Suicide Island

There is an island, upon which a tribe resides. The residents either have blue eyes, or brown eyes. Yet, it is taboo to talk about eye color in any way. Thus, one resident can see the eye colors of all other residents, but has no way of discovering his own (there are no reflective surfaces).  If a resident does discover his or her own eye color, then he or she must commit ritual suicide, at 12-noon, the following day, in the village square, for all to witness.

Note: Everyone on the island is perfectly logical. On this island, there are 100-blue-eyed people, and an unknown number of brown-eyed people.

On day zero, a traveler comes to the island, and says aloud for all residents to hear:

“On this island, there is at least one person with blue eyes.”

Having said that, he departs; on day 100, all 100-blue-eyed people commit suicide simultaneously. Explain the logic behind these suicides.

Corollary: Why is the traveler important?

Identification of the problem

One crux to this problem, could be the fact that the people on the island are perfectly logical. If the people weren’t logical, they wouldn’t be able to make conclusions about other people’s eye colors, or, about their own eye color. This would make the traveler completely irrelevant.

The traveler is important, because he or she starts the count, and gets things rolling. If the traveler hadn’t come to the island, and made the statement about at least one person having blue eyes, none of the people on the island would have been able to figure out their own eye color, and they would all still be living happily ever after.

If either of the two situations above weren’t true, or didn’t happen; everyone on the island would still be alive. The people on the island only died because they were both perfectly logical, and, the traveler came.

Analysis

This problem can be solved using multiple cases, with smaller, more manageable numbers. It’s hard to consider the entire problem at once, but if we break the problem down, it becomes much easier to understand.

Case 1: To start off, let’s assume that there’s only one blue-eyed person on the island. Let’s name this person, Mark. This will make the problem a little easier to understand. The traveler comes to the island, and states that he sees at least one person on the island with blue eyes. Mark, the person with blue eyes, looks around the island, and notices that everyone else on the island has brown eyes. From this, Mark concludes that he must have blue eyes, and, therefore, kills himself at noon the next day.

Mark can make this conclusion because the traveler stated there was at least one person with blue eyes on the island; however, he can tell that everyone else on the island has brown eyes, by looking at them.

The other people on the island notice that Mark killed himself, and therefore, must have figured out his eye color. When the other people on the island see, that Mark has blue eyes, they put that together with what the traveler stated. Since they are perfectly logical, they also realize that this means Mark must not have seen anyone else on the island with blue eyes.

This information leads everyone else on the island to conclude that they have brown eyes. At noon, the next day, everyone remaining on the island kills themselves.

Case 2: In case two, we will have two blue-eyed people. We’ll name them Mark, and Bob. After the traveler comes to the island and makes his statement, both Mark and Bob look around the island at everyone else.

Mark looks at Bob, and notices that Bob has blue eyes, but everyone else on the island has brown eyes. Mark assumes that Bob will kill himself at noon the next day, but when Bob fails to kill himself, Mark deduces that Bob must see someone else on the island with blue eyes. Because Mark is perfectly logical, he knows that this means he must have blue eyes.

At the same time, Bob notices that Mark has blue eyes, but everyone else on the island has brown eyes. Bob also assumes that Mark will kill himself at noon the next day. When Mark fails to kill himself, Bob concludes that Mark must see someone else on the island with blue eyes. Bob realizes that he must be the other person with blue eyes, and both men kill themselves at noon the next day.

When the rest of the people living on the island notice Mark and Bob kill themselves, on day 2, they realize that they must all have brown eyes, and, therefore, kill themselves at noon on day 3.

Case 3: We’ll use three people in this case. At this point we can continue to use the previous case to help us solve the problem. We already know that Mark was waiting for Bob to kill himself, and vice versa. Let’s add the third person into the problem. We’ll name him, Jim.

In this case, Jim is also waiting for Mark and Bob to kill themselves; however, just as in previous cases, when Mark and Bob fail to kill themselves on day 2, Jim realizes that there must be another person on the island with blue eyes. He can conclude this because neither Mark, nor Bob, have killed themselves, and, therefore, they must see another person on the island with blue eyes.

Just as in previous cases, Jim looks around, and notices that everyone else on the island has brown eyes, and, therefore, he must be the other person with blue eyes. There are a few things that are happening here:

  1. Mark is waiting for Bob, and Jim, to commit suicide
  2. Bob is waiting for Mark, and Jim, to commit suicide
  3. Jim is waiting for Mark, and Bob, to commit suicide

Because none of the men are killing themselves, they conclude that they all have blue eyes, and, therefore, they all kill themselves at noon on day 3. When the three men kill themselves, the rest of the people living on the island instantly know that they all have brown eyes. After learning their eye color, the rest of the people on the island commit suicide at noon the next day.

Case N: From the previous cases, you should be able to notice the patterning that is arising. This information can be used to construct an algorithm to help solve the larger problem.

In case 1, we had one blue-eyed person. That person killed themselves on day one, and everyone else killed themselves on day two. In case 2, the two-blue-eyed people killed themselves on day two, and everyone else killed themselves on day three. Similarly, in case 3, the three-blue-eyed people killed themselves on day three, and everyone else, on day four.

We’ve used a minimum of three cases to prove that a pattern exists. Now that we know there is a pattern, we can solve the finial problem. We could list out every case, up until day one hundred, or, we could take a more logical approach to reach the solution.

The case-number itself, directly reflects the number of blue-eyed people in the problem. Case 1 had one blue-eyed person; Case 2 had two blue-eyed people, and so on. This means we can substitute the letter N (meaning number of blue-eyed people, in this case) for the number of blue-eyed people in our problem.

Case (N): We can break Case N down into two sub-cases: N0, will be the day in which all the blue-eyed people will die, and N1, will be the day in which everyone else on the island dies. If we substitute N, for the number of blue-eyed people, we can figure out when everyone is going to die.

The sum of N, plus its sub-case, will provide the answer we need:

  • Case N = {x | x ϵ N and x is the number of blue-eyed people}.
  • Case N+0 = Blue-eyed people die.
  • Case N+1 = Brown-eyed people die.

Heuristics

  • We used multiple cases to help break a large problem down, into more manageable pieces.
  • We constructed a simple algorithm to help solve the problem.

Looking Back

I think this is a silly problem. First, it states that everyone is perfectly logical. Why would anyone that is perfectly logical, kill themselves over eye color? If the people really were perfectly logical, they would use their logic to put together a raft, and sail away from the island, while waving their fists, and telling the rest of the inhabitants to “stuff” their crazy laws.

But on the bright side of things, if you happened to be lucky enough to have brown eyes, you get to live one day longer then everyone that had blue eyes. On the down side, you had to watch 100 blue-eyed people commit suicide, which is probably the real reason, why all the brown-eyed people killed themselves, the following day. Either that, or they all died from heat stroke, while attempting to dig 100 graves.

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