Five Seamen and a lot of Coconuts

Number Theory Level 5

Five seamen were marooned on an island as a punishment for stealing supplies from their ship.

After many hours of searching for food, they found a coconut grove and spent the entire day harvesting coconuts. Tired of the days work, they decided to sleep, without counting the coconuts, and to do the sharing the next day.

The first seamen woke up during the night and divided the coconuts into five equal shares and found one extra coconut, which he threw into the sea. Then, he hid one share and combined the other four into one heap, without waking the others.

A little while later, the second seamen woke up and did the same dividing exercise. He too found one extra coconut, which he threw away, hid his share and combined the rest into one heap.

The same operation was repeated by the rest three seamen, without one another's knowledge.

The next day, the five men saw a heap that was definitely much smaller. But, unwilling to admit their mistake, each kept quite and divided the remaining coconuts equally among themselves.

So, how coconuts were actually harvested on the previous day, if it is assumed that the seamen spent 10 hours harvesting coconuts with each harvesting no more than 250 coconuts in an hour.

The answer is 3121.

1 solution

Janardhanan Sivaramakrishnan
May 17, 2015

Let the original number of coconuts be $C_0$ .

If we assign $C_i$ as the number of coconuts 'left' by the i-th seamen after his 'distribution', we get the equation.

$C_i=\frac{4}{5}\left(C_{i-1}-1\right)$

Thus, we get the number of coconuts left in the heap on the second day as

$C_5 = \frac{1}{5^5}\left(1024 C_0 -8404\right)$

and the final share for each seamen as

$S = \frac{C_5}{5} = \frac{1}{15625}\left(1024 C_0 - 8404\right)$

This gives the Diophantine equation

$1024 C_0 - 15625 S = 8404$

Using Euclidean Algorithm we get the solutions as

$C_0 = 15625n+3121, S = 1024n+204$

The solutions are positive for $n=0,1,2...$

However, for $n=1,2,..$ , the values of $C_0 > 15000 = 12 \times 5 \times 250$ which is the maximum possible number of harvested coconuts.

Hence, using $n=0$ , we have the solution as $\boxed{C_0 = 3121}$

Five Seamen and a lot of Coconuts

The answer is 3121.

1 solution

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