My thoughts on the Market Scale Problems

I first started on the one-pan scale problem first using 5 weights to sum up from 1 to 31. The trial and error process began with trying out sum of weights for the first few whole numbers. Since I know the highest number achievable should be 31, having two weights both using 1 is irrational. Hence, I started with the first two weights as 1 and 2 and was able to reach 3 through that. For next whole number of 4, I instinctively assigned the third weight as 4 and was able to reach whole number of 7. The problem begins to look like a combinations of weights using exponents of 2. Using this rationale, I assigned fourth and fifth weight as 8 and 16 respectively which the sum of five weights are 31. I knew then that the solution is a series of exponents of 2. I proceeded to validate this solution accordingly. Moreover, each weight has a binary status of 0 and 1, so the total combination is 2^5-1 which also comes out to be 31.

Next, I worked on the two-pan scale problem with only 4 weights to sum up from 1 to 40. Based on the hint of an exponent series from the one-pan scale, I thought of a series of exponents which should provide the largest sum of 40. After plugging in 3, the weights of 1, 3, 9, 27 happens to add up to 40. I then validated this result using + and - of each weight. This has similar approach where technically each weight has a status of (0, 1, -1), but the value of 27 can only be (0,1).

For the classroom, I would incorporate this exercise in learning of combinatorics. While it presents a good problem to learn about usage of exponents of 2 and 3, the part which intrigues me is the discussion of how you can generate the total number of combinations by thinking about combinatorics. For example, the one-scale is just 4 unique numbers each with a status of (0,1) for product sum which leads to 2^n-1. For the two-scale problem, it's trickier with the limitation that it must be a whole number (non-negative), but it warrants further discussion on how to constraints can be represented.

Finally, in connection with number theory, I recall that any number can be presented in a base system. Let it be base 2 or base 3. This is a good exercise to present the numbers in a binary or ternary system.