# 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…)

## 5 thoughts on “A Boxes Riddle”

1. chazisop |

There is a better one.

2. I assume the last step should say that B needs to find the box, not A?

I’m pretty sure A and B can always win.

• Thanks. Fixed.

Also, this is indeed the answer.

3. Fledgling |

“Z tells A the number N”

You mean Z tells B, right?