After trying a brute-force approach, I started exploring specific number patterns to find a more structured solution. First, I considered using consecutive even numbers like 2, 4, 6, and so on, thinking they might build up neatly. However, I quickly realized that even though they could add up to small values easily, they weren’t flexible enough to reach the larger numbers like 40 efficiently. Next, I turned to powers of 2—such as 2, 4, 8, and 16—remembering how the ancient Egyptians used them in their number system. While powers of 2 are powerful for addition-based systems, I found them limiting on a two-pan scale, where subtraction is just as important as addition. It was only when I began experimenting with powers of 3—specifically 1, 3, 9, and 27—that everything clicked. These weights, used with the idea that they can be placed on either side of the scale, allowed for a combination of positive and negative contributions. This meant I could weigh any amount from 1 to 40 grams, not just by adding up, but by balancing weights in a way that also takes subtraction into account. The powers of 3, it turns out, perfectly fit the logic of the balanced ternary system, which is ideal for this type of scale.
I like how you described your thought process—moving from even numbers, to powers of 2, and finally realizing that powers of 3 might work. It shows persistence and good reasoning. However, you didn’t really explain how you arrived at the exact powers of 3 or how you verified that they can measure every number from 1 to 40. I can see that you reached the right insight, but the process of finding and confirming it isn’t fully shown.
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