Who Gets the Job?

Date: 12/11/2001 at 08:25:00
From: Pete
Subject: Interview - who gets the job

Three candidates are all equally qualified for a job.  The boss does 
not know which to pick, so he devises a plan to see who is the most 
clever. 

The three candidates are blindfolded, and the boss puts either a red 
or a black dot on the forehead of each of them. Candidates can see the 
other two candidates' dots, but not their own. The boss tells them to 
raise their hands if they see a black dot. Whoever can figure out the 
color of his own dot gets the job.

Date: 12/11/2001 at 09:21:21
From: Doctor Ian
Subject: Re: Interview-who gets the job

Hi Pete,

Let's call the three candidates A, B, and you.  Each one might have a 
black dot (b), or a red dot (r).  So these are the possibilities:

    A    B    you
    -------------
    b    b     b
    b    b     r
    b    r     b
    b    r     r
    r    b     b
    r    b     r
    r    r     b
    r    r     r


Now, let's use a '*' to mark anyone who should have his hand up 
(because he sees a black dot): 

    A    B    you
    -------------
    b*   b*    b*
    b*   b*    r*
    b*   r*    b*
    b    r*    r*
    r*   b*    b*
    r*   b     r*
    r*   r*    b
    r    r     r

Let's order the possibilities by the number of people who have their 
hands up:

    A    B    you
    -------------
    b*   b*    b*       3 hands up   -> 0 or 1 red dot
    b*   b*    r*
    b*   r*    b*
    r*   b*    b*

    b    r*    r*       2 hands up   -> 2 red dots
    r*   b     r*
    r*   r*    b

    r    r     r        no hands up  ->  3 red dots

Now, if no one has a hand up, you know right away that you're all 
wearing red dots, which means that you're wearing a red dot too, so 
you can have the job. 

If two of you have hands up, it means that two of you are wearing red 
dots, so by looking at the other two guys, you can tell whether yours 
is red or black, and you can have the job. 

What if there are three hands up?  Well, if there are three hands up, 
there is only one red dot. If you can see it, you don't have it, so 
you can have the job. 

    A    B    you
    -------------
    b*   r*    b*       3 hands up, you see a red dot
    r*   b*    b*

    b*   b*    b*       3 hands up, you don't see a red dot
    b*   b*    r*

What if you don't see it? You're in kind of a pickle. Your own dot 
could be red, or it could be black. 

But suppose you wait a while, and neither of the other candidates has 
said anything. Assuming that the other candidates are smart enough to 
figure all this out, what does _that_ tell you?  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/