The market puzzle

The wrong idea

I started the scale problem with a brute force method of just trying whatever. Originally I thought "oh we must need a weight of 1 gram" but then I became fixated on the fact that two subsequent numbers could also give us the ability to measure 1 gram. 

For example: 2g weight + 1g spices = 3g weight   

This led me down a rabbit hole of trying pairs. Spoiler alert! This was not the right solution, but I'll show my initial thoughts anyway because they helped me get to an answer in the end. Let's start with 2,3. If we start with 2, 3 we can get all numbers from 1 to 5, except for 4. So now we need a 4 weight.

1 = -2+3

2 =  2

3 =    3

4 = ...

5 =  2+3

Adding that weight we get

4 =       4

5 =   2+3

6 = 1+2+3

7 =     3+4

8 = 1  +3+4

9 =   2+3+4

10= 1+2+3+4

And now we're stuck again and need an 11g weight. But the maximum value this weight can give us is 2+3+4+11 = 20. Not even close!

I then thought maybe our subsequent weights needs to be higher value, and tried that but it was difficult to know what the smaller weights should be, and the sums I could get seemed random. I thought alright it's time to try a different approach, let's throw out the idea that we can do this without a 1g weight.

A new approach

Starting with a 1g weight I thought what is the next smallest number n that would "fill in all the gaps" between 1 and n+1? Here you might try 2, 3, 4 (and I did!) and find that the answer is 3 which gives:

1 =  1

2 = -1 + 3

3 =      3

4 =  1 + 3

But now we're stuck and need to add a weight that gives us 5 grams. Fortunately, we've started a pattern and so finding which weight comes next is simpler than before. We know we can make up to 4 grams already, so we now need a value n such that n-4=5, i.e. n=9.

5 = -1 - 3 + 9

6 =    - 3 + 9

...

13  =  1 + 3 + 9

And now we're stuck again and need to add a weight that gives us 14 grams. Similar to before we need a value n such that n-13=14, i.e. n=27.

14 = -1 - 3 - 9 + 27

...

40 =  1 + 3 + 9 + 27    Yay!

So our weights are 1, 3, 9, and 27 grams.

What about one pan?

We can use what we've learned to find 5 weights that can sum to 31 grams. Let's start with a 1g weight. Now we want the next smallest number that will fill in the gaps between 1 and n+1. Just as before, we try 2 and 3. Because we now only have one pan, we can't "subtract" digits anymore. So this time we find that 2 works, giving:

1 = 1

2 =   2

3 = 1+2

And now we're stuck again. Since we can't subtract anymore, the next weight we need is 4 grams. Now the maximum we can get with our weights is:

7 = 1+2+4

And now we need an 8 gram weight to give us a maximum of:

15 = 1+2+4+8

So we need a 16 gram weight, giving us a maximum of:

31 = 1+2+4+8+16    Yay!

So our weights in this case are 1, 2, 4, 8, and 16 grams.

Extending these ideas

There are a few interesting topics that could be introduced with this puzzle. It could be a way to connect with combinatorics. For example, the number of choices we have for each weight in the 2 pan puzzle is 3: left pan, right pan, or no pan. This means with 4 weights we have 3^4=81 possibilities. But didn't we say it can measure up to 40? Why are we now getting 81? Well, with the four weights, we can get 40 on the left pan, 40 on the right, and 0 (no weights are used), summing to 81 possibilities. This would be a great exercise for students to try, and then you could ask them why the second problem with just one pan sums to 31? Well here we have only 2 possibilities for each weight: used or not used. With 5 weights that is 2^5=32 possibilities. But didn't we say we can measure up to 31 grams? Why do we now have 32 possibilities? The answer is of course that no weights can be used to measure 0 grams.

It could also be a fun way to introduce geometric sequences. You could ask the students if they notice any patterns in the weight values and what they think the next weight would be? Then see if they could use the combinatorial logic discussed above to determine how many grams this set of weights could measure. This would work for both the two and one pan problems. You could also relate the series of numbers (and the number of choices for the weights in each problem) to different bases. 1, 3, 9, 27, ... can be related to base 3 and 1, 2, 4, 8, ... to binary. Lots of applications!

A flower market in Mexico City

Does the story matter?

It was absolutely shocking to learn that mathematical word problems date back to at least Babylonian times, and quite interesting indeed that those problems had the same issues with realism that many word problems today. It is little insights like these into the lives of people that came 4000+ years before us that really make me feel connected to these individuals that lived so long ago.

In my own learning I was the student who didn't read the words to the problem, but just extracted what I thought was the meaning based on the numbers and partied on. This happened from childhood into university, and I really had to make an effort at the end of an exam to review each problem (and really read each word!) to ensure I (1) hadn't missed anything and (2) answered what the question was actually asking. It always made me wonder what is the point of having a word problem to begin with when, just as we mentioned in class, the story doesn't matter? To me it always felt like the abstractions were the interesting bit, and the words were just "in the way".

I've come to realize that perhaps that was part of the lesson? Being able to filter through the noise of "reality" in order to apply the appropriate mathematics tool is a useful skill. One that I've used in my career as a data scientist many times. I worked for a biotech company where I'd work closely with biologists to investigate their questions through data. These started with a lot of noise! And very often an ill-defined question that we'd collaborate to refine. Despite being the student who didn't want to bother with the story, this back and forth was often my favourite part of the job. 

A tree at Horne Lake that looks like it belongs in a story

Babylonian algebra hurts my head

I was impressed when reading how Babylonians could solve higher order equations by making them "look like" quadratics (for which they had a formula) using substitution. In some ways that feels one step away from our modern algebraic symbols. It does not quite fit the definition of syncopated algebra (though this term is new to me and I'm still not completely clear on the definition), but it feels like a sort of syncopation in the sense that they were seeing "x^2 hidden in x^4" for example so they could use their familiar tools.

I will say that following the descriptions of the Babylonian rhetorical algebra made my brain hurt. It was such a new way of thinking about these kinds of problems and really difficult to try follow their steps in earnest without resorting back to my modern algebraic tools, and in some cases I had to in order to understand what they were doing. Example 4.8 was one such case. I was really confused by the Babylonian step 1 and had to go through the algebra to grasp what they were doing so quickly. It's so interesting to think about how the ways in which we learned math shape our understanding in such a way that it becomes difficult to even think in another way, at least for me.

This reading gave me a re-appreciation for our modern algebraic tools!

The shape of my week

I'm not sure if it is because of the way one of my elementary school teachers chose to write out the days of the week or some conceptualization I developed on my own, but I've always pictured a week in my head like this:

    Sunday    Monday

                    Tuesday

                    Wednesday

                    Thursday

    Saturday  Friday

A funny not-quite-cyclical-but-kind-of series of days that make up the week, with Saturday and Sunday getting their own special place. Over the years I've asked my friends how they picture a week and no one has ever had this answer or even could relate to my visual. Interestingly, in my previous career this image felt especially relatable as I was always trying to rush through my week to get to the weekend, where I could spend my "real" time, the time that mattered so much more to me, the time separate from the rest.

I really related to the sundial notion of an hour described in the Scientific American article that had variable length depending on the season. This was a byproduct of the tool itself, but to me it is also how time sometimes *feels*. In fact, I often resent modern aspects of time of schedules. A friend once said to me that he can't believe that no matter the weather or occasion, you could go forward 10 years, 20, 50, and if it's a Wednesday we'll probably be indoors and at work.

Two somewhat tangental things:

• I wish the articles talked more about why 7 days per week. There didn't seem to be much of a focus on 7 at all and I'm curious!
• I was surprised to learn that the language of minutes and seconds was first used as a subdivision of degrees, not time. I knew of the terminology but had always assumed that it was first used in the time sense, likely because that is just the order in which I learned the usages. How very me-centric to think the terms must have developed chronologically to how I learned them!

Some gentle waves on Pender island

Babylonian multiplication table (with fractions!)

 Here are five values that multiple to 45 written in base 60:

1,30 * 30 = 45

4 * 11,15 = 45

5 * 9 = 45

5,20 * 8,26,15 = 45

8 * 5,37,30 = 45

Learning while unlearning

The thing that surprised me most was the sheer quantity of influences between regions (often far from one another) that did not share culture or even a language. It is one thing to have a rough idea of influence and quite another to see these connections mapped out in Figure 1.4 (page 14). Moreover to recognize that this is a lower bound as there were surely many more connections for which we simply have no record. This seems to demonstrate the universality of a subject like math.

Another thing that struck me was that despite the reading being published in 1991, my undergraduate history of math course (likely taken around 2012) taught us that our modern numerals were Arabic, not Indian. This myth was clearly dispelled at least 20 years prior to my taking that class but the material was out of date. It was a good reminder for me to fact-check items from my own education that I've assumed were truths.

Finally, I was surprised to read that the concept of place value was discovered upon four independent occasions as mentioned on page 22. The reason it surprised me was because notation could have developed into absolutely anything but there was something about place value (perhaps the convenience) that people were drawn to. It reminded me of the concept of "carcinization" in evolutionary biology where unrelated crustaceans evolved to have a crab-like body. There's just something advantageous about a crab-like body, and apparently place value notation as well. 

Not a crab... yet

(A starfish I saw on a beach in Sechelt)

To math is human

Before reading the analytic survey my thoughts were that math history should be considered an important aspect of teaching math. Grounding math in its historical context is a more honest way of learning. It helps combat the misconception that "math is all solved". It shows by example that math is a human endeavour* (I have some more thoughts on this particular claim in my final paragraph). It shows that math exploration can be (and is often) VERY messy! And that the struggle and mess of it all is okay and expected. These are a subset of my initial thoughts prior to reading the article. As for the how of it all, I was not so sure! Luckily the analytic survey showed us that there is a fruitful landscape of historical math that we can pull from for our lessons, and we don't have to start from scratch.

Upon reading the article, I had three distinct moments of pause: one during the objections, one argument, and one example:

• Objection: What struck me first was the first objection listed in the article on page 203 that "history is not math". Perhaps it is true that history is not necessarily math; yet math is certainly connected to history and history itself has shaped math. To me this objection was a shallow perspective of what might be considered math.
• Argument: On page 209 the authors talk about how history of math can help with the question of "What's the point?" by revealing the conditions that made the math necessary. This was not something I had thought of before reading the article, yet it was so compelling I wanted to edit my pre-reading list to include it! 
• Example: By showing how different cultures and people frame the same problem and providing a vast array of proofs, the Pythagorean theorem historical package example presented on page 218 seemed to encompass all my pre-reading/gut feelings around what makes history of math worthwhile. This really convinced me the value of historical packages as it shows the students that math is more than "one right answer".

*One sentence I originally typed, and then erased, and then typed again was "math is a human endeavour". Given the recent news (that many of us have already chatted about together) that OpenAI released an AI-generated solution to the Navier–Stokes Millennium Prize Problem, I am not sure if this claim remains true. Though the data used by OpenAI was certainly human-made, the proof itself was not, and I'm not sure what this means about the future of math research, and how we might continue to relate to a proof not authored by humans. Of course computers and other technology have aided us in math research in the past, but this feels different as AI was not so much used as an aid, but was instead the author. It seems a few of us are struggling with or are curious about this question, and I'd love to hear more thoughts from others, especially as it relates to math (recent) history.

A picture I took of some spring flowers, just to lighten the mood!

Hello (again!)

Hello from Pender Island