Cracking the Coding Interview - Fooo

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
FOLLOW UP
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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