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# Bake it

## Question

Host ask each of 4 guests to make
a single slice on a whole round cake,
so that they all have a 1/5 piece to take.
How they'll do that for fairness sake?

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quick question for clarification. Will the host participate in this too or is there only four participants?

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Are we looking for a way that the four friends get a fair sized piece of the cake?

Or, are we looking for a way to divide the cake exactly into fifths?

To clarify what I mean, for two people to get a fair division of a piece of cake,

the first one cuts it into two pieces; the second person chooses which piece to take.

We don't get an exact 1/2 division necessarily, but it is a fair division.

The OP seems clear that it's fairness we're looking for, so my question is probably not necessary.

It's an interesting puzzle.

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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

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The host will not cut..but he shall eat too. Fair enough for all.

The 4 visitors will cut the cake one by one, then every one put his piece

on a plate and start the party..

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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

Person 1 can be unfair to Person 5 by cutting a very small first piece.

Or is it the fact that Person 1 was chosen at random that makes it fair?

Would it make the process fair, if an intentionally small piece went back to the cutter, if no one else chose it?

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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

Not sure that this is what TSLF is aiming at. I think he means "How do you divide a round cake equally into 5ths using 4 cuts". But I've been known to be wrong before...

Edit... or more specifically, how do you ensure that each of 5 people gets an equal amount.

Edited by fabpig
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The host expect the plates to have a piece of cake with similar weight, size, shape and flavor (for fairness).

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If the pieces of cake must have the same size and shape, then I get the feeling this isn't one of those questions of the "one person cuts and another chooses" variety. It's asking how a certain number of slices can result in equally shaped pieces if the guests can cut with absolute precision.

The first guest makes a cut through the center with a 36 degree bend at the center (shown in black). The other guests make cuts to the (former) center at 72 degree angles from previous cuts (shown in other colors).

Of course, that's only if guests are allowed to make cuts with a sharp angle.

An alternative solution if cuts must be straight without any angles is that they lay the cylindrical cake on its side so the axis of the cylinder is parallel to the plate, then cut the cylinder perpendicular to its axis into five equally sized, short, circular pieces. But that alternative might not be acceptable if the cake has icing on the top since one of the criteria is that each piece has similar flavor.

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@Plasmid - by bending the knife. (allowed) it can also be done with 3 cuts if the 2nd cutter wants to cut as the first cutter,leaving the 3rd cutting just half as much as the other while the last guest complains for not being able to cut.

in fairness for the guests ..the host weighs & cut the whole 500 grams cake and took his 100 grams part from it.. now that makes it easier for his friends..to have same size and shape and flavored dessert..

Note: he used a straight knife that the other should use to..

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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

Person 1 can be unfair to Person 5 by cutting a very small first piece.

Or is it the fact that Person 1 was chosen at random that makes it fair?

Would it make the process fair, if an intentionally small piece went back to the cutter, if no one else chose it?

If person 1 cuts a small slice, no one will take it and person 1 gets the small slice

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You're right. I didn't read to the end of your solution, and I think you've solved it.

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You're right. I didn't read to the end of your solution, and I think you've solved it.

Apart from:

The host expect the plates to have a piece of cake with similar weight, size, shape and flavor (for fairness).

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If exact results are required I agree.

I can't imagine how without a scale or other way to verify.

If OP requires similar and fair, it has the same flavor as the I cut - you choose n=2 case.

With regard to specifics,

1. Flavor - seems automatic. Sounds like it might even be tongue in cheek (pun intended.)
2. Size and weight go together - each person does a visual inspection.
3. Shape is satisfied by cutting wedges from the center.
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I get the logic... but the OP still says to create 5 equal pieces with four cuts. And each person has 1/5

Edited by fabpig
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If exact results are required I agree.

I can't imagine how without a scale or other way to verify.

If OP requires similar and fair, it has the same flavor as the I cut - you choose n=2 case.

With regard to specifics,

1. Flavor - seems automatic. Sounds like it might even be tongue in cheek (pun intended.)
2. Size and weight go together - each person does a visual inspection.
3. Shape is satisfied by cutting wedges from the center.

The issue here is fairness for all the 4 guests/cutters ..and it is still fair for all if the remaining piece on the tray has the same flavor,volume ,weight yet not the same shape like the pieces on the plates.

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If exact results are required I agree.

I can't imagine how without a scale or other way to verify.

If OP requires similar and fair, it has the same flavor as the I cut - you choose n=2 case.

With regard to specifics,

1. Flavor - seems automatic. Sounds like it might even be tongue in cheek (pun intended.)
2. Size and weight go together - each person does a visual inspection.
3. Shape is satisfied by cutting wedges from the center.

The issue here is fairness for all the 4 guests/cutters ..and it is still fair for all if the remaining piece on the tray has the same flavor,volume ,weight yet not the same shape like the pieces on the plates.

OK, so the question is, and we're all anxious to know ... has the puzzle been solved?

And if not, what is missing from dgreening's solution?

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Now that we've established that "1/5 piece" does not mean "1/5 of the cake", the easiest solution would be to cut 4 diameters at 45degrees to each other. Each person takes 1/8 of the cake, leaving 3/8.

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The host may give all his guests similar knives so they can cut at the same time ,

and take the piece nearest to them altogether,exerting the same amount of effort

in slicing the cake to have a piece that has the same flavor (quantity of layers),the same size (weight or volume or fifth of the cake)

and the same shape (solid structure) like the others for their identical plates..That's how the phrase "fair and square"got originated.

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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

Person 1 can be unfair to Person 5 by cutting a very small first piece.

Or is it the fact that Person 1 was chosen at random that makes it fair?

Would it make the process fair, if an intentionally small piece went back to the cutter, if no one else chose it?

If person 1 cuts a small slice, no one will take it and person 1 gets the small slice

@dgreening "Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2"- can not do that with single cut..

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Each guest cuts at 1/5 of the height.

Four cuts give 5 pieces, equal in size and shape
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Each guest cuts at 1/5 of the height.

Four cuts give 5 pieces, equal in size and shape

I think LVan Toren has it. NIce!

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yes, as proposed by plasmid, it might work ..unless the image shows different layers of
different flavors not different food colors.

There would be a problem of who chooses first..still not fair for the four guests..

A geometric problem:

"How to divide the circle with 4 straight lines to make

4 identical parts that are 1/5 of the circle area?"

L= length of line segment

A = area of circle

R= radius of the circle

D= diameter of circle

Edited by TimeSpaceLightForce
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This is a variation on the problem of 2 people trying to split something. [where A cuts and B chooses which half].

One solution that solves this problem, but can be used for any practical value of N people

Arrange people in some order [ example: we will use sitting around a table and that things are passed clockwise].

• Person 1 [arbitrarily chosen] cuts a piece of cake, puts it on a plate and passes it to Person 2
• Person 2 can either keep the piece [if it looks good] or pass it along to Person 3
• ...
• Person 5 can either keep the piece [if it looks good] or pass it along to Person 1 [who must keep it]

After the 1st slice has been kept by someone. The knife is passed to the next person [that does not have a piece already] and the same process occurs. After someone takes a slice, they will not longer participate.

This process continues until someone cuts the 4th slice and the remaining person gets to choose that slice or what is left.

With this process, every "cutter" is encouraged to cut a slice that is fair, but not too big

Not sure that this is what TSLF is aiming at. I think he means "How do you divide a round cake equally into 5ths using 4 cuts". But I've been known to be wrong before...

Edit... or more specifically, how do you ensure that each of 5 people gets an equal amount.

Ahh k-man, where were you when we needed a modicum of common sense? You're THE man.

Oh and, BTW, I did so inform y'all thusly.

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