'It's pure logic captain'

This is a classic logic problem for you to think about. You may find it helpful to represent the various scenarios in a visual way.

A school has 1000 students and 1000 closed lockers numbered 1 to 1000.
The first student walks along the corridor and opens every locker.
The second student walks along and closes every second locker.
The third student walks along and changes the state of every third locker (ie. an open locker is closed and a closed locker is opened) .
Remaining students walk along, and the k^th student change every k^th locker, until all 1000 students have walked along the row of lockers.

Questions:

Question 1:

1^st student = O O O O O O O O O O O O O O O O O O O O

2^nd student = O C O C O C O C O C O C O C O C O C O C O C

3^rd student = O C C C O O O C C C O O O C C C O O O C C C

Closed = \(3 \times 166 + 3\) = 501 closed

Question 2:

4^th student = O C C O O O O O C C O C O C C O O O O O C C O C

Closed = \(5 \times 83 + 2\) = 417 closed

Question 3:

100^th locker = \(2^{2} \times 5^{2}\) = 9 students = odd number = OPEN

Question 4:

1000^th locker = \(2^{3}\times 5^{3}\) = 16 students

but not 125^th , 200^th , 250^th , 500^th , 1000^th students

= \(16\) – \(5\) = 11 students = odd number = OPEN

Question 5:

Closed = even number of factors (eg 6 = 1, 2, 3, 6)

Open = odd number of factors (eg 16 = 1, 2, 4, 8, 16 or any square)

Open = n² lockers, for integer n = 1 to 31

By Mike Baxter

‘It’s pure logic captain’