Captain Hook plundered a safe containing 2520 gold coins. During the night, each of his pirates secretly took some coins. The first took 1/2 of the coins, the second took 1/3 of the remaining coins, the third took 1/4 of what was left, and so on. When Captain Hook opened the safe, he found only 252 coins. How many pirates did Captain Hook command?

To find out how many pirates Captain Hook commanded, we need to work backward from the remaining coins to the original amount. Since Captain Hook found only 252 coins left, we can deduce how many coins each pirate took.

We begin with the last pirate, working our way to the first:

1. Let's assume that the last pirate is the nth pirate. The nth pirate took 1/(n+1) of the coins, leaving 252 coins inside. 2. Before the nth pirate took his share, there were 252 * (n+1) coins. 3. Working backwards, we continue this logic for each pirate, where the previous number of coins would be the amount left multiplied by (n-i+2), where i is the ith pirate.

The series of fractions that each pirate took goes as follows: 1/2, 1/3, 1/4, ..., 1/n

Remember that the safe started with 2520 coins.

We don't know the exact number of pirates, but we can reverse the operation each pirate performed:

Let's say the last pirate (nth pirate) left x coins: x = 252 * (n+1)

For the pirate before him (n-1), we have: x * n/(n-1)

Continuing this pattern until we reach the first pirate, reversing each action, we should end up with the original amount of coins, which is 2520.

We need x to satisfy this equation:

x * n/(n-1) * (n-1)/(n-2) * ... * 3/2 * 2/1 = 2520

Our known value is x = 252 * (n+1)

Instead of computing this directly for each n, we observe a pattern: With each pirate, we multiply by a fraction that consists of the current number of the pirate divided by one less than that number. When we reach the first pirate, we would have multiplied by 2/1—because the first pirate took half of the coins. This means that the product of the fractions will cancel out all the numerators with denominators and eventually we'd be left multiplying by 2 (from the first pirate's 1/2).

For simplicity, from the last pirate's action, we have:

252 * (n+1) * 2 = 2520

So, compute:

252 * (n+1) = 1260

Divide both sides by 252:

n+1 = 5

Subtract 1 from both sides:

n = 4