100 Blue-eyed Islanders Logic Puzzle with Solution

Submitted by Atanu Chaudhuri on Mon, 02/12/2024 - 03:38

100 Blue-eyed Islanders Logic Puzzle

The Classic Logic Puzzle of 100 Blue-eyed Islanders: Step by Step Easy to Follow Solution

One of the 100 blue-eyed logicians imprisoned in an island had to correctly tell his eye color for all to be free. But none can see his own eye color. Things changed when a visitor announced, "At least one of you has Blue eyes."

The 100 Blue-eyed Islanders Logic Puzzle

100 perfect logicians, all with blue eyes, were imprisoned by a monster on an island with the strict rule of no communication between themselves. The monster, however, provided them a slim hope.

He told them, "After exactly one month, I will call you one by one at midnight and ask color of your eyes. If one of you can answer correctly, all can leave the island the same night. Remember, each of you would have the option to keep silent when not sure. If you give me a wrong answer, I will throw you into the sea."

None could see color of own eyes though they could see color of eyes of all others. They remained on the island without any hope.

Meanwhile, before the month was over, a visitor was allowed on the island by the monster on one condition, he must not tell any prisoner color of own eyes. Without violating the condition, the sympathetic visitor told the logicians, "At least one of you has blue eyes", and left.

When questioning time came, the monster called the first prisoner at midnight and asked color of his eyes. What happened next? Could any of the prisoners leave the island?

Time to solve: 20 minutes.

Hint: Think like a logician. Start simple.

Solution to the 100 Blue-eyed Islanders Logic Puzzle: Step by Reasoned Step

We are not perfect logicians, but we can think logically and also use common sense strategies.

First step is to define the problem precisely.

Problem definition:

There are 100 blue-eyed islanders, each a perfect logician.
Each islander could see the eye color of all others but not their own.
They could not communicate their observations to anyone.
A visitor announced, "At least one of you has blue eyes."
All 100 islanders would leave the island the night one of them deduced his or her own eye color.

Take up the simplest case. Assume instead of 100, there are two blue-eyed prisoners.

Logic Analysis for the Simplest Case of Two Blue-eyed Prisoners

Each blue-eyed person could see one other blue-eyed islander.

Each of them would think: "If I don't have blue eyes, the other friend would see no blue-eyed person, would be sure of own eye color as blue, answer correctly and should leave the first night."
When no one left the first night, the second prisoner reasoned, "The other friend can see one Blue-eyed person just like I see, but could not leave simply because, to him I could be the only one with Blue eyes and not him. As he saw my Blue eyes, I am sure of my eye color now. I being the last to question, no one is left after me to confuse me."
The second prisoner answers correctly the second night and both left the island the same night.

Use this insight for next steps in logic analysis and decision making.

The Case of 100 Blue-eyed Islanders Puzzle: Solution

The first night, when questioned about his (or her) eye color, the prisoner knew there were 99 other islanders with Blue eyes, but could not be sure of his (or her) own eye color.

None left the first night.

Same pattern repeated every night till the 98th night. On the 98th night also, the prisoner on the questioning dock wasn't sure and kept silent.

None of the first 98 prisoners left the island or was thrown into the sea. The situation for the 98th prisoner was same as the first night prisoner:

Each could see 99 persons with Blue eyes, but could not be sure of own eye color.

What Happens on the 99th Night?

The 99th prisoner knew previous 98 prisoners could not be sure as each could see 99 other prisoners with Blue eyes but could not be sure of own eye color. The 99th prisoner also saw 99 others with Blue eyes, but even with immensely developed reasoning skill, could not come to a positive conclusion and kept silent.

The 99th prisoner also didn't leave the island.

Next morning when the prisoners found the 99th prisoner also unable to leave the island, the last to be questioned 100th prisoner reasoned:

Just as I could see 99 others with Blue eyes, each of my previous friends also could see 99 others with Blue eyes.
That means: I was among the 99 persons with Blue eyes my previous friend saw, because none else is left to be questioned.
Eureka! I am now sure of my eye color as Blue.

The Freedom at 100th Midnight for All 100 Imprisoned Blue-eyed Islanders

The last of the 100 prisoners answered correctly on the 100th midnight, and

All islanders, free at last, left their island prison the same night.

Sum-up and Generalization

In this beautifully constructed logic puzzle, the visitor's statement set in motion a series of deductive reasoning and corresponding actions among the prisoner islanders. For 2, or 100, or 1000 islanders, the logical reasoning and actions of all islanders would remain same.

All but the last person reasons:

The visitor said there is at least one Blue-eyed person, but didn't mention any other eye color. I see all others with Blue eyes. I might be the only one with an eye color not Blue and I don't know what my eye color can be.

Reasoning the same way, all but the last one kept silent when questioned, and didn't leave the prison island.

The last prisoner found all others unsure, and reasoned,

Like me, all could see only Blue-eyed persons, but thought he or she might be the only one with a different eye color and couldn't leave.

This implies, I must be among the Blue-eyed persons my immediate predecessor saw, as no more person is left after me to be questioned.

My eye color must then be Blue.