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BGonline.org Forums
100 islanders logic puzzle - what about a silent oracle?
Posted By: Tom Keith In Response To: 100 islanders logic puzzle - what about a silent oracle? (Ian Shaw)
Date: Monday, 24 August 2009, at 3:59 p.m.
Look at the situation from each of the 10 people who have blue eyes. Those 10 see only 9 others with blue eyes and are hoping that's all there is. What they know for sure is there are either 9 or 10 people with blue eyes. Suppose it is only 9. Where does that take us (still from a blue-eyed person's point of view)?
1. Assume 9 have blue eyes. The 9 blue-eyed people would each see 8 others with blue eyes. Those 8 would know that the true number of blue eyes is either 9 or 8. Suppose it is 8.
2. Assume 8 have blue eyes. The 8 blue-eyed people would each see 7 others with blue eyes. Those 7 would know that the true number of blue eyes is either 8 or 7. Suppose it is 7.
3. Assume 7 have blue eyes. ...
4. Assume 6 have blue eyes. ...
5. Assume 5 have blue eyes. ...
6. Assume 4 have blue eyes. ...
7. Assume 3 have blue eyes. ...
8. Assume 2 have blue eyes. The 2 blue-eyed people would each see 1 other with blue eyes. That 1 person would know that the true number of blue eyes is either 2 or 1. Suppose it is 1.
9. Assume 1 has blue eyes. The 1 blue-eyed person would see no one else with blue eyes. He would know that the true number of blue eyes is either 1 or 0. ... until the Oracle makes his statement. Now the 1 blue-eyed person would know he is the one. And the next morning (Day 1), he would kill himself.
So, by the afternoon of Day 1, when nothing happens, it becomes clear that Assumption 9 is wrong.
Now the 2 with blue eyes under Assumption 8 know the truth of their eye color because their assumption (hope) that there was only 1 blue-eyed person has been proven wrong. The next morning those 2 must kill themselves. So, by the afternoon of Day 2, when nothing happens, it becomes clear that Assumption 8 is wrong.
Now the 3 with blue eyes under Assumption 7 know the truth of their eye color because their assumption (hope) that there was only 2 blue-eyed people has been proven wrong. The next morning those 3 must kill themselves. So, by the afternoon of Day 3, when nothing happens, it becomes clear that Assumption 7 is wrong.
... By the afternoon of Day 4, when nothing happens, it becomes clear that Assumption 6 is wrong.
... By the afternoon of Day 5, when nothing happens, it becomes clear that Assumption 5 is wrong.
... By the afternoon of Day 6, when nothing happens, it becomes clear that Assumption 4 is wrong.
... By the afternoon of Day 7, when nothing happens, it becomes clear that Assumption 3 is wrong.
Now the 8 with blue eyes under Assumption 2 know the truth of their eye color because their assumption (hope) that there was only 7 blue-eyed people has been proven wrong. The next morning those 8 must kill themselves. So, by the afternoon of Day 8, when nothing happens, it becomes clear that Assumption 2 is wrong.
Now the 9 with blue eyes under Assumption 1 know the truth of their eye color because their assumption (hope) that there was only 8 blue-eyed people has been proven wrong. The next morning those 9 must kill themselves. So, by the afternoon of Day 9, when nothing happens, it becomes clear that Assumption 1 is wrong.
Now the 10 with blue eyes know the truth of their eye color because their assumption (hope) that there was only 9 blue-eyed people has been proven wrong. The next morning those 10 must kill themselves. And they do.
Each of the assumptions is a hypothetical assumption. But even hypothetical assumptions have value if we can show that they lead to a contradiction. When we know an assumption is false, we can use that information as part of a larger argument.
Initially it seems like useless information because even the green eyes know the minimum number of greens is at least 9. But, day by day, the argument allows us to increase our knowledge about what the minimum number of green eyes is. And eventually the green eyes learn the truth of their situation.
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