You are in a storage room with 3 wooden crates. One is labeled APPLES, one is labeled ORANGES, and one is labeled APPLES & ORANGES. You are told that every single label is wrong. Inside each crate is either only apples, only oranges, or a mix of apples and oranges, but none of the crates currently contains what its label says. You are allowed to reach into exactly one crate and pull out exactly one piece of fruit (without looking inside). From that one fruit, you must figure out how to correctly relabel all three crates.
From which crate should you take a fruit, and how can you determine the correct label for each crate?
Answer:
The short answer is that you should select a piece for fruit from the crate labeled APPLES & ORANGES. But what to do with that fruit requires a fuller explanation!
The key to this puzzle is remembering that all three labels are wrong. That means the crate labeled “APPLES & ORANGES” cannot contain a mix of fruit — it must contain either only apples or only oranges. Because of that, reaching into this crate will immediately tell you exactly what’s inside. If you pull out an apple, then that entire crate is full of apples. If you pull out an orange, then it contains only oranges.
Once you know the true contents of that crate, you can then determine the contents of the other crates in the following way. Suppose you drew an apple. The crate you opened now becomes the “APPLES” crate. The crate labeled “ORANGES” cannot actually contain only oranges (because its label is wrong), and it cannot contain apples (since you’ve already identified the crate full of apples). It must be the mixed crate. The remaining crate therefore has to contain only oranges. If instead you had drawn an orange from the first crate, the same logic applies with apples and oranges reversed.