this post was submitted on 04 Feb 2024
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The first person decides where the first apple would be cut, such that he would be equally happy with either taking the larger portion or dividing up what remains.
The second person either says he wants that larger portion, in which case he's done, or he says he wants to divide up what remains, in which case the first person takes the larger portion and is done.
The third person decides where the second apple would be cut.
Either the first or second person (depending on the outcome of the second step) picks either the larger portion of the second apple or the two smaller portions.
The third person takes what remains.
The two apples are cut and their pieces distributed.
This doesn't guarantee a solution, since it's possible that the third person would have wanted the larger portion of the first apple. It only works if we assume that the first person, when given the motivation to cut fairly, does so with high precision. Otherwise the first person can, by screwing himself over, also screw over the third person.
Neat, but you failed since it requires more than one slice
No, you line the apples up under the knife and then slice once. Each apple can be moved relative to the knife and independently of the other apple.
I should have been more specific when I said things like
because what I meant by that is "permanently claims the larger portion as his own". The apple is not cut (once) and no one actually gets their pieces in their hands until step 6.