Kinesys materializes on a tiny tropical island on Abakos, surrounded by a turquoise sea and buoys numbered in perfect order. On the beach, three sailors shipwrecked from another vessel are squabbling over a pile of coconuts. A monkey watches them hopefully.
On Abakos every division of goods must be recorded, and the three need to know how many coconuts there were at the start. The trouble is that nobody counted them, which on this planet is practically a crime.
Dord is appalled: “Half a coconut? Who on earth takes half a coconut?” Liz sits down on the sand: at least the sun is out. M00N immediately offers to help, and gives the monkey a smile too.
The Riddle
Three sailors find a pile of coconuts. The first takes half of them plus half a coconut. The second takes half of what’s left plus half a coconut. The third also takes half of what remains plus half a coconut. Exactly one coconut is left, and the sailors give it to the monkey. How many coconuts were in the pile to begin with?
Hint
Don’t start from the original pile—start from the end: one coconut is left, the monkey’s. How many coconuts did the third sailor have in front of him?
Solution
The answer is 15 coconuts.
It’s best to work backward. If one coconut is left at the end (the one given to the monkey), the third sailor found a pile of 3 coconuts and took 2 of them, that is, half the pile (one and a half coconuts) plus half a coconut, leaving 1.
Likewise, the second sailor had 7 coconuts: he took 4 (three and a half plus half a coconut), leaving 3 for the third.
Finally, the first sailor had 15 coconuts and took 8 (seven and a half plus half a coconut), leaving 7 for the second. No coconut was ever cut in half.
Source: Martin Gardner, Mathematical Puzzles and Diversions (Italian edition: “Enigmi e giochi matematici”)