Cracking the Coding Interview - Fooo

Solutions to Chapter 6 | Brain Teasers
6 6 There are one hundred closed lockers in a hallway A man begins by opening all one
hundred lockers Next, he closes every second locker Then he goes to every third
locker and closes it if it is open or opens it if it is closed (e g , he toggles every third
locker) After his one hundredth pass in the hallway, in which he toggles only locker
number one hundred, how many lockers are open?
SOLUTION
Question: For which rounds is a door toggled (open or closed)?
1 4 9
Cracking the Coding Interview | Concepts and Algorithms
pg 60
A door n is toggled once for each factor of n, including itself and 1 That is, door 15 is toggled
on round 1, 3, 5, and 15
Question: When would a door be left open?
Answer: A door is left open if the number of factors (x) is odd You can think about this by
pairing factors off as an open and a close If there’s one remaining, the door will be open
Question: When would x be odd?
Answer: x is odd if n is a perfect square Here’s why: pair n’s factors by their complements
For example, if n is 36, the factors are (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) Note that (6, 6) only
contributes 1 factor, thus giving n an odd number of factors
Question: How many perfect squares are there?
Answer: There are 10 perfect squares You could count them (1, 4, 9, 16, 25, 36, 49, 64, 81,
100), or you could simply realize that you can take the numbers 1 through 10 and square
them (1*1, 2*2, 3*3, , 10*10)
Therefore, there are 10 lockers open