The challenge
Three sealed boxes are labelled APPLES, ORANGES, and MIXED. Every label is wrong. You may draw one fruit from one box without looking inside. How can you label all three boxes correctly?
Mislabelled Fruit Boxes Logic Puzzle
Solve this medium logic challenge, then compare your reasoning with the hint, checked answer, verification, and variation.
Hint
Start with the label that gives the strongest guarantee when it is wrong.
Start with the label guaranteed to be false
Draw from the box labelled MIXED. It cannot be mixed. If you draw an apple, that box is APPLES. The box labelled ORANGES cannot be oranges and cannot be apples, so it is MIXED. The remaining box is ORANGES. Reverse the fruit names if you draw an orange.
Forced assignments
- Draw from the box labelled MIXED because that label is guaranteed to be false.
- Assign that box to the single fruit type you draw.
- Use the two remaining false labels to place the mixed box and the final single-fruit box.
The less informative first draw
Drawing from APPLES or ORANGES can leave ambiguity because a sample could come from the mixed box. Start where a false label creates the strongest guarantee.
Check all three false labels
Check the finished assignment against all three original labels. Each one must be wrong, and all three content types must be used once.
Test where uncertainty is smallest
When all labels are false, the most specific false label creates the best test.
How confident were you?
Use the global condition first
Use the condition that every label is wrong before drawing anything. The box marked MIXED must contain a single fruit type, so one sample identifies its entire contents and forces the remaining assignments.
Relax the all-labels-wrong condition
If exactly two labels were wrong rather than all three, would one draw still guarantee a complete relabelling? Explain which uncertainty remains.
Study constraint satisfaction for problem solving for the reusable method behind this challenge. Then explore the logic puzzles and deductive reasoning learning path for connected examples and practice.