12 Marbles

Written on 3:23 am by Vja Students

You have twelve marbles. Eleven of the marbles are of equal weight, but one is heavier or lighter. You have a balancing scale you can use to find this marble and figure out if it weighs more or less than the others. What is the minimum number of weighings required to do this?

Ans: The problem can be solved in three weighings.

  • Weigh four marbles against four others, leaving four on the table.
    • If both sides are equal, all eight marbles on the scale can be eliminated. Put three of the four from the table onto one side and three from the eliminated batch on the other.
      • If both sides are equal, the odd marble is the last one; weigh it with any other marble to see if it's heavier or lighter.
      • If the side with the marbles still under consideration moves up or down, weigh one of those three marbles against one of the others, and the third marble is set aside.
        • If both sides are equal, the third marble is the odd one, and it is heavier or lighter depending on whether or not the scales moved down or up in the previous weighing.
        • If the scales move, the odd marble is the one that moves in the same direction that the three marbles under consideration moved in the previous weighing. If it moves up, it's lighter; if it moves down, it's heavier.
    • If the scales move, take one marble from each side and switch them. One one side only, remove the other three and set them aside for later. Replace them with three marbles from the four left on the table (now known not to be the odd one).
      • If the two sides are equal, the odd marble is among the three set aside. Weigh one against another, and set the third aside.
        • If the sides are equal, the odd marble is the third one, and it is heavier or lighter depending on which way the scales moved in the first weighing.
        • If the scales move, the odd marble is the one that moved in the same direction as it did in the first weighing, and it is heavier or lighter depending on whether it went down or up.
      • If the two sides move in different directions as in the first weighing, the odd marble is one of the two that switched places. Weigh one of the two against any of the other ten.
        • If both sides are equal, the odd marble is the one left out. It's heavier or lighter depending on which way the scales moved in the second weighing.
        • If the scales move, the marble on the scales that's under consideration is the odd one, and it is heavier or lighter depending on whether it went down or up.
      • If the two sides move in the same direction as in the first weighing, the odd marble is one of the three that hadn't moved from its side. Weigh one of the three against another, and set the third aside.
        • If the sides are equal, the odd marble is the third one, and it is heavier or lighter depending on which way the scales moved in the previous weighings.
        • If the scales move, the odd marble is the one that moved in the same direction as it did in the previous weighings, and it is heavier or lighter depending on whether it went down or up.

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