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W _ Y Clue
THE ONE LABELED BOTH. SINCE YOU KNOW THAT THE BOX IS LABELED INCORRECTLY, IT MUST HAVE ALL BLACK MARBLES OR ALL WHITE MARBLES. IF A BLACK MARBLE COMES OUT, YOU KNOW THAT HAS TO BE THE BLACK MARBLES BOX, SO YOU PUT THE "BLACK" LABEL ON IT. THE BOX LABELLED "WHITE" MUST THEN BE LABELLED "BOTH" BECAUSE YOU KNOW IT IS ALSO LABELED INCORRECTLY, AND THEREFORE CAN'T BE WHITE. YOU WOULD LABEL THE LAST BOX "WHITE" BECAUSE THAT IS THE ONLY COLOR/BOX COMBO LEFT. IF THE FIRST MARBLE YOU PULLED OUT FROM THE BOX LABELED "BOTH" IS A WHITE MARBLE, THEN YOU SOLVE THE PROBLEM IN THE SAME GENERAL WAY IT IS NOT POSSIBLE TO GUESS ONLY THREE FRUITS CORRECTLY, THE FOURTH FRUIT WILL BE CORRECT TOO! SO NOBODY HAS GUESSED THREE FRUITS CORRECTLY AND SO 10 (123 - 43 - 39 - 31 = 10) PEOPLE HAVE GUESSED FOUR FRUITS CORRECTLY YOU WOULD ONLY NEED TO TAKE OUT ONE MARBLE BECAUSE WE KNOW THAT ALL OF THE LABELS ARE INCORRECT. SO YOU PULL ONE MARBLE OUT OF THE BOX LABELED "MIXED." IF RED COMES OUT, YOU KNOW THAT HAS TO BE THE ALL-RED BOX, SO YOU PUT THE RED LABEL ON IT. THE BOX LABELLED "BLUE" MUST THEN BE LABELLED "MIXED" BECAUSE YOU KNOW IT IS ALSO LABELED INCORRECTLY, AND THEREFORE CAN'T BE BLUE. YOU WOULD LABEL THE LAST BOX "BLUE" BECAUSE THAT IS THE ONLY COLOR/BOX COMBO LEFT. IF THE FIRST MARBLE YOU PULLED OUT FROM THE BOX LABELED "MIXED" IS A BLUE MARBLE, THEN YOU SOLVE THE PROBLEM IN THE SAME GENERAL WAY YOU WILL NEED TO OPEN A MINIMUM OF 2 BOXES. FIRST, OPEN THE BOX LABELLED APPLE; IF IT'S LABELLED CORRECTLY WE'RE DONE (THOUGH THIS IS PURELY BASED ON LUCK), OTHERWISE WE'LL FIND EITHER BANANAS, CARROTS, OR DATES. IN ANY OF THESE CASES, WE KNOW EITHER THAT THE BOXES LABELLED BANANAS, CARROTS, OR DATES MUST ALSO BE MISLABELED. IF THE FIRST OPENED BOX HAS: BANANAS INSIDE, THEN THE BOX LABELLED BANANA IS INCORRECT AND EITHER CARROTS OR DATES ARE CORRECT. CARROTS INSIDE, THEN THE BOX LABELLED CARROTS IS INCORRECT AND EITHER BANANA OR DATES ARE CORRECT DATES INSIDE, THEN THE BOX LABELLED DATES IS INCORRECT AND EITHER BANANA OR CARROTS ARE CORRECT. NO MATTER THE CIRCUMSTANCE, AFTER OPENING ONE BOX WE CAN IDENTIFY TWO REMAINING BOXES THAT MIGHT BE THE CORRECTLY LABELLED BOX. OPEN EITHER ONE OF THEM. IF IT'S THE CORRECTLY LABELLED BOX, WE'VE FOUND IT. OTHERWISE THE REMAINING UNOPENED BOX IS CORRECTLY LABELLED AND WE'VE FOUND IT. WE DON'T NEED TO OPEN IT TO DOUBLE CONFIRM