1) Suppose Euclid is the truth-teller.
Then it must be that Euclid was telling the truth when he said he's not the greatest one.
We also know that Plato is not the greatest one, because Socrates, who is a liar according to our assumption, says that he is.
Note that Plato asserts that Socrates is wrong. Since our assumption indicates that Plato's claim is a lie, it must be that Socrates is right. This, however, isn't the case, because it violates our assumption that Socrates is not telling the truth.
This contradiction leads to the conclusion that Euclid is not the truth-teller.
2) Suppose Socrates is the truth-teller.
Since Euclid is lying, we know that his claim is wrong and thus Euclid is the greatest.
Remember that we're assuming that Socrates is telling the truth. So his claim that Plato is the greatest must be true.
However, it cannot be that Plato and Euclid are both the greatest.
The assumption that Socrates is the truth-teller also leads to a contradiction.
3) Finally, suppose Plato is the truth-teller.
W know that this has to be the case, because the other two assumptions didn't work.
Since Euclid is lying, his claim that he's not the greatest is wrong -- in other words, Euclid is the greatest.
Since Socrates is also lying, we know that Plato is not the greatest.
Plato is telling the truth, and his claim that Socrates is wrong is correct.
We find no contradiction here, and Euclid is the greatest.
Answer: Euclid
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