### Brain Teasers

# Powerful Weights and Measures

Long ago, in a land far away, there lived a primitive culture whose founding chief had eleven fingers. This chief had a strong affinity for the number eleven because of his special circumstances and thus founded a numbering system based on the number eleven. The number 121 had very special meaning, as it was eleven times eleven.

The system of weight measures that he devised were also based on the number eleven. The base unit was called a stone, for a convenient rock that was handy. The next unit was the lump, being eleven stone in weight. Eleven lumps were called a mass.

So: 11 stone = a lump and 11 lumps = a mass.

The problem was, using 10 lump and 11 stone weights to measure out any weight from 1 stone to 1 mass required carrying around quite a lot of weight! The chief, being aware that this was beyond his math skills, asked his advisors for help. The request was to come up with a minimum number of weights that would allow for the weighing of any measure between one stone and one mass using a set of scales. What was their solution?

The system of weight measures that he devised were also based on the number eleven. The base unit was called a stone, for a convenient rock that was handy. The next unit was the lump, being eleven stone in weight. Eleven lumps were called a mass.

So: 11 stone = a lump and 11 lumps = a mass.

The problem was, using 10 lump and 11 stone weights to measure out any weight from 1 stone to 1 mass required carrying around quite a lot of weight! The chief, being aware that this was beyond his math skills, asked his advisors for help. The request was to come up with a minimum number of weights that would allow for the weighing of any measure between one stone and one mass using a set of scales. What was their solution?

### Hint

The title has a slight hint.The number eleven has nothing to do with the set of weights.

### Answer

The minimum number of weights is 5.weight 1 = 1 stone = 3^0

weight 2 = 3 stone = 3^1

weight 3 = 9 stone = 3^2

weight 4 = 2 lump 5 stone = 27 stone = 3^3

weight 5 = 7 lump 4 stone = 81 stone = 3^4

All five added together weigh one mass, and allow the weighing of any measure down to one stone.

The progression from one stone to one mass would be:

1, 3-1, 3, 3 +1, 9-3-1, 9-3, 9-3 + 1, 9-1, 9, 27-9-3-1, 27-9-3, ....

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