### 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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## Comments

idk... I thought it was very confusing... I didn't know what to do... somebody help me out here...

nice

i didnt have the patience to figure it out

i didnt have the patience to figure it out

May 07, 2006

Although the binary system of weights requires FEWER weights than the base-11 method, the total mass of the weights is the same (namely, 1 mass), thereby defeating the purpose. However, if the scale used had arms of unequal length, lighter weights could be used to weigh heavy objects...(technicality)

Aye, 'tis true. Either a variable, or offset scale would allow for less weight total. The real puzzle was to find out how to count from 1 to 121 with the least number of different units.

The problem should have read

"The problem .... around quite a lot of weights!"

Forgot the s.

The problem should have read

"The problem .... around quite a lot of weights!"

Forgot the s.

"8.2" for situation- "0.3" for setup

Hisses for the setup, because in a numbering system based on eleven, the number 121 is equal 144 base 10.

Let's make it an explicit part of the answer. This is not chem class; there is no worry about contaminating what you measure. The weights can go on either side of balance.

Hisses for the setup, because in a numbering system based on eleven, the number 121 is equal 144 base 10.

Let's make it an explicit part of the answer. This is not chem class; there is no worry about contaminating what you measure. The weights can go on either side of balance.

Binary? Shouldn't it be tertiary? (Or something to that effect. Anybody know any latin?)

Nice... once I understood it. The word weight was thrown about so loosely in the teaser that it wasn't clear to me that he was looking for the equivelant of a few dumbells. You may want to clarify (or rather, eliminate) the point that's already been made about the scale being offset.

I thought the problem was easy to understand (and easy to do), but I had the same issue with the fact that no matter what, the total weight would still be one mass.

That did add some confusion.

That did add some confusion.

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