Problem #4: Weights

Given a two-pan-balance and weights of 1, 3, 9, 27, 81, 243, and 729 grams, what weights can you measure? If you want to upgrade your set of weights, what is the next weight you might buy?

After rewriting the different

weights I had it was easy to see that they were all powers of 3 (0-6) so I answered the second question very quickly. The next weight in the pattern woul

d have to be 3^7 which is 2187 grams.

I then went back to the first question. From first glance I was only thinking about addition- putting a weight or set of weights on one side and using an unknown weight to balance that out. I was also thinking about only using two weights for some reason. So I wrote out all the possible combinations of two weights on one side. I came up with 21 different possiblities and also figured out the smallest

I could measure was 1 gram and the largest unknown would be 1093 grams.

All this talk about combinations started me thinking about put

ting more than two weights on the one side. So using the combination formula, I figured out the number possibilities with 1 to 7 weights on the scale. Using only one side of the scale, there are 127 different unknown weights I could measure. SInce I didn't take the time to figure out what each of these possiblities were, this didn't really help me that much :/

I stepped away from the problem for a bit and returned with a new take on this problem.

For some reason when I first sta

rted this problem, I only thought about putting the known weights on one side and trying to balance them out with an unknown (addition). But my new idea led me to subtraction (weights on both sides!). To come to this conclusion, I wrote out all the weights I knew up to 36 and tried thinking about ways I could "fill in the blanks". I drew myself some pictures to help out.

Since this wasn't very orgainized in my head I decided to make a table. I listed numbers 1-20 and came up with combinations of how to measure them, some with addition, some with subtraction. There was no pattern I could see in how I could come up with the numbers and also no pattern on when they were added or subtracted.

From this table I want to say that there is a way to measure all the possible number combinations (from 1 gram to 1093 grams), but I know I can't just do that without trying all of those numbers.

I know this isn't a conclusive proof, I'm just a bit at a loss on where to go next... I'll be sure to revisit this later this week.

In thinking about this a little later, I thought about maybe proving that this works for all prime numbers 1-1093. Since all other numbers are divisible by something else, I figured I could start to make a case around that. In looking up how many numbers are prime, I realized that was a horrible idea (there's around 100). So I don't really know where else to go with this. I know I can't just say because I've proved it for the first 20 numbers, I must work for the rest, but I am completely stumped...