So there's a puzzle I heard, and I think I know the answer to, but I wanted to share my logic to see if I am correct. I am also interested in a more mathematical way to solve it which escapes me. Here's the setup:
There are 4 boxes labelled A,B,C and D. Each box contains one of the 4 letters and exactly one box contains the same letter as labelled on it. What is the minimum number of boxes required to open in order to know exactly which box contains the letter matching its label?
I am fairly confident that the answer is 2, because on the first choice I either find the correct box or I don't. If I don't then I know that box is not correct (obviously) but it also can't be the box with the label matching the letter in the one I opened. So then I just choose any of the other two boxes, and either I get the correct box, or I know it is the other.
When I'm asking for a more mathematical method, I feel like there should be a way to do this with permutations and combinations but I am stuck on how. My idea was to determine the number of possible arrangements and see how many possibilities go away with each choice. I thought the number of arrangements should be $4!$, but then I couldn't get a number for the reduction of possibilities of each choice.
Is this another way to solve this problem and if so, how does it work? If this isn't, what is another, more formal mathematical way of considering this situation and finding the solution?