The Isle of Logicians and The Lying Guest

by Theo Hupkens

On an island, far from here, there live intelligent creatures. These creatures are called Logicians because they always reason strictly logically. Logicians either have blue eyesor green eyes . They are quite happy with their lives, although they have a small problem: if a blue-eyed Logician ever discovers the color of his eyes he has to jump off the cliff at midnight. Fortunately a Logician does not know the color of his own eyes. Logicians can see the color of the eyes of all other Logicians, but they are not allowed to do or say anything that could give another Logician the chance to discover his eye color. Nobody can see the color of his eyes from the mirror image of the ocean, because the color of the ocean is a deep greenish blue.

One day a guestarrives at the island. When leaving the island, he says: "I have seen at least one blue-eyed Logician." Obviously the guest did not know anything about the problem with blue eyes.

It is now sure that one night every blue-eyed Logician will jump off the cliff.

Here's the reason:

Suppose there is only one blue-eyed Logician. He knows all other Logicians have green eyes, because he can see that. So he has to conclude that he is the one with the blue eyes. Now say there are two blue-eyed Logicians. Both blue-eyed Logicians see one Logician having blue eyes. So they conclude they do not know whether or not they have blue eyes. But at midnight nobody jumps. Now they start reasoning: if that other blue-eyed Logician they see did not jump that must mean that he saw another Logician having blue eyes and that other Logician must be me. So the second day both Logicians know they have blue eyes.
What if there are even more Logicians (say three)? The third day - if nobody jumped the night of the second day - everybody knows there must be at least 3 Logicians have blue eyes and so on. So if X Logicians have blue eyes, then the night after the X-th day they will all jump off the cliff.

A lying guest

Suppose the guest lies about the number of blue-eyed Logicians he has seen. If he mentions a number that is way too large, all Logicians will know immediately that he lies and so they still don't know the colors of their own eyes. They will wait until midnight and than force the guest to jump off the cliff. However, if the guest mentions a number that is exactly one higher than the number of blue-eyed Logicians...

A devilish dilemma

.. then all green-eyed Logicians conclude that they have blue eyes and will jump off the cliff. All blue-eyed Logicians know that the guest is lying, but they do not know if the green-eyed Logicians know this too. There is a chance (say 50%) that they have green eyes (WE know they have blue eyes, but they do not know this). If so, all other green-eyed Logicians will know the guest is lying and nothing will happen. However if they have blue eyes, the green-eyed Logicians will think they have blue eyes and jump off the cliff. If that happens then the blue-eyed Logicians know their eye-color and will have to jump too. But they are not allowed to do or say anything that could help any Logician discover his eye-color. So what should they do? Do nothing and take a risk of 50% that the Logicians become extinct, or violate the rule? A devilish dilemma.

Many thanks to Iris