What we have here is a riddle published by the TED-Ed channel. A band with ten members is disastrous at organizing themselves, their manager has lost the last screw he had left, and he locked them all in a room. The way to escape and keep playing is that each member must find their instrument, obeying a series of very strict rules. Do you think you can help them?
The Puzzle
Let's go again: A very popular band is composed of ten members. They are great on stage, but their organization before each concert leaves much to be desired, since they keep losing their instruments. The manager, or "artistic representative," got fed up with this problem, and in a psychotic attack locked everyone inside an isolated, without windows, recording studio, and as if that were not enough, with their hands tied. Any sane person would beat the manager to a pulp at that point, but the man presents the band with a special challenge: Outside the studio there are ten boxes. Each one contains an instrument and has a drawing on its surface, although that drawing does not necessarily represent its contents.
The manager, drowned in madness, decides to execute his plan: He will let each band member leave one at a time, and see the contents of at most five boxes, before security drags him to the bus. What they cannot do is touch the instruments, nor communicate what they saw to their peers, and that includes signs, whistles, shouts, or marks on the boxes. Here comes the most interesting part: All the band members are forced to find their instruments, or otherwise they won't be able to play that night, and the record label will abandon them. What would be the best course of action for them? You can watch the video below, and enable the subtitles (in English) if you need a better reference.
The Strategy
Needless to say, the numbers are not good for the musicians. Each one on their own has a 50 percent chance of finding their instrument in five boxes, but if "all" must succeed, the overall rate drops to 1 in 1024. However, the drummer proposes a special strategy that turns that dramatic "1 in 1024" into a very appreciable 35 percent: Each musician first opens the box with the drawing of "their" instrument. If it is there, they don't need to do anything extra; otherwise, they follow with the box that has the drawing of the instrument they just saw. The rest is repeating the pattern until they find their instrument, or exhaust the fifth option.
In essence, the drummer's idea is that the band members deliberately enter a loop, something much more effective than picking boxes at random (in other words, there is a 35 percent chance that all loops end in five boxes or less). The end of the video explains how that 35 percent is calculated, and also highlights that if we increase the number of musicians, the percentage begins to drop toward 30 percent. That is, the drummer's solution can fail (and it's not the only one, because at the end they could come out and try their luck), but if that 35 percent is the best their band has in that situation, they should take it... right?
TED-Ed
