Ian (lovingboth) wrote,

This is the anti-death penalty bleeding heart liberal BBC for you...

Three people, one of whom is blind, are each given a hat from a supply of three white and two red ones. Without anyone being able to see what hat they have, can any of them deduce what colour they have?

In the BBC's Today programme version they're prisoners and will all be given their freedom if any can state it correctly. As stated by them the answer is trivial: if all of them say 'white', at least one of them must be correct and there's no punishment if anyone is wrong! But you're expected to work out how the third, blind, one can be correct after the first two say that they don't know.

OK, what are the possibilities? Normally with three either/ors, they'd be 2x2x2 = eight of them, but they can't all be red, so that leaves seven:

W W W / W W R / W R W / W R R / R W W / R W R / R R W / R R R

The point of this puzzle is that we don't need to know which of the seven it is, just what the last letter is.

When the first one says they don't know, we know that the two they can see are not both wearing red, otherwise they'd know they had white:

W W W / W W R / W R W / W R R / R W W / R W R / R R W / R R R

Obviously, the second person must also be able to see that the other two aren't wearing red..

W W W / W W R / W R W / W R R / R W W / R W R / R R W / R R R

So that means four out of the remaining five possibilities have the blind one wearing white, but how do you eliminate the WWR combo to make it four out of four?

Imagine that it is WWR. The first one sees W&R. Hmm, their hat could be either. If the second one also sees W&R, they could announce that they have a white hat - if they had a red one, then the first person would have seen two reds and thus known their hat colour. So if the first two can't answer, it can't be WWR:

W W W / W W R / W R W / W R R / R W W / R W R / R R W / R R R.

.. so without looking at the other two, the third person knows they have a white one.

The WP version sensibly adds mass execution if any get it wrong, and also restricts people's vision so that the second person can only see the third one's hat...

This entry was originally posted at http://lovingboth.dreamwidth.org/546076.html, because despite having a permanent account, I have had enough of LJ's current owners trying to be evil. Please comment there using OpenID - comment count unavailable have and if you have an LJ account, you can use it for your OpenID account. Or just join Dreamwidth! It only took a couple of minutes to copy all my entries here to there.

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