Where to stand among 1024 pirates

Question

Pirates numbered 1 to n stand in a circle. Pirate 1 shoots pirate 2, pirate 3 shoots pirate 4, and so on round the circle, skipping the dead; whoever is next in line after the last shooter continues, always shooting the next living pirate. This goes on until one pirate is left. 1. With 1024 pirates, which position survives? 2. With 1000 pirates? Find a rule that works for any n.

Accepted Answer

With 1024 pirates, position 1 survives. With 1000 pirates, position 977. In general, write n = 2 to the m + k with k as small as possible; the survivor is 2k + 1. Powers of two. With 1024 pirates, after one full round all the even-numbered pirates are dead, 512 remain, and the shooter is pirate 1 again, exactly as at the start but with half the circle. Repeating, pirate 1 shoots first in every round and is never the target. After ten rounds pirate 1 is alone. So for any power of two, position 1 survives. Other numbers. With n = 1000, write 1000 = 512 + 488, so k = 488. After the first 488 sho…

Where to stand among 1024 pirates

Your answer

Discussion

Learn the concepts

Related questions