THIS IS THE answer to yesterday’s pirate puzzle.

The question: 

Sharing has been, in one way or another, the basis of much of the world’s conflicts whether big or small. Mathematics can be used to analyse sharing and also to develop ‘fair’ systems of sharing.

Captain Blackbeard was trying to calm his greedy crew. “Men,” he said. “We’ve a mighty fine haul today: a pretty little chest of doubloons.”

The men were straining and pushing to get at the coins.

We’ll divide them using our own pirate’s code. Those who died in the raid get nothing. The loot will be divided as follows: the most senior man will get 10 coins and one tenth of the remainder, the next man will get 9 coins and one-ninth of the remainder and so on down the line.

Thus they began with Blackbeard himself the first looked after. When the third man was paid the fourth declared, “There’s only 49 doubloons left to do us. That’s only seven each!”

1. How many pirates are there?

2. How many coins were there to start with?

3. How much did the last pirate get?

The answers and method

1. There are 10 pirates

2. There are 100 coins to start

3. The 10th man gets one coin

Explanation

1. There are three pirates paid and the fourth says there is 49 coins left: 7 each.

That means 7 people left after 1st 3 = 10

(b) If there are 49 coins after the third person we must work backwards.

The 3rd person gets 8 coins and 1/8 of what’s left.

49 is left after this.

The sum X left before 1/8th is extracted is 49.

Therefore 49 = 7/8X

X = 56

So 3rd person gets 8 coins and 7 coins = 15 coins.

Therefore there is 49 + 15 = 64 coins after second pirate

Working backwards again…

Second pirate gets 9 coins and 1/9th of remainder

Therefore 64 represents 8/9 of remainder after he takes 9 coins.

So remainder is 72 plus his 9 coins is 81

81 left after Blackbeard gets his share which is 10 coins and 1/10 of remainder.

81 = 9/10th of remainder so remainder is 90 (Blackbeard gets 9 of these)

And 10 coins = 100 coins at start

3. The 10th man (working the other way if you don’t notice a pattern) gets 1 coin.