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It seems this will go on forever: if A and B aren't wearing the same colour, neither of them will ever find out each other's colour and C and D will just sit there snickering at them. Either I've made a logical error or there's a mistake in the OP.

The same argument goes on. No new information is gained by anyone.

The same argument goes on. No new information is gained by anyone.

In Cases 2 and 3, A and B never find out their colours; C and D will just sit there snickering at them forever. But we're told in the OP that eventually "only one of them failed to name the color of his own hat", so we can deduce the following two pieces of information:

• the first two logicians to speak have the same colour of hat, as do the last two
• the third logician is the last one not to know his own colour.

This is the final answer.

B is still wearing black, and argues as follows. If B were wearing red, then A would have seen 1 white, 1 black, 1 red and not known his own colour; B would have said what he did; C would have seen 2 white, 1 red and known he was wearing black; D would have seen 1 white, 1 black, 1 red and known he was wearing white (if D was wearing black, A would have seen 2 black, 1 red and known his own colour); and finally A would still not have known his colour (if A was wearing black, everything would have happened the same way it did!). So B says, "I don't know."

A is still wearing white, and argues as follows. If A were wearing red, then as above, B, C, D, and A (second time round) would still have responded as they did, and then B (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if B were wearing white, then we have RWBW and everything goes through as before!) So A says "I don't know" for the third time.

B is still wearing black and argues as follows. If B were wearing red, then as above, the first and second rounds would have gone as they did, and then A (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if A were wearing black, then we have BRWB and everything goes through as before!). So B says "I don't know" for the third time.

B is still wearing black, and argues as follows. If B were wearing red, then A would have seen 1 white, 1 black, 1 red and not known his own colour; B would have said what he did; C would have seen 2 white, 1 red and known he was wearing black; D would have seen 1 white, 1 black, 1 red and known he was wearing white (if D was wearing black, A would have seen 2 black, 1 red and known his own colour); and finally A would still not have known his colour (if A was wearing black, everything would have happened the same way it did!). So B says, "I don't know."

C and D still know their colours.

A is still wearing white, and argues as follows. If A were wearing red, then as above, B, C, D, and A (second time round) would still have responded as they did, and then B (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if B were wearing white, then we have RWBW and everything goes through as before!) So A says "I don't know" for the third time.

B is still wearing black and argues as follows. If B were wearing red, then as above, the first and second rounds would have gone as they did, and then A (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if A were wearing black, then we have BRBW and everything goes through as before!). So B says "I don't know" for the third time.

It seems this will go on forever: if A and B aren't wearing the same colour, neither of them will ever find out each other's colour and C and D will just sit there snickering at them. Either I've made a logical error or there's a mistake in the OP.

B is still wearing black, and argues as follows. If B were wearing red, then A would have seen 1 white, 1 black, 1 red and not known his own colour; B would have said what he did; C would have seen 2 white, 1 red and known he was wearing black; D would have seen 1 white, 1 black, 1 red and known he was wearing white (if D was wearing black, A would have seen 2 black, 1 red and known his own colour); and finally A would still not have known his colour (if A was wearing black, everything would have happened the same way it did!). So B says, "I don't know."

C and D still know their colours.

A is still wearing white, and argues as follows. If A were wearing red, then as above, B, C, D, and A (second time round) would still have responded as they did, and then B (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if B were wearing white, then we have RWWB and everything goes through as before!) So A says "I don't know" for the third time.

B is still wearing black, and argues as follows. If B were wearing red, then A would have seen 1 white, 1 black, 1 red and not known his own colour; B would have said what he did; C would have seen 2 white, 1 red and known he was wearing black; D would have seen 1 white, 1 black, 1 red and known he was wearing white (if D was wearing black, A would have seen 2 black, 1 red and known his own colour); and finally A would still not have known his colour (if A was wearing black, everything would have happened the same way it did!). So B says, "I don't know."

A is still wearing white, and argues as follows. If A were wearing red, then as above, B, C, D, and A (second time round) would still have responded as they did, and then B (seeing 1 white, 1 black, 1 red) would still not have known his own colour (if B were wearing white, then we have RWBW and everything goes through as before!) So A says "I don't know" for the third time.