Weights Problem

What 8 numbers add up to 3280 and allow you to combine them to make every number less than 3280.

Inference will allow you to figure out some of this.

We need 1 to measure 1
We need 2 to measure 2 (No need to have two 1s)
1 plus 2 gets us to 3
4 will get 4, 5, 6, 7
8

So there is a pattern of
1, 2, 4, 8, 16, 32, 64, 128, 256

But when we get to 8 numbers we can't add them up to 3280.

So we have to take a different approach. Assuming we have a scale, there is nothing that says we can only put the weights on one side of the scale.

If we are weighing a 3 kg piece we can counterbalance by adding a 1 kg stone to that side and a 4 kg on the other side. We then figure the 8 numbers that added or subtracted from each other will give every number 1 to 3280.

As I interpret the problem I see no solution, even with a balance scale.

3^8 is 3280

The power of 3 will provide your answer

I spoke too soon. If you use a balance scale the weights would be:

1, 3, 9, 27, 81, 243, 729, 2187



Edited to note that Parados beat me to it.

Doesn't really explain why powers of 3 works. Is the proof by mathematical induction? Why should it be powers of 3?

No, but the sum of the first eight powers of three is 3280.


Because a stone can be in the left pan, the right pan, or neither pan.

Here's something I put together for a smaller version of the problem:
http://www.able2know.org/forums/viewtopic.php?p=3168646#3168646

This doesn't prove the powers of three result, but it does show you how to generate any given weight. In order to produce a weight, it must be the difference of two numbers whose base three representations only have ones in it. Let's look at 42. 42 in base three is...

01120

That's a direct representation of how you could measure this in base three: use one 27kg, one 9kg and two 3kg weights. This can't be measured directly since we don't have two 3kg pieces. To eliminate the two, we need to add 01110 to produce 10000. That means that you can get 42 by using the 81kg piece on one side (10000) and using the 27kg, 9kg and 3kg weights (1110) on the other.

Trying again with 1234 kg, in base three:

01200201

Needed adder
01100100

Result
10001001

So you can get 1234kg by putting the 2187kg, 27kg and 1kg weights on one side and the 729kg, 243kg and 9kg weights on the other.