Is that enough slices?
A man is hired to cut cake at a party. He was informed the day before the party that
2
8
people could definitely attend the party but one person wasn't sure if he could make it or not. To make sure everything would be fair, he decided the cut the circular cake into slices such that if either
2
8
or
2
9
people attended the party, the cake could be distributed evenly to all attendees. At first he thought he needed
2
8
∗
2
9
=
8
1
2
slices but his wife informed him that he was being an idiot.
What is the minimum number of slices the cake cutter needs to slice the cake into such that either 2 8 or 2 9 people can attend the party and the cake can still be distributed evenly?
The answer is 56.
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2 solutions
You demonstrated a way that the minimum is achieved, but you did not prove that it is the minimum.
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I'd like to see a proof that 56 is the least please.
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Well you know he is going to need at least 28 pieces, so let's have him cut those 28 equal pieces first. Then he needs to make AT LEAST one cut in each of the big 28 pieces he has because otherwise that piece of cake cannot be served because it is too big for one person, should 29 people attend.
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@Nathan Ramesh – Well, how do you know that you have to cut 28 equal pieces? You can cut 29 unequal pieces, and still have the cake given out equally.
For example, 8 4 1 , 2 8 1 , … 2 8 1 , 4 2 1 works.
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@Daniel Liu – I know but that involves the 28th piece be split into a 1/84 and 1/42.
Let's say that on the day of the party, the cake has been distributed to 28 attendees. The minimum amount of slices for this to occur is 28. All of a sudden, before any cake is eaten, the 29th attendee arrives, and the cake must be redistributed. To make every piece 1/29 sized, each piece in the hands of the first 28 attendees is too large and must be cut down into two smaller pieces. It doesn't matter what size the smaller pieces are; each of the 28 slices must be divided into 2 so that each attendee gets 1/29 of a cake. Therefore, the minimum amount of slices possible is 56.
There is a very simple way to prove that there are at least 56 pieces
Hint: Consider (any) arrangement that will feed 28 people. Each collection must be made of at least 2 pieces.
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Oh, yes! If the 28 people were to be given the cake equally, and the 29th visitor came, then they'd all need to give him/her a portion of their cake, effectively doubling the number of required pieces.
Which was already stated above. Sorry, didn't see, don't mind me...
Awesome solution
Can you read the question again... cuz it says that the cake should be distributed evenly to all the attendees...
Where is the combinatorics ?? :P
Excellent solution
Cutting the cake into 29 slices, then one of those into another 28 means if 29 attend, they each get one of the original pieces. If 28 attend, they each get one of the original pieces plus one of the 1/28th pieces of the last slice
but if 29th person attend the party he will get 28 slices and other will get only one in terms of slices he will get more than other and this will become unfair to other ???
I have a query- I did it this way - As we have three axis, if we cut one cut along each of the axis ( although it cant be practically done, nowhere in the question is given to assume so), we will get 8 pieces. If we increase the no. Of cuts to 9 on two axis and on one axis 8 then, the amount of pieces increases to 900. So our answer should 26. I know I am wrong but can anyone help me out where??
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The question is the number of slices they need to slice, I.e. the number of pieces of cake, not the number of cuts, so 26 pieces would not be enough to distribute amonst 28 or 29 people, whilst as you say, 26 cuts would easily produce enough pieces.
Well at first I thought this was a wrong question but I was wrong . The important thing is to notice that all the slices need not be of equal size . Since the maximum number of people who can come to the party are 29 . Therefore cut the cake into 29 equal sizes . Now cut the 29th piece into 28 Small equal pieces . We get that 1 big piece = 28 small pieces .If there are 28 people give 1 peice to each from the 28 big pieces and 1 each from small piece. If there are 29 people give 28 big pieces to 28 people and the 28 small pieces to the 29th man . In both case we get an equal distribution . Hence the answer is 28 big + 28 small = 56 total pieces .