4 numbers have to at least add up to 40,
Lets try a smaller example, 1- 10
OK, so clearly 1, 2, 3, 4 works here, even with a one pan solution.
Lets do to 1 - 20.
I don't necessarily know I need 1 yet, but lets say 1 because I feel like it would fill in the small nooks from either side.
So, since I need the number to add to 20, some numbers must be bigger than the numbers in the original set.
Im going to skip 2, because I can get 2 with 3 - 1.
I can also get 4 from 3 + 1
Can't get 5 yet, but lets say that instead of just taking 5, I can subtract the 4 we can already create. I know that we can create negative values because there is two sides to the scale.
I can get if i subtract 9 and 4, so lets see if we can make 6 as well from 1, 3, and 9.
Yes, its 9 - 6.
7 is 9 - 6 + 1
8 is 9 - 1
9 is 9!.
Let's see how far this can go.
10 is 9 + 1
11 is 9 + 3 - 1
12 is 9 + 3
13 is 9 + 3 + 1
14 is ?? but like last time, we know that we can subtract 13 from the next number, so lets try doing 27 as the last number in our set. It seems to work nicely since 1 + 3 + 9 + 27 is 40.
so as we just set, 14 is 27 - (1 + 3 + 9)
we already know that we can make every number from -13 to 13, so we know that we can reach every number from 14 to 40!.
On a one pan scale, you would need powers of 2 to get every number, because you don't have the luxury of subtracting, so you need a binary representation to get all integers. so it would be 1, 2, 4, 8, 16.
To extend, I would ask what the general rule is for reaching up to any integer m, and what is the number of weights you need to get it? I think this would get them to experience the practice of solving a smaller problem and generalizing it to address a wider range of problems.
Your explanation feels very natural—it shows how you were thinking through the problem, not just presenting the answer. I like how you started with a smaller example and gradually built up the logic, realizing why subtraction expands the range and why powers of 3 work for the two-pan case. The reasoning feels intuitive and exploratory, exactly what the assignment asked for. I also liked the way you connected this to generalization and teaching ideas—it shows higher-level thinking.
ReplyDeleteHonestly, your work was fun to read, though a bit informal in style—it reads like you’re “thinking out loud,” which actually makes your reasoning easy to follow.