Math Help -> Puzzles -> Pirates -> Answer

Pirate Puzzle

A handful of pirates distributed some golden coins among themselves, such that:

	the share of Pirate 1 plus half the total share of the rest (all pirates except #1)
	= the share of Pirate 2 plus 1/3 of the share of the rest (all pirates except #2)
	= ...
	= the share of Pirate N plus 1/(N + 1) of the share of the rest (all pirates except #N)

Captain Cutthroat, the puzzle-loving pirate, informed his captive Jolly Rogers of this strange method of distribution. Moreover, he promised to set Jolly free if he could correctly tell him the share of each pirate.
"The total number of coins is less than 1000, and more than 50% of us received an odd number of coins."
Jolly Rogers got to work immediately, and after some time, asked the Captain, "Can you give me a little hint?"
"The least a pirate got is this number."
Jolly Rogers was still not sure until the Captain boasted, "And I got more than 10 times that fellow."

Can you guess the number of pirates and the share of each?

Source: http://www.ken.duisenberg.com/potw/archive/arch01/010925sol.html and Sudipta Das

Suppose there are three pirates...

Let x be the share of pirate 1,
let y be the share of pirate 2, and
let z be the share of pirate 3.

x + (1/2) y + (1/2) z = c
(1/3) x + y + (1/3) z = c
(1/4) x + (1/4) y + z = c

This is a system of 3 equations in 3 unknowns. Let's solve it for x, y, and z in terms of c using Cramer's Rule.

The coefficient matrix is

é
ê
ê
ë

1	1/2	1/2
1/3	1	1/3
1/4	1/4	1

ù
ú
ú
û

The determinant of this matrix is 17/24.

The determinant of

é
ê
ê
ë

c	1/2	1/2
c	1	1/3
c	1/4	1

ù
ú
ú
û

is 5c/24, so x=5c/17.

Similarly, y=11c/17, and z=13c/17

If x, y, and z are all integers, then they can be 5, 11, and 13, respectively, or multiples of these numbers. As the Captain has boasted, "And I got more than 10 times that fellow," it follows that there can't be just three pirates. None of these pirates shares is more than 10 times any other shares.

So let's consider 4 pirates. The coefficient matrix is

é
ê
ê
ê
ê
ë

1	0.5	0.5	0.5
0.333	1	0.333	0.333
0.25	0.25	1	0.25
0.2	0.2	0.2	1

ù
ú
ú
ú
ú
û

The determinant of this matrix is 37/60.

The shares, as calculated using Cramer's Method, of the four pirates are 1c, 19c, 25c, 28c. The ratio of these shares satisfy the captain's boast, which is good. Also, 50% of the numbers are odd, as long as c is odd. But there are quite a few odd values of c for which the 4 pirates' shares total less than 1000 coins.

Even after the captain's hint ("The least a pirate got is this number."), Jolly Rogers wasn't sure that there were four pirates. That must mean that the number mentioned by the captain satisfied a 3-pirate or 4-pirate solution. In other words, the number was an odd multiple of 5. It wasn't until the last hint ("And I got more than 10 times that fellow.") that Jolly Rogers knew that four pirates were involved. Only one odd multiple of 5 yields a solution with the total of the 4 shares still under 1000 coins.

That number is c=5.

So the four shares are 5, 95, 125, and 140.

This works. But is there another solution? What if there were 5 pirates, or 6?

If the system of equations for 5, or 6, or in fact any larger number of pirates, is solved, we see that the smallest share is a negative number, which is impossible. Therefore, the highest -- and only -- number of pirates that solves this puzzle is four.

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