Logic Problem!?

Question

In ancient times, four mathematicians discovered this game of logic. The four men were Hermos, Xentos, Appino and Tenedos. Hermos seated the other three men one behind the other, so that Xentos saw Appino and Tenedos, and Appino saw only Tenedos. Tenedos, who was in the front, saw neither Appino or Xentos.
Hermos had five hats which he showed to the three men. Three of the hats were grey, and two were white. Then Hermos put a hat in each man and tossed the remaining two aside.
“what color is your hat Xentos?” asked Hermos. Xentos did not know. Appino did not know the color of his hat either when asked. Tenedos, who could not see any hats at all , gave the right answer when he was asked the color of his hat, what was the color of his hat? How did he figure it out?

PLEASE HELP ME!

Answer 1

If X saw two white hats in front of him (there are only two), then he would have known that his hat was one of the three remaining gray hats. But this is not the case because he saw either one gray and one white... or he saw two grays... in front of him.

Then A then knows he is wearing a gray or white hat... but that either he and T were wearing different colors or were wearing both gray. If A sees a white in front of him, he knows his hat is gray. But this is not the case, so he still doesnt know which color his hat is.

Then T concludes that his hat is gray through process of elimination.


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See. There are only two white hats... so if X sees WW in front of him, he knows his is gray. But this doesnt happen... so he doenst see WW in front of him.

There are three possibilities that he does see.
GG
GW
WG.

If, then, A sees W in front, he knows his is gray. But this is not hte case...

So, the only possibilites left are the two with G in front

Answer 2

Xentos' couldn't have seen two white hats, otherwise he would know that he had gray.

Appino hearing this looks forward, if he saw a white hat ahead on Tenedos, he would know he must have gray because Xentos didn't see two white, but he doesn't say.

Tenedos now knows that he can't have white, and must have gray.

Answer 3

if T and A had W hats X would guess he has a G had, so T and W do not have both W hats. knowing that A would have guessed he had a G hat if he was seeing a W hat on T. so T has a G hat.