A Boxes Riddle

I just recalled a nice mathematical riddle. I can’t remember where I originally read it, but it was likely in one of the blogs that are in the blogroll to the right.

Riddle. A game takes place where person A and person B are on the same team while person Z is their adversary. There are 100 boxes and 100 notes containing the numbers 1 to 100 (that is, each number is on exactly one note). The game goes as follows:

  • First, only Z and A are in the room. Z places one note in each box.
  • A sees the actions of Z and may afterwards pick two boxes and switch the notes in them (with each other). He may only perform one such switch.
  • Z sees the actions of A and then chooses a number N between 1 and 100.
  • A leaves the room while B enters it (they cannot exchange information during this process). Z tells B the number N.
  • Finally, B needs to find the box that contains the note with the number N. For this purpose, B may open up to 50 boxes.
If A chooses the 50 boxes at random, his probability of success is obviously 0.5. Do A and B have a strategy for increasing their chances? Or does Z have a strategy for which 0.5 is the best possible?


Feel free to write the answer as a comment. However, I think that it would be nice not to provide a full explanation here (that is, only to write whether it’s 0.5 or what better probability you can get).

(Later edit: I seem to be getting senile! I just noticed that I wrote the same riddle in a post two years ago…)