{"id":521,"date":"2016-06-11T02:35:18","date_gmt":"2016-06-11T02:35:18","guid":{"rendered":"http:\/\/www.laurentlessard.com\/bookproofs\/?p=521"},"modified":"2016-09-03T18:52:05","modified_gmt":"2016-09-03T23:52:05","slug":"the-blue-eyed-islanders","status":"publish","type":"post","link":"https:\/\/laurentlessard.com\/bookproofs\/the-blue-eyed-islanders\/","title":{"rendered":"The blue-eyed islanders"},"content":{"rendered":"

Today’s Riddler<\/a> problem is another classic. The current incarnation of the puzzle is about error-prone mathematicians, while the classic version is about blue-eyed islanders<\/a>.<\/p>\n

A university has 10 mathematicians, each one so proud that, if she learns that she made a mistake in a paper, no matter how long ago the mistake was made, she resigns the next Friday. To avoid resignations, when one of them detects a mistake in the work of another, she tells everyone else but doesn\u2019t inform the mistake-maker. All of them have made mistakes, so each one thinks only she is perfect. One Wednesday, a super-mathematician, whom all respect and believe, comes to visit. She looks at all the papers and says: “Someone here has made a mistake.”<\/p>\n

What happens then? Why?<\/em><\/p><\/blockquote>\n

Here is the solution:
\n[Show Solution]<\/a><\/p>\n

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On the tenth Friday, all mathematicians will resign. To understand why this happens, we’ll use a recursive approach similar to what we did for the puzzle of the pirate booty<\/a>.<\/p>\n

For simplicity, we’ll say a mathematician is flawed<\/em> if they made a mistake, and perfect<\/em> otherwise. Let’s look at the world through the eyes of Alice, one of the ten mathematicians. Alice doesn’t know whether she is flawed or not. She does, however, know how many of her colleagues are flawed.<\/p>\n

According to the problem statement, all the mathematicians are flawed. So Alice sees that all nine of her colleagues are flawed. Let’s solve a simpler version of the problem first. What if none of Alice’s colleagues were flawed? In this case, the news that “someone here has made a mistake” can only mean one thing: that “someone” must be Alice! So Alice will resign on Friday.<\/p>\n

What if exactly one of Alice’s colleagues is flawed? Two possibilities:<\/p>\n