The Tri-Container Mislabeling Paradox — Daily Brain Teaser & Logic Puzzle

The Scenario

Imagine an empirical laboratory containing three identical, opaque storage canisters. Each canister has been affixed with a mechanical label denoting its contents:

1. Canister A: Labeled "Pure Rubies"
2. Canister B: Labeled "Pure Sapphires"
3. Canister C: Labeled "Mixed Gems (Rubies & Sapphires)"

You are informed of one critical constraint: Every single canister is currently mislabeled. Not a single label reflects the true internal contents of its respective container.

Your objective is to ascertain the true contents of all three canisters with absolute deductive certainty by drawing specimen gems one at a time, without looking into the canisters.

The Questions

1. What is the minimum number of specimens you must draw to correctly relabel all three canisters?
2. From which canister must your initial sample be extracted?

Hint

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Focus your analytical attention on the information density of the container labeled "Mixed Gems." What does the premise of 100% mislabeling imply about a container bearing a mixed label?

Solution & Logical Deduction

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Answer:

1. Minimum draws required: 1 specimen.
2. Initial canister choice: You must draw from the canister labeled "Mixed Gems."

Analytical Breakdown:

1. Step 1: The Initial Selection
   Draw one gem from the canister labeled "Mixed Gems." Because we know with certainty that all labels are false, this canister cannot actually contain a mixture. It must be either 100% Rubies or 100% Sapphires.
2. Step 2: Deducing Canister C
   Suppose the single gem drawn from "Mixed Gems" is a Ruby.
   • This proves conclusively that Canister C contains Pure Rubies.
3. Step 3: Deductive Elimination for Remaining Canisters
   We now have two canisters unverified: Canister A (labeled "Pure Rubies") and Canister B (labeled "Pure Sapphires").
   • Canister B cannot contain "Pure Sapphires" (because its label is false).
   • Canister B cannot contain "Pure Rubies" (because we just proved Canister C contains the Pure Rubies).
   • By process of elimination, Canister B must contain the Mixed Gems.
4. Step 4: Final Assignment
   Remaining is Canister A (labeled "Pure Rubies"). Since the other two containers are assigned, Canister A must contain Pure Sapphires.

(Note: Symmetrically, if the initial gem drawn from "Mixed Gems" had been a Sapphire, the exact same chain of elimination resolves all three canisters in one turn.)