Skip to content
Related Articles
Open in App
Not now

Related Articles

Puzzle 76 | Toggling prison cell

Improve Article
Save Article
  • Difficulty Level : Medium
  • Last Updated : 18 Jan, 2023
Improve Article
Save Article


There was a prison consisting of 1000 cells numbered from 1 to 1000.

  • Each cell can be marked with ‘+’ or ‘-‘sign. Initially, all cells were marked with ‘-‘ sign.
  • From days 1 to 1000,the jailor toggles marks on the cell from + to – or vice versa.
  • On the i-th day ,the signs on cells that are multiples of i get toggled.
  • Now in the process of verification on 1001-th day, all cells marked with + signs are opened.

Can you identify the cell numbers with ‘+’ sign?

Solution: Cell numbers are 1, 4 ,9, 16, 25, 36 and so on.


A cell gets toggled as many times as the number of divisor it has. For example let’s take cell number 20, it gets toggled on days 1, 2 ,4, 5, 10 and 20.

  • Now we can see that  divisors come in pairs like 20=1*20=2*10=4*5. We can see that total number of divisor is even. But this trend is not followed if the number is a perfect square.
  • In perfect square , total number of divisors is odd. We see that cells are initially marked with – sign and only cell numbered as perfect square gets changed to + sign as it is having odd number of divisors.

These cells are 1, 4 ,9, 16,25,36,49 and so on.

This puzzle is contributed by Ankur Chaudhary. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

My Personal Notes arrow_drop_up
Related Articles

Start Your Coding Journey Now!