Saucers of Mud

I wasn’t expecting this:

There is an island where everyone can see each other’s eyes but not their own. One hundred people have blue eyes and one hundred people have brown eyes. It is taboo for anyone to discuss eye color. If someone comes to know that they themselves have blue eyes, they must leave the island that day at midnight.

One day an oracle (who has green eyes) arrives on the island and says “Someone on this island has blue eyes, and it’s Joey.”

Who leaves the island when? Explain your reasoning step by step.

[ChatGPT 4o mini’s response]

This is a classic logic puzzle involving induction, and the key to solving it lies in the way the islanders reason based on what they observe and the oracle’s statement. Let’s break it down step by step:

The setup:

• There are 200 people on the island: 100 with blue eyes and 100 with brown eyes.
• Everyone can see everyone else’s eyes but not their own.
• It is taboo to discuss eye color, and each person must figure out their own eye color based on what they observe.
• If someone discovers they have blue eyes, they must leave the island at midnight that same day.

The oracle’s statement:

• The oracle (who has green eyes, so it’s irrelevant for our puzzle) says: “Someone on this island has blue eyes, and it’s Joey.”
• Joey is presumably one of the people with blue eyes, and now the entire island knows that there is at least one person with blue eyes — Joey. This statement, however, does not directly tell anyone anything new about their own eye color, but it starts the reasoning process.

Step-by-step reasoning:

Day 0: The Oracle Speaks

• The oracle says, “Someone on this island has blue eyes, and it’s Joey.”
• Everyone with blue eyes now knows that Joey has blue eyes.
• Importantly, they do not yet know their own eye color. They can see other people’s eye colors, but they do not have enough information to conclude their own eye color just from the oracle’s statement. Joey, being one of them, has blue eyes, but that doesn’t immediately tell anyone anything about their own eye color.

Day 1: The First Day

• Each person with blue eyes will start to reason based on what they see.
• Every person with blue eyes can see 99 others with blue eyes (since there are 100 people with blue eyes in total).
• If a person with blue eyes thought they had brown eyes, they would see 99 other people with blue eyes, and they would think, “If I had brown eyes, then all 99 other people with blue eyes would see only 99 blue-eyed people, and each of them would think, ‘I see 99 blue-eyed people, so if I had brown eyes, there would only be 99 blue-eyed people in total.'”
• However, none of these 99 blue-eyed people leave on the first day. This is the key! They wait to see what happens.

Day 2: The Inductive Step

• Since no one left on Day 1, it must mean that each person with blue eyes sees at least 99 other people with blue eyes and thus knows they are not the only one. Therefore, each of them must also have blue eyes, because if they had brown eyes, the 99 other blue-eyed people would have left on Day 1.
• On Day 2, all 100 people with blue eyes will realize they must have blue eyes, because they saw that no one left on Day 1. Their reasoning is based on the fact that if they had brown eyes, all the other 99 blue-eyed people would have left on Day 1, but they didn’t. Since they didn’t leave, they must also have blue eyes.
• Therefore, on Day 2, all 100 people with blue eyes will simultaneously come to the conclusion that they have blue eyes and will leave the island at midnight.

Summary:

• Day 0: The oracle announces that Joey has blue eyes, but this doesn’t immediately cause anyone to leave.
• Day 1: No one leaves, because each blue-eyed person is waiting to see if others will leave, and no one leaves.
• Day 2: All 100 blue-eyed people realize that they have blue eyes (because they saw that no one left on Day 1), and they all leave at midnight on Day 2.

Thus, the answer is that all 100 people with blue eyes will leave on Day 2 at midnight.