Problem of the Week: The Locker Problem
Problem Statement: In this problem, we are asked to determine which lockers will be
open at the end of a process where each successive person opens and closes lockers
based on multiples of the next number. Louise closed all the even numbered lockers –
multiples of two; Jeremy closed (or opened as necessary) all multiples of three, and this
process continued until there was a student who only changed locker 100.
Process: For this problem, I began by working with lockers from one to twenty. I
started here for two reasons: Avi said that this problem was easy and related to the
divisor problem, whose answer was related to perfect squares; and Josh and Keith said
that you didn’t have to go all the way to one hundred to see the pattern. By the way,
they did not tell me the answer – they knew that I was frustrated because I kept making
each POW more difficult than it needed to be.
X2
Locker
1
O
2
C
3
O
4
C
5
O
6
C
7
O
8
C
9
O
10
C
11
O
12
C
13
O
14
C
15
O
16
C
17
O
18
C
19
O
20
C
O = open locker
C = close locker
X3
X4
X5
X6
X7
X8
X9
X10
C
O
C
O
C
C
O
C
C
O
O
O
C
C
O
O
C
O
O
C
O
C
O
C
O
O
Solution: Based on my observations of the table above, I noticed that 1, 4, and 9 are
the only open lockers if I work with just twenty lockers. What these numbers have in
common is that 4 and 9 are perfect squares. Their positions (open /closed) changed an
odd number of times. Since there are no more perfect squares in my sample set of
data, I need to prove my solution by creating a second set of data that includes a few
more perfect squares.
Locker
15
16
17
18
19
20
21
22
23
24
25
10
O
C
O
O
O
O
O
C
O
C
C
11
12
13
14
15
C
16
17
18
19
20
21
22
23
24
25
O
C
C
C
C
C
O
C
C
O
C
O
After twenty-four students take turns opening and closing lockers based on the multiples
of the next subsequent number, the lockers that are open are the four perfect squares
(4, 9, 16, and 25 plus 1 as a special case).
This means that if there were one hundred lockers to begin with and students opened
and closed the lockers based on the established system, the only lockers that would be
opened at the end would be the perfect squares to 100 4, 9, 16, 25, 36, 49, 64, 81,
100, as well as 1.
If this method continues to work, and there were 1000 lockers then the ones that would
be opened would be those already named as perfect squares as well as 121, 144, 169,
225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and
961. The squares of numbers larger than 31 would be larger than 1000.