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# How many weights are required to balance any number of ounces up through 2000 on

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Aficionadaposted 5 years ago

How many weights are required to balance any number of ounces up through 2000 on a balance scale?

A merchant has a precision balance scale, set to weigh exact ounces only, to use in case of an apocalyptic electrical failure. What is the maximum number of weights needed to be able to weigh any number of ounces from one through 2000 ounces? What specific weights are they?

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JThomp42posted 5 years ago

They can always go back to the trusty old weighing system. Outdated scales.

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calculus-geometryposted 5 years ago

You can do it with 8 weights: 1, 3, 9, 27, 81, 243, 729, and 2187 -- the powers of 3 up to 3^7 = 2187.

Some quantities can be achieved by piling the appropriate weights on one side of the scale, while others require you to place different weights on both sides of the scale.  For example, 10 ounces can be made by putting the 1 and 9 on the same side since 9 + 1 = 10.  But to make 11, you need to put the 9 and 3 on one side and the 1 on the other, since 9 + 3 - 1 = 11.

To balance 2000 ounces, you place the 2187, 81, and 3 ounce weights on one side and the 243, 27, and 1 ounce weights on the other.  This works since 2187 + 81 + 3 - 243 - 27 - 1 = 2000.

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tussinposted 5 years agoin reply to this

That's more efficient than using 11 powers-of-2 weights: 1, 2, 4, ..., 1024, which was how I first thought to do it. Since you can balance on both sides, there's more freedom.

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Don Fairchildposted 5 years agoin reply to this

Brilliant! Thinking out side the box ehh. This technique would work on binary weights too.  2048 on one side, 32 + 16 on the other side.

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Tusitala Tomposted 5 years ago

If I am interpreting the meaning of your quesitons correctly, you're asking for the names of the various weights, right?  For example, ounces, pounds, hudred-weight, and tons.  These were the common ones used before this country (Australia) went metric.

As I recall it it was 16 oz to 1 pound,  112 pounds to one hundred weight, and twenty hundred weights to one ton.   The term 'stone' was often used, but generally only to describe physical body weight,  e.g. 11 stone 6 lbls.

The old time scales were virtually a fulcrum with a tray on either end.  You put the goods to be measured on one tray and the brass weights on the other.  When you got a balance, you added up the weights on the scale and that was the weight of the goods you were purchasing, say, three pounds of sausages, on the other.

Make sense?

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