Friday, 20 May 2011

The one about three intelligent men

A friend asked me the other day if I knew the riddle about three men who had to compete in working out whether a black or a white patch had been stuck on their forehead. I remembered it, vaguely, and told him I would look it up and tell him how it went.

But I then found that it was not so easy to google something that contained no distinctive words. I tried various combinations, patch+ three+ black+ intelligent and so on, and eventually found a number of variations of this old puzzle, but none of them explained it well.

Finally I remembered that I had first seen it in a marvellous book my late sister gave me for my eleventh birthday: "The Complete Home Entertainer". It has a particularly fine section called Fun with Matches, Coins and String and this kept me happily occupied for a very long time, but I finally found  the puzzle I wanted under Brain Twisters. I could have scanned it and posted it as a picture but it was on more than one page so that would have been tedious. Instead I typed out a prĂ©cis of the question and answer, and here it is:

The three most intelligent applicants for a job are given a test.

The interviewer tells them to shut their eyes and he will then stick either a black or a white patch on the forehead of each of them. He says "When I ask you to open your eyes, anyone who can see a black patch must raise his hand. If he can deduce whether his own patch is black or white he must lower his hand".

He then stuck a black patch on each forehead and told them to open their eyes. Three hands shot up and almost immediately one came down.

"Yes? What colour is your patch?"

"Black, sir"

"Correct! The job is yours!"

How had he worked it out?

Call the candidates X, Y and Z. X says to himself:

"Suppose my patch is white. Then Y has his hand up because he can see Z's black patch."

But also he sees Z with his hand up and, being intelligent, he asks himself: "Why is Z's hand raised? Answer: because he can also can see a black patch which must (Y will argue) be mine. IF MY PATCH WERE WHITE, the intelligent Y (and the intellgent Z) would work this out in no time. But neither has done so, since both still have their hands up. Therefore the problem is not as simple as that. Therefore my patch is black".

So X got the job.

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