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How Old Are the Guys - Answer

Boyd must be a liar because only a liar would say he is 81 (a perfect square). Therefore, his age must be 64 (8 x 8) or under 50.

It then follows that Chester is also a liar because Boyd can't be 61.

Dwight's second statement indicates that he is truthful. Therefore, Anton is 57.

Anton is 50 or older, therefore he tells the truth. Eugene is then 27 and truthful (3 x 3 x 3).

From Eugene's first statement, Boyd must be 64. From his second statement, Dwight is 27 or 50 or older. But since Chester is a lair, he is under 50 or 64 or 81. The only consistent option is that Chester is 81 and Dwight is 51.