "inner" cake.Before we make any actual cuts, we return the ignored 28 portions to the picture and then make those three cuts extended to the true perimeter. The 24 triangles pictured previously increase in altitude to 4 units. By rules of similar triangles, the base increases by the same 1/3 (33 1/3%) to 1 1/3 units. The area of 8 extended triangles is 8 * ½ * 4/3 * 4 = 64/3 portions, which is exactly one-third of the cake.
(The perimeter of a 7-by-7 square can be formed by connecting the centers of the X's in the "ignored" 28 portions. Applying the above methods but now grouping 28 triangles into foursomes, we get 7 equal size pieces.)
Hide Hint Show Hint Hide Answer Show Answer
What Next?
View a Similar Brain Teaser...
If you become a registered user you can vote on this brain teaser, keep track of which ones you have seen, and even make your own.
Solve a Puzzle
Comments
I enjoy working with figures, odd shapes and logic problems! This one was a piece of cake. As if you haven't heard that enough times. Keep them coming! Totally awesome teaser.
I must have missed something. It seemed to be so simple I thought I was missing something and I was.
The bottom line is that you have a perfectly square 8 X 8 cake and you need to divide it among 3 partners. Cut two lines- bottom left corner to halfway point at top edge then second line from bottom right corner to that same halfway point on top. Voila! 3 equal pieces. The halfway point is easily found because of the squares and Xs marked on the cake.
Told you I missed something...
The bottom line is that you have a perfectly square 8 X 8 cake and you need to divide it among 3 partners. Cut two lines- bottom left corner to halfway point at top edge then second line from bottom right corner to that same halfway point on top. Voila! 3 equal pieces. The halfway point is easily found because of the squares and Xs marked on the cake.
Told you I missed something...
scallio -
Thanks for comment, but the wedge between your corner cuts is twice as big as either of the other two. It is triangle with a base and a height of 8 and, therefore, an area 32 square.
As a physical "proof" of the described method, one could cut a square from a heavy sheet of paper. (It doesn't have to be 8 inches to a side, though it could.) Mark the center. Measure the perimeter. (For this, a ruler is permitted.) Make first cut from center to ANY point along edge. Measure along edge to find point one-third of perimeter away. Cut from center to that point. From that point measure another one-third of perimeter and cut again. (The edge from here to starting point will be yet another one-third.) Weigh pieces with postal scale to see they are equivalent.
Thanks for comment, but the wedge between your corner cuts is twice as big as either of the other two. It is triangle with a base and a height of 8 and, therefore, an area 32 square.
As a physical "proof" of the described method, one could cut a square from a heavy sheet of paper. (It doesn't have to be 8 inches to a side, though it could.) Mark the center. Measure the perimeter. (For this, a ruler is permitted.) Make first cut from center to ANY point along edge. Measure along edge to find point one-third of perimeter away. Cut from center to that point. From that point measure another one-third of perimeter and cut again. (The edge from here to starting point will be yet another one-third.) Weigh pieces with postal scale to see they are equivalent.
well it is easy but long ja but kind of anoying because there is a stupid noise in my speakers which is very annoying
fun one I got this one!!
Very nice... very very nice.
Fantastic teaser!
I spent way longer to get this than it should have taken. Finally I had the "aha!" moment after I gave up on the 3-way divide and tried the 7-way.
This is now one of my favorite teasers on the site!
I spent way longer to get this than it should have taken. Finally I had the "aha!" moment after I gave up on the 3-way divide and tried the 7-way.
This is now one of my favorite teasers on the site!
To post a comment, please create an account and sign in.
Follow Braingle!