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![enter image description here](https://i.stack.imgur.com/PLG2v.jpg)

The chef ask each of the 4 judges to make

a single slice on the whole round cake,

so they'll all have a 1/5th piece to take.

How the judges do it for fairness sake?

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  • 3
    $\begingroup$ What do the judges consider fair? That each is satisfied that they have at least 1/5? Or (a more stringent condition) that each is satisfied that they have at least as much as any other judge has? $\endgroup$ – Rosie F Sep 22 at 6:33
  • $\begingroup$ should have a plate with as much as possible similar piece of 1/5 cake with all the other plates. $\endgroup$ – TSLF Sep 22 at 6:52
  • $\begingroup$ Do judges have infinite accuracy in terms of deciding where to cut and measuring how big each slice is? $\endgroup$ – JS1 Sep 22 at 22:27
  • $\begingroup$ Lets say the eyes can tell when two object have the same area else do the math and mark. I think a geometry tag is what lacks here $\endgroup$ – TSLF Sep 23 at 1:52
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The judges can cut the cake like this:

Let X be the cake's volume. Each judge, in turn, uses a vertical cut to remove a piece of volume X/5 off of the remaining cake. The final piece of the cake also has volume X/5 and goes to the chef.
enter image description here
Since the cake as depicted is homogeneous in each layer, the outcome is fair.

Edit: New solution for updated rules:

> The rule is simple 4 cuts to make 4 equal or same object (piece of cake)
enter image description here
Maybe you mistyped and wanted 5 equal objects?
enter image description here
Or maybe you really only wanted 4 equal objects, each of which should contain 1/5 of the cake?
enter image description here

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  • $\begingroup$ All the judges plates must contain a piece with same shape and size for fairness $\endgroup$ – TSLF Sep 22 at 12:45
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    $\begingroup$ Ah, a secret rule. Before I retry, are there any other secret rules you didn't put in your post beforehand? It would help if you restate the complete unambiguous problem in plaintext after your poem (scenario, objective, and rules), instead of leaving us to guess which kind of problem you're thinking of. $\endgroup$ – Magma Sep 22 at 12:52
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    $\begingroup$ Because right now, it's completely unclear from your post whether it's a cake-cutting problem or a dissection problem, and if cake-cutting problem, then whether it should be proportional, envy-free, coalition-resistant, equitable or exact, but if dissection problem, whether it's 2D or 3D, whether the cuts must be straight or connected or both, whether the goal is equal area or volume or shape, and whether the goal should be reached or only approximated as well as possible (in which case you should specify how to measure the quality of an approximation). $\endgroup$ – Magma Sep 22 at 13:25
  • $\begingroup$ sorry no cake cutting tag here. The rule is simple 4 cuts to make 4 equal or same object (piece of cake) $\endgroup$ – TSLF Sep 22 at 14:35
  • $\begingroup$ I've edited my solution. $\endgroup$ – Magma Sep 22 at 15:31
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Maybe I've missed something, but:

Judge A makes an arbitrary cut from the center to the edge.

Then

B makes a cut that B believes cuts the cake into a 3:2 split. A decides whether the split is {B C D} {X A} or {A C D} {X B}, where X is the chef. Thus, B is motivated to make a true 3:2 cut so that A's decision cannot change B's fate.

Then,

C cuts the bigger part into what he thinks is a 2:1 split. {B or A - whichever is not paired with X} decides whether C gets the single slice or whether {B or A} gets the single slice. C is motivated to make a true 2:1 split, since he or she will get whatever the worst outcome is if not.

Finally,

Judge D aligns the remaining 2-slice pieces in such a way that he or she can (to the best of his or her ability) cut them into four equal slices with one cut. There are 2 judges left who do not have a slice, are not D, and are not X - one of them decides which slice D gets. The remaining slices are assigned randomly. D is motivated to make 4 equal slices because D will get the smallest one.

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  • $\begingroup$ The initial cut by A seems unnecessary. B, C and D can all accomplish their goals without a radius. $\endgroup$ – hdsdv Sep 22 at 10:39
  • $\begingroup$ Judge A's radius cut is necessary for judge D to align the two 2/5 slices. But the aligning task and cutting twice as much as the other can be an issue of effort unfairness. Who would want to cut last if he can be the first cutter? $\endgroup$ – TSLF Sep 22 at 12:11

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