Question
In Transum prison there are 50 prisoners in cells numbered 1 to 50.
On day 1, the guard turns the key in every lock to open every cell.
On day 2, the guard turns the key in every cell which is a multiple of 2. This locks all the even numbered cells.
On day 3, the guard turns the key in every cell which is a multiple of 3, locking or unlocking them.
On day 4, the guard turns the key in every cell which is a multiple of 4, locking or unlocking them.
This continues for fifty days. The prisoners whose cells are open after the 50th day are set free. Which prisoners will be set free?
On day 1, the guard turns the key in every lock to open every cell.
On day 2, the guard turns the key in every cell which is a multiple of 2. This locks all the even numbered cells.
On day 3, the guard turns the key in every cell which is a multiple of 3, locking or unlocking them.
On day 4, the guard turns the key in every cell which is a multiple of 4, locking or unlocking them.
This continues for fifty days. The prisoners whose cells are open after the 50th day are set free. Which prisoners will be set free?
Answers
GPT-4o mini
To determine which prisoners will be set free at the end of the 50 days, we need to analyze the pattern of how each prisoner's cell is toggled (opened if closed, closed if opened) based on the days the guard operates.
Each prisoner's cell, numbered \( n \), is toggled on each day \( d \) that is a divisor of \( n \). For example, cell 12 is toggled on days 1, 2, 3, 4, 6, and 12. Therefore, cell \( n \) will be toggled a number of times equal to the number of divisors of \( n \).
A key insight is that:
- A cell will end up open if it's toggled an odd number of times (because it starts closed, and an odd number of toggles will leave it open).
- A cell number has an odd number of divisors if and only if it is a perfect square. This is because divisors typically come in pairs (e.g., \( d \) and \( \frac{n}{d} \)), but if \( n \) is a perfect square, one of those divisors will repeat (the square root).
The perfect squares within the range of \( 1 \) to \( 50 \) are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
Thus, the prisoners in cells numbered with these perfect squares will be set free.
Therefore, the prisoners who will be set free are in cells **1, 4, 9, 16, 25, 36, and 49**.
Each prisoner's cell, numbered \( n \), is toggled on each day \( d \) that is a divisor of \( n \). For example, cell 12 is toggled on days 1, 2, 3, 4, 6, and 12. Therefore, cell \( n \) will be toggled a number of times equal to the number of divisors of \( n \).
A key insight is that:
- A cell will end up open if it's toggled an odd number of times (because it starts closed, and an odd number of toggles will leave it open).
- A cell number has an odd number of divisors if and only if it is a perfect square. This is because divisors typically come in pairs (e.g., \( d \) and \( \frac{n}{d} \)), but if \( n \) is a perfect square, one of those divisors will repeat (the square root).
The perfect squares within the range of \( 1 \) to \( 50 \) are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
Thus, the prisoners in cells numbered with these perfect squares will be set free.
Therefore, the prisoners who will be set free are in cells **1, 4, 9, 16, 25, 36, and 49**.