Shopping Shuffle II
You have four boxes labeled Apples, Bananas, Carrots, and Dates. Each box is closed.
While you know all four types of food are there and each box only contains one type of food, you also know that only one of the boxes is labeled correctly.
How many boxes do you need to open to be guaranteed to know where the dates are? (You don't have to have opened the box with the dates, just know where they are.)
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1 solution
The first box we would open would be the one labeled "Dates". If the dates are inside then we are done. If not, then, without loss of generality, say the bananas are inside. We would then open the box labeled "Bananas". We know the dates won't be inside this box, for if they were then the remaining two boxes would either both be labeled correctly or both incorrectly, contradicting the given that precisely one of the boxes is labeled correctly. So say, again without loss of generality, the carrots are inside. We then know that the box labeled "Carrots" is incorrectly labeled, implying that the final box, in this case labeled "Apples", is the one correctly labeled box, and thus that the dates will be inside the "Carrots" box.
So after opening 2 boxes we are guaranteed to know where the dates are, and this is the minimum number sought, as the opening of one box will at best, unless we get lucky and find the dates on the first try, reveal only two of the incorrectly labeled boxes and the definite contents of only one of them.