Mislabeled Bags

One bag contains only apples, one only oranges, and one a mixture. Every label is wrong. What is the minimum number of fruits you must draw to identify all three bags?

Goal: fewest guaranteed draws
Label says 🍎 APPLES
❓
Cannot actually be apples.
Label says 🍊 ORANGES
❓
Cannot actually be oranges.
Label says 🍎 + 🍊 MIX
❓
Must be a pure bag.

Possible worlds

2 possibilities

The “every label is wrong” rule leaves only these two assignments.

World A
“Apples”
🍊 Orange
“Oranges”
🍎+🍊 Mix
“Mix”
🍎 Apple
World B
“Apples”
🍎+🍊 Mix
“Oranges”
🍎 Apple
“Mix”
🍊 Orange

Your deduction

No fruit drawn yet.
Try the bag labeled Mix. Since its label is wrong, it cannot really be mixed. So a single fruit tells you its entire contents.
🍎

The one-draw decision tree

Minimum = 1
Draw from bag labeled “MIX”
Choose either branch to see how the remaining two bags become forced.
Why not draw from the other bags first? A fruit from a bag labeled “Apples” or “Oranges” might come from either a pure bag or the mixed bag, so one draw does not always distinguish the two possible worlds. The bag labeled “Mix” is the only guaranteed one-draw solution.