Figure 1. A balance (Richard via Picassa Web)

There aren’t many mathematics blogs that are either interesting or accessible enough (meaning “easy to understand”)  to attract my attention. However, one that I do take a peak at on occasion is The Math Less Traveled by Brent Yorgey,  a PhD student in programming languages at the University of Pennsylvania. Brent is also a former math teacher at Woodrow Wilson SHS in Washington DC.

The Math Less Traveled offers various musings on mathematics and programming and an occasional interesting problem. One such problem appeared there recently. It is known as the broken weight problem. As Yorkey points out, it’s an old problem (circa 1612!) and it appears in an interesting book called 100 Great Problems in Elementary Mathematics: Their History and Solution by Heinrich Dorrie (which also happens to be available at the Pikes Peak Library in Colorado Springs).

According to Yorkey, it goes like this:

A merchant had a forty pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used to weigh every integral weight between 1 and 40 pounds. What were the weights of the pieces?

Okay, I said to myself, I can probably figure that out. It might not be pretty, but I can figure it out!

My Thinking on the Broken Weight Problem

My first thought was that the sum of the eights must equal 40. That seems useful. Then I got thinking that some addition and subtraction was going to be used to generate the combinations required to weigh every integral value from 1 to 40. Uh-huh.

Next, I realized all the numbers were probably all odd, since only even numbers can be generated by adding and subtracting even numbers, but both odd and even numbers can be made by addition and subtraction of odd numbers. So far so good!

My next thought is that the average of  the 4 numbers was 10. So I figured the required numbers would probably lie on either side of 10 and be not too far from 10.

To summarize: sum of 40, not far from 10, all odd.

So 9 and 11 are in play, and so are 7 and 13. HOWEVER the difference of 9 and 11 is 2 and so is the difference of 11 and 13. I got to figured that each consecutive pair would have to have a difference different from any other pair so as the maximum number of sums and differences would be obtainable. I had a hunch that prime numbers would be involved in some way (I am not sure why, it was just a hunch). What the heck, I’ll try 1, 9, 13 and 17. They have differences of 8, 4, and 4 respectively and their sum is 40 (but I didn’t check the differences or else I probably would have tried something else). I avoided 15 for no other reason than it isn’t prime.

Does that work? Nope, you can’t set up a scale to measure a weight of 2! Choosing 1 was probably fairly limiting also. Try again.  I’ll pull the 13 down to an 11 and push the one up to a 3, thus preserving the sum of 40. So I’m working with 3, 9, 11 and 17. The sum is 40, they are on either side of 10, they are all odd, and I’ve got a couple primes in there to boot.

Unfortunately, try as I might (using this admittedly ugly method) I cannot get a weight of 4 to balance (nor 7 for that matter). Must. Think…

Three Weeks Later

I finally got it. I’d say I spent an additional 2-3 hours on it overall. I posed the problem to a colleague at school and he had in a few days! He almost showed me his result but I stopped him before I could see it. I wanted to figure it out on my own. But, I did run my reasoning by him to see if I was on the right track.

• The 4 weights sum to 40. Check.
• Since the average is ten, they must be close to 10. Not exactly.
• They must all be odd. Check.
• I forgot to ask if they were all prime.

He let slip that three of them were in fact less than ten. Oh well I guess I’ll take that clue.

The breakthrough came when I realized hat one of them must be 1. Because, when all four pieces are on one side to balance an object of weight = 40, the only way to measure a weight of 39, would be to remove one piece and that piece had to be 1.

So I made a list of all combinations of three odd numbers (including 1) less than ten, plus one other to add up to 40. This led to

• 1, 3, 7, 29
• 1, 5, 7, 27
• 1, 3, 9, 27
• 1, 5, 9, 25

At this point the answer was staring me right in the face but I still didn’t see it.  I ruled out 1, 5, 9, 25 because it had two differences equal to 4 (5 – 1 and 9 – 5), and I had previously decided  it would be advantageous to have the differences between all four numbers to be not the same. On a hunch, I then chose the one that had the smallest difference  between 9 and the largest number and this was 1, 3, 9, 27 (27 – 9 = 18). Then I started testing, working my way down from 40. It worked!

Then I saw the pattern. To find the correct N number of pieces, choose weights with values of (N-1)⁰, (N – 1)¹, … (N-1)^n-1. In this case N = 4 pieces and 3⁰ = 1, 3¹ = 3, 3² = 9 and 3³ = 27.

I tested this idea with N = 3 and this works for finding 3 weights that can weigh any integral value up to 7. I assume it would work for N = 5 with weights of 1, 4, 16, 64 and 256 to weigh any integral weight up to 341. Perhaps I’ll try to write a computer program to test it.

Note added in addendum: As Brent points out in a comment, integral values from 1 – 341 cannot be weighed with weights of 1, 4, 16, 64 and 256. Obviously! There is no way to balance a weight of 2 (or 6, and many others).

A New Pattern Discovered

As it turns out, my colleague hit on that (N-1)⁰, (N – 1)¹, … (N-1)^n-1 idea pretty quickly. Smart! But before I saw that for my self I actually found another pattern that works, at least for the one case for which I tried it.

Figure 2. Testing from 121 to 41 with 5 weights (1, 3, 9, 27, 81)

What if? I thought, I added another term to the sequence 3⁰ = 1, 3¹ = 3, 3² = 9 and 3³ = 27? That would be 3⁴ = 81, which would give five weights with a sum of 121. I worked it down for each weight from 121 to 41 (since I already knew it would work from 40 on down) and sure enough it could be done (See Figure 2 which shows the work from 80 to 41).

So it appears that one could perhaps sometimes find two different total weights that could be weighed with N pieces. Even though 341 fails with 1, 4, 16, 64 and 256, 15 can be done with 1, 2, 4, and 8 (using a similar idea to find that 121 can be done with 1, 3, 9, 27, 81). And guess what? Integral weights from 1-31 can be weighed with weights of 1, 2, 4, 8 and 16.

I have come to think that no base greater than 3 can be used to generate a sequence that works. This is because there is no way to combine the resulting pieces to balance a weight of 2.

There are probably some other interesting things to investigate along these lines.

Note: This is a repost of mine  from the 20inthe21st Ning, a Ning created for D20 teachers who want to improve use of 21st century skills in the classroom.

We are all searching for those “magic bullets” that will somehow create a better learning environment and encourage intrinsically motivated children. That’s a terrible analogy (magic bullet) in this case isn’t it?! I wish I could think of a different way to put it. Hmmm… anyway.

There Ain’t No Magic Bullet

Here’s what I have been doing lately (which I have to say, is quite different from what I was doing only a month or so ago) and it seems to make for a happier classroom and a lot more on-task behavior. This is how I run my 7th grade prealgebra classes. I hope to start using it in Algebra soon as well.

(1) I stopped giving detentions for being off task, talking, pulling others off task, or not completing work. Detentions don’t work and make schools into minimum-security prisons. Prison is not a happy motivating place.

(2) I stopped “going over the homework” and stopped asking things like “what questions do you have? Do you need to see anything done or have it explained?”

(3) The kids work in groups (which they did before, but not like this). They call me over when everyone in the group has the task/problems done. I read off the answers to the group and they circle the ones they got wrong. Then I leave and tell them to call me back when they are ready to try again on those.

(4) I usually answer questions with questions, but I will occasionally explain some thinking, give some ideas, or offer up a clue or an example on the board.

(4) The group does not move on to the next thing until they have ALL solved/answered/completed EVERY part of the assignment correctly.

(5) Not all groups are working on the same investigation. They go at their own pace. The fact that they don’t get to go on to the next investigation until the current one is done has proven to be huge. Much more cooperation, more more effort.

(6) There is no more homework (almost)! It’s not needed because much more gets done in class. The kids are so much happier and they are so much more motivated by this! It’s a 180-turn from what I thought was “the best way to educate.”

(7) The only time there is an outside class assignment is at the end of a unit or section when I assign practice/review problems. This now happens about once every 2-3 weeks. Even so, some of the kids get the majority of them done in class

(8) I don’t have to spend any time grading this work, although I do have to go over the answers about 10 times for each lesson, instead of just once. I don’t mind. It’s way better, since when am going over the problems with the group I sit right there with them and interact with them on a much more personal level. With this system, I can be assured everyone gets the answers right so there is nothing to grade. I still enter the assignment in IC, but only have to check them off as they are done.

(8) I randomly ask learners in the group to explain how they got their answer. If they can’t do it, I leave and come back when there are ready to explain it.

(9) If any group or person gets way ahead or completes the learning objective far in advance of the others, I turn them loose on a small packet of assorted interesting and challenging problems.

(10) One thing I still do is make a lot of noise, tell stupid jokes, alternate between talking excessively loud and extremely quiet, switch between acting deadly serious and being a complete idiot, playing air guitar, and shouting out encouragement.

I’m still fine tuning it, but right now I am having more fun, and making more progress than ever before. And, if feedback from from the kids is any indication, so are they.

The biggest challenges are:

(1) Getting the composition of the groups right.

(2) What to do when a kid is absent and rejoins a group that has moved on.

(3) What to do with a brilliant kid who is frustrated with his peers who can’t keep up or “get it” as fast.

I implemented this scheme too late to have any affect on CSAP performance this year. Even though I think CSAP is an incredible waste of time and resources, I am curious to see what the effect will be next year after a full year of this way of learning.

If I could sum this whole process up it would be like this: We are no longer covering the material, the students are learning the material.

Exactly how relevant this material is to their lives and futures is, of course, a topic for another discussion.

ADDENDUM of April 10, 2010: Based on one classes quiz scores on the material being studied (it’s just one class out of three mind you) it has made no difference at all. How can that be?

Although I have to say, many of the wrong answers and apparent failure to have learned much at during the past three weeks of work seems to mostly stem from a solid lack of understanding of, or facility with, the previous six years worth of material and not necessarily what this quiz was on (order of operations, distributive property, division of integers). Very discouraging.

Which leads me to consider the possibility that the whole idea behind what me and my school are trying to accomplish with math education is hopelessly broken and needs wholesale change.

Addendum of May 26, 2010: Another quiz. This comes after 3-4 weeks of self-paced study of similarity. I have not analyzed the results yet but they are more encouraging than they were last time. The students who usually do well, did well, and those that don’t… well.