Wednesday, February 16, 2011

Logical puzzle of blue-eyed islanders

An island has 1001 people with assorted eye colours (having different eye colours). No one knows their own eye colour, but can see the eyes of everyone else (rest 1000 people), also they are highly logical people, so if it is possible to derive the eye colour, they will know it. Out of 1001,

  • 100 have blue-coloured eyes
  • 900 have red-coloured eyes
  • 1, the leader has Green-coloured eyes

The rule is that, If a person come to know his own eye colour today, then he will leave the island the next morning. No one ever speaks to anyone or give any hint about the eye colours. The Guru, however, is allowed to speak one sentence in his life-time.

One fine day, the Guru exercised his right to speak, and said:

“I can see some blue-eyed men”

What will be the repercussions of his statement?

1 comment:

Shashank Mani said...

Its easy to drive the solution using mathematical induction.

If there is only 1 blue-eyed person on the island, then he will obviously leave the island the next morning.

If there are 2 blue-eyed people, Each of them will see only one blue-eyed person (other than him) and will expect him to leave the island the next morning. But when the other person does not leave next morning, he will ask to himself “If I don’t have blue eyes than the other person would have left”. So on the 2nd day both of them will leave the island.

If there are 3 blue-eyed people, then each of them is able to see 2 blue-eyed persons and will expect them to leave after 2 days. But when they will not leave on the 2nd day, he will conclude that he also has blue eyes. Hence, all the 3 will leave on the third day.

Similarly, Since there are 100 blue-eyed persons, each of them is able to see 99 blue-eyed person (other than himself) and will expect them to leave the island on the 99th day. When they does not leave each of them will conclude they have blue eyes, and hence all of them will leave on the 100th day.