Cracking the Coding Interview, 4 Edition - 150 Programming Interview Questions and Solutions

Solutions to Chapter 6 | Brain Teasers
6.4 A bunch of men are on an island. A genie comes down and gathers everyone together
and places a magical hat on some people’s heads (i.e., at least one person has
a hat). The hat is magical: it can be seen by other people, but not by the wearer of
the hat himself. To remove the hat, those (and only those who have a hat) must dunk
themselves underwater at exactly midnight. If there are n people and c hats, how
long does it take the men to remove the hats? The men cannot tell each other (in any
way) that they have a hat.
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SOLUTION
Prove that your solution is correct.
This problem seems hard, so let’s simplify it by looking at specific cases.
Case c = 1: Exactly one man is wearing a hat.
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Assuming all the men are intelligent, the man with the hat should look around and realize
that no one else is wearing a hat. Since the genie said that at least one person is wearing
a hat, he must conclude that he is wearing a hat. Therefore, he would be able to remove it
that night.
Case c = 2: Exactly two men are wearing hats.
The two men with hats see one hat, and are unsure whether c = 1 or c = 2. They know, from
the previous case, that if c = 1, the hats would be removed on Night #1. Therefore, if the other
man still has a hat, he must deduce that c = 2, which means that he has a hat. Both men
would then remove the hats on Night #2
Case General: If c = 3, then each man is unsure whether c = 2 or 3. If it were 2, the hats would
be removed on Night #2. If they are not, they must deduce that c = 3, and therefore they
have a hat. We can follow this logic for c = 4, 5, …
Proof by Induction
Using induction to prove a statement P(n)
If (1) P(1) = TRUE (e.g., the statement is true when n = 1)
AND (2) if P(n) = TRUE -> P(n+1) = TRUE (e.g., P(n+1) is true whenever P(2) is true).
THEN P(n) = TRUE for all n >= 1.
Explanation
» » Condition 2 sets up an infinite deduction chain: P(1) implies P(2) implies P(3) implies ...
P(n) implies P(n+1) implies ...
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