### Brain Teasers

# Weight In Gold

During the California Gold Rush, an assessor in San Francisco used a balance to measure out quantities of gold dust. He had a set of weights to place on the balance. The assessor needed to be able to measure out gold dust in exact ounces, no fractions, for every number of ounces starting at 1.

If the assessor has four weights, what is the largest number of ounces of gold dust he can measure?

Note: The answer is not 15 ounces

If the assessor has four weights, what is the largest number of ounces of gold dust he can measure?

Note: The answer is not 15 ounces

### Hint

Weights can be placed on the balance plate with or without the gold dust.### Answer

The maximum weight is 40 ounces.The weights are powers of 3: 1 ounce, 3 ounces, 9 ounces, 27 ounces.

Example:

7 ounces – Plate A: 9 oz. and 1 oz. weights. Plate B: 3 oz weight and gold

In any weighing, each weight can be placed in one of three ways: on the plate with the gold, on the plate without the gold, on the desk next to the balance. So given n weights, there are 3^n possible combinations.

Throw out the case where all weights are on the table. That leaves 3^n – 1 cases.

Half of those cases would weigh negative gold, so eliminate half of these.

Thus, the number of valid combinations for n weights is (3^n – 1) / 2.

For four weights, (3^4 – 1) / 2 = (81 – 1) / 2 = 80 / 2 = 40.

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