Marble scales

We can do it in 2 weighings.

If we only have 3 marbles, we can determine the heaviest with one just weighing: Put a marble on each side of the scale, and leave one aside.

• If the scale is unbalanced then the heavier side is the heavy marble.
• Otherwise the marble left aside is the heavy marble.

To solve the problem for 8 marbles, first weigh 3 marbles on each side of the scale, leaving 2 aside:

• If the scale is unbalanced, take the 3 marbles on the heavy side and determine the heavy marble with one weighing as above.
• Otherwise weigh the 2 remaining marbles to determine which is heavier.

We can do it in 3 weighings.

First note, that if we have 3 marbles with 1 odd marble then we can find the odd marble in one weighing if we know that the odd marble is heavier or lighter. Simply weigh 2 of the 3. If the scale is unequal then we’ve identified the odd marble, otherwise the odd marble is the remaining marble.

First group the marbles into 3 groups of 4. Weigh 2 of the groups:

• If the groups are of equal weight:
  • Mark all eight marbles weighed as good.
  • Take 3 of the 4 remaining marbles and weigh them against 3 good marbles:
    • If the two sides wiegh the same then the last marble is the odd marble. Weigh it against a good marble to determine if it heavier or lighter.
    • If the two sides are unequal, then we know that the odd marble is in the group of 3, and we know whether it is heavier or lighter. This is just the 3 marble case which can be solved in one weighing.
• If the groups are unequal:
  • The 4 remaining marbles are good.
  • From the scale we have a heavy group of 4 and a light group of 4.
  • Now will weigh 3 marbles from the light group and one from the heavy group against 3 good and 1 from the light group:
    • If the side with 3 light and 1 heavy is heavier then the odd marble is either the 1 from the heavy group, or the 1 from the light group on the light side. Comparing either of these with a good marble will identify the odd marble.
    • If the side with 3 light and 1 heavy is lighter, then one of the 3 from the light group is the odd marble. This is just the 3 marble case which can be solved in one weighing.
    • If the scale is equal then the odd marble is one of the 3 remaining marbles from the heavy group. This is just the 3 marble case which can be solved in one weighing.