I will start with a puzzle, as background food for thought: what is the minimum number of weights, each weighing an integral number of pounds, required to be able to measure with a balance scale any integer weight from 1 pound to 30 pounds, inclusive?
When I was a kid, one of my mathematics classes participated in the Atlantic-Pacific Mathematics League. This was nearly 25 years ago, but it seems the league is still going strong, with little or no change to the format. Each month students get 6 problems and 30 minutes in which to solve them. I remember looking forward to those problems each month. The above puzzle was one of those problems. Or at least, it is my best recollection of the exact statement of the problem.
What is the correct answer? I will wait to discuss details until the end of this post. Suffice it to say here that the answer that I submitted was incorrect… or so I was told.
I was reminded of this episode while reading an article published in the most recent College Mathematics Journal, written by Alif Anggoro, Eddy Liu, and Angus Tulloch. The article is interesting for a couple of reasons. First, the authors are seventh and eighth graders. Second, the subject of the article involves a similar situation, where a problem seems to have multiple solutions, depending on assumptions that are not made explicit in the problem statement.
The article describes a dialogue between the students and their teacher, relating to the following problem: provide the next row of numbers in the following triangular array:
I usually dislike “complete the sequence” problems on sight, because they are under-determined. As the students correctly point out, “any five numbers should be acceptable.” In this case, the intended solution is the next row in Pascal’s triangle, or (1, 4, 6, 4, 1). However, the students came up with (1, 4, 5, 4, 1), and they make the case in the article that the rule or pattern that yields this next row is even simpler than that for Pascal’s triangle. I will leave it to you to either read the article or work out what that rule might be.
This raises all sorts of interesting questions. What is meant by “simpler”? The students’ pattern indeed yields a closed form expression for the entry that is simpler to compute than the binomial coefficient of Pascal’s triangle. However, the associated recurrence relation seems slightly more complex than its counterpart.
Resolution of these issues is not really the point of this post. My point is simply to applaud the teacher’s encouragement of these questions. The students were sufficiently interested in and excited about their proposed solution, and motivated to defend it at least in part by the teacher’s questions and arguments, that they investigated it further, proving several nice properties of “their” triangle in the process.
Coming back to the balance scale puzzle, I do not remember having quite the same engaging experience. Here, the intended correct answer to the problem was five weights, of 1, 2, 4, 8, and 16 pounds, allowing measurement of any weight up to 31 pounds. Looking back, I suppose the problem was meant to have a very specific purpose, to “teach” students about binary numbers, perhaps, just as the other students’ problem was intended to “teach” them about Pascal’s triangle.
The incorrect answer that I submitted was four weights of 1, 3, 9, and 17 pounds. With these weights, you can measure any weight up to 30 pounds… with the caveat that sometimes you have to put weights on both sides of the scale. For example, to measure the weight of a 2-pound object, you must put the object and the 1-pound weight on one plate of the scale, and the 3-pound weight on the other plate.
When I described my solution, my teacher responded to the effect of, “You can’t do that.” Never mind that the problem didn’t state that I couldn’t do that. I was supposed to learn about binary numbers, my solution did not reflect that, and the discussion stopped there.
That was a sad day, I think, because there were so many interesting paths where the problem and its multiple solutions could have led. The intended “binary” solution works up to 31 pounds; why did my solution, which essentially assumed more “expressive power” in the use of weights, only work up to the required limit of 30 pounds? Do other sets of weights do even better? Is there a provably “best” set of weights?
As it turned out, I had the right idea, but a less-than-elegant implementation. The 30-pound limit in the problem statement was a distraction, and I didn’t yet see the nicer “ternary” pattern emerging that would have allowed measurement of weights up to 40 pounds.
This is all a roundabout message to the teachers, parents, etc., out there: when a student asks, “Why can’t I do this?”, please do not respond with, “Because the plan was for you to do this.” There is almost always interesting mathematics buried in those questions. Instead, walk down the path with them: “Well, let’s see what happens if you do that.” Quite often, it’s not just the student that ends up learning something.