Logical puzzle three mislabeled crates of apples and oranges how can you relabel all correctly by looking at only one fruit from one crate?

Introduction / Context:
This is a classic logic puzzle about mislabeled boxes. You have three crates: one actually contains only apples, one contains only oranges and one contains a mixture of apples and oranges. Due to a faulty labeling machine, each crate now has an incorrect label. Your challenge is to figure out the correct label for all three crates by examining just a single fruit taken from only one crate. This tests logical deduction and the ability to use a minimum amount of information to resolve all possibilities.

Given Data / Assumptions:
• There are three crates, each originally labeled apples, oranges and apples and oranges.
• Every crate is labeled incorrectly after the machine error.
• One crate truly has only apples, another only oranges and the third has a mixture of both fruits.
• You may take out and inspect only one fruit, and it must come from only one crate.
• The goal is to work out the correct label for all three crates using this single observation.

Concept / Approach:
The key idea is to start with the label that gives the most restrictive information. Since every label is wrong, the crate labeled apples and oranges cannot actually contain a mixture. Therefore it must be either all apples or all oranges. By drawing one fruit from this crate, you immediately identify what it really contains. Once you know that, you can deduce the contents of the remaining crates by elimination and by using the rule that none of their incorrect labels can match their real contents.

Step-by-Step Solution:
Step 1: Focus on the crate labeled apples and oranges. Because all labels are wrong, this crate cannot contain a mixture. Step 2: Therefore the crate labeled apples and oranges must contain either only apples or only oranges. Step 3: Take one fruit from this crate. If you draw an apple, then this crate is the all apples crate. If you draw an orange, it is the all oranges crate. Step 4: Suppose you draw an apple. Then the crate labeled apples and oranges is truly the apples crate. The crate labeled apples cannot contain only apples (its label is wrong) and it cannot be the mixture crate (because we still need a pure oranges crate), so it must be the all oranges crate. The remaining crate, labeled oranges, must therefore be the mixed apples and oranges crate. Step 5: If instead you draw an orange, the same reasoning works with apples and oranges swapped, and you can still label all three crates correctly.

Verification / Alternative check:
You can verify this strategy by testing both possible outcomes from the crate labeled apples and oranges. In both cases, knowing the identity of that crate forces the identities of the other two crates because each label must be wrong. No ambiguity remains. Also, no other crate gives you this immediate certainty, because if you start from a crate labeled apples or oranges, that crate could actually be either the mixture or the opposite single fruit crate, leading to more confusion instead of certainty.

Why Other Options Are Wrong:
• Taking one fruit from the crate labeled apples and relabeling only that crate does not let you fully determine the other two crates with certainty.
• Taking one fruit from each crate violates the rule that you can inspect only one fruit from one crate and also wastes information in this puzzle setting.
• Relabeling randomly based on labels without using the guaranteed wrong label rule cannot logically solve the puzzle; it becomes pure guesswork.

Common Pitfalls:
A common mistake is to forget that every label is incorrect. Some people try to start from the crate labeled apples or oranges and do not realise that the mixed label is the most informative. Another pitfall is thinking you must see more than one fruit to be sure, which is not necessary if the logic is applied correctly. The important lesson is to look for the crate whose wrong label gives you the strongest deduction when combined with a single sample.

Final Answer:
You should draw one fruit from the crate labeled apples and oranges. Based on whether it is an apple or an orange, you can deduce all contents and relabel every crate correctly, so the correct strategy is to take one fruit from the crate labeled apples and oranges and relabel based on that single draw.