How many slices of cake?

Algebra Level 2

Nine friends bought three cakes, each cake was cut into eight slices. Everyone ate cake and nothing left. Knowing that each of the friends ate only whole slices of the cake, we can be sure that

Everyone ate two slices, at least One of them ate at least three slices Someone ate four slices One of them ate only one slice

Luiz Claudio by Luiz Claudio , 19, Brazil

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3 solutions

Vedant Saini
Apr 1, 2019

There are 24 slices of cake and 9 people to eat them.

By simple application of the Pigeonhole Principle, we can conclude that at least 1 person ate at least 3 slices of cake.

9 Helpful 0 Interesting 1 Brilliant 6 Confused

So were 8 of them eating even amounts of cake or just random amounts each? I'm confused because if there were 21 slices left after the person ate the 3 wouldn't the rest have to eat different amounts?21÷8= 2.625

Epic Burger - 2 years, 1 month ago

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Each person can eat a different number of whole slices, along the lines of your "random amounts each", as long as each amount is a whole number of slices.

Carl Jones - 2 years ago

Good question but answers may be not sure

Bhukya Kushwanth - 1 year, 11 months ago
Om Patel
Nov 25, 2019

We can prove that the correct answer is accurate by indirect proof. Assuming everybody ate most 2 slices leaves at most 18 slices eaten. Since we need 24 slices to be eaten, this is impossible. Therefore, at least one person had at least 3 slices of cake.

1 Helpful 0 Interesting 1 Brilliant 0 Confused
Carl Jones
May 14, 2019

Cool, I learned about the Pigeonhole Principle today! Not knowing that ahead of time, I was able eliminate the first three options by counter-example.

  1. Everyone ate two slices. False. Have one person eat 16 slices, and the other eight eat 1 slice each.

  2. One of them ate only one slice. False. Each person can eat 2 slices, with 6 slices remaining. It doesn't matter how those get eaten.

  3. Someone ate four slices. False. Each person can eat 2 slices, with 6 slices remaining. Then six different people eat those as their third slice. No on eats 4.

  4. One of them ate at least three slices. True. Assume that this is false, that no one ate at least three slices. Everyone ate at least 1 slice. That leaves 15 slices for the nine people. Everyone can eat only 1 more slice, so that no one eats at least 3 slices. This leaves 6 slices, with no way for them to be eaten. This contradicts the setup. The assumption is wrong, and this option is true.

Also the Pigeonhole Principle!

1 Helpful 1 Interesting 0 Brilliant 1 Confused

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