In tackling this problem, I was able to determine that 1 and 3 must be among the four weights, while the other two weights fall between 10 and 30. However, because I was using an elimination-by-exhaustion approach, I didn’t explore all the possible combinations. For a one-pan scale, the five weights would be 1, 2, 3, 6, and 21.
This puzzle can serve as an introduction to combinatorics, demonstrating how different combinations can be assembled to achieve a desired outcome. Students can explore how many possible combinations exist, which work, and which don't. This encourages them to practice elimination and grouping methods, while strengthening their understanding of numerical expressions and the concept of equality.
The puzzle also ties into number theory, illustrating how numbers can be manipulated in various ways. I started to imagine possibilities beyond one- or two-pan scales. Could there be three, four, or even more pans for measuring a larger number of weights? I believe there could be a pattern that forms an arithmetic series.
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Hi Caris. Thanks for your start on this puzzle, but I think you have more work to do on it, for both the two-pan and one-pan balances! The 1 and 3 gram weights work for amounts from 1 to 4 grams, but there are two more weights to work out -- and the elimination-by-exhaustion approach will not take that long as the total amount weighed is limited to 40 grams. Similarly, you have further work to do on the one-pan balance. 1, 2, 3, 6 and 21 don't work for all the amounts, and they are redundant for others. For example, you don't need the 3 gram weight as you can put the 1 and 2 together on the one pan to make 2, and you don't need the 6 gram weight as the 1, 2 and 3 grams together add up to 6. The weights you propose don't work for the amounts from 13 to 20 (unless I'm missing something?), as you cannot subtract weights on the one-pan balance -- only add them.
ReplyDeletePlease do some more work on this puzzle and add it to the post as an **EDIT, so that I can see you completed thinking and work on this!