Yay — we’re home! And despite the fun of hanging out at beaches (Oregon) and cities (Cincinnati, Chicago) with a total of three dozen family and friends over the past month, it feels mighty good to be home. But our classes start in just over three weeks (wait, really? Somehow it feels like it should be a lot further away than that), so it’s back to work sooner rather than later. As in, tomorrow.

However, there’s no better way to get into the work spirit than with a Carnival! The Carnival of Maths #55 is over at Maths at SBHS, the Class blog for Maths at Sowerby Bridge High School, UK. One of the things I really like about this edition is that I didn’t recognize several of the blogs with contributing articles, so there are a whole bunch of new places to check out. Woo hoo! (Plus, the post below it was pretty funny, about estimating the general shape of a graph of the number of bras left on a fence over time in New Zealand.)

The next Carnival of Mathematics will be hosted by Someone at Some point in the future. Stealth Carnivaling at its best!

As part of our limited-internet limited-posts July, we fell behind on all the Carnivals. Rather than skip them completely, which would be sad, we’ll do a quick summary.

Earlier this month was Math Teachers at Play #11, hosted by Sue Van Hattum at Math Mama Writes…. (Incidentally, her current post on Myths about Math is also interesting to think about. I see a lot of those beliefs when I teach our Thinking Mathematically course, which is one of my favorite classes to teach [usually taken only to satisfy the general ed requirement]).

Finally, today isMath Teachers at Play #12, hosted by Jason Dyer over at The Number Warrior. (I love the photo in one of the links of an elevator in Serbia with negative numbers!)

This is unrelated to Carnivals, but we were thrilled to be included on Online Universities’ list of the 100 best websites for Mathletes. They have several groups of info, including major math organizations [primarily in the US], a list of contest sites, and a section on Career Tools and Guides. Thanks!

There was a show on TV late last night, but I don’t know what the show was or, for that matter, what channel it was on. Two men and two women had been trying to lose weight in some boys vs. girls competition, so they added up the weight loss for each pair, divided by their previous weight, and compared. They guys had lost more weight, but the gals had lost a higher percentage so they won.

Comparing percentages instead of absolute amounts made sense, but I’m not sure what the fairest way to make that comparison is. Suppose you had a team with a 100 and a 200 pound person, and another team with two 150 pound people. If each person lost 5% of their body weight then the total for each team would be 15 pounds, or 5%.

But what if someone lost 1 pound? If it were the 100 pound person, that seems more significant than the 200 pound person, but they’re counted the same because those two are on the same team. Likewise, if the 200 pound person lost an extra 10 pounds and one of the 150-pound people lost 9 pounds, the first team would win even though the second individual lost a higher individual percentage because it’s the team total that is used for the percentage, and both teams weigh 300 pounds.

I was thinking that it might make more sense to compute the percentages and then average them. The problem is that it might be easier for a 200 pound person to lose even 5% of their weight than a 100 pound person so even that is imperfect — ease and health of weight loss don’t quite follow a linear scale. But more importantly, I suspect, it might just be one step too confusing for a late night show.

The other day our 9-year old said to me, “A is 1 inch from B, and B is 1 inch from C. How far apart are A and C?” It turns out this was a joke, and the answer was supposed to be 1 inch (I think he got it from Calvin and Hobbes, though I don’t remember seeing it there), but of course I couldn’t help mentioning that the answer could be anywhere from 0 to 2 inches.

It got me thinking — what would the “average” distance be? And I think the average would be the square root of 2. My logic goes something like this: I think of the line segment AB being fixed, and then BC rotating around B in a circle. There is symmetry between the angle ABC being acute and being obtuse, so I think the “average” distance AC would occur when ABC is a right angle. In that case, the distance AC would be √2.

So here’s my question:

1) Are there any other reasonable ways of coming up with an answer (whether or not it is the same)?

2) I want the answer, √2, to somehow be an average of the two extremes 0 and 2. But it’s not the arithmetic mean (1), the geometric mean (0), or the harmonic mean (0). That bothers me. Is it some other kind of average? (Yes, I’m looking for a process that gives me the answer I want, which is kind of backwards but there it is.)

This section should be entitled “Ways Other People told me about” because it pretty much comes from other people’s recent comments, with a couple of extras thrown in. But the best part? We’re up to 25 ways!

In an effort to appear organized, I created a single page with links to all of the ways to multiply. If anyone writes up any additional ways, post a link in the comments and I’ll add them.

14 is 1110 in binary and 12 is 1100. Multiply the two in binary: 1100 x 1110 (think of 1110 as being on the bottom — I can’t draw it that way since the formatting won’t work). The partial products, in order, are 0000 (0), 11000 (24), 110000 (48), 1100000 (96). The nonzero products are copies of the top number, 12, shifted left — doubled — an appropriate number of times. Adding the partial products gives 10101000 (168).

(20)per Repiego, described by Pat Ballew of Pat’sBlog in the comments here. This is listed by Pacioli in the Trevisio, and is essentially breaking a number into factors and then mulitplying by each of those in turn. So to multiply 14 by 12, for example, you might multiply 14 by 2 (getting 28), then multiply that product by 2 (getting 56), then multiplying THAT by 3, to get a product of 168.

(21) This method from Pappus, also described by Pat Ballew in this article. It starts off:

I will instead use 257 x 62 to shorten the process.

Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
_______257
_________26
_______ 4

The bottom number is shifted, more pairs are multiplied and added, and you can read about the rest of the process here.

(22) This method from YouTube, which Jason Dyer of The Number Warrior pointed out in this post and in the comments here. It’s like a visual depiction of grid multiplication. It’s listed as Mayan multiplication, but I think that’s likely to be false: In Victor J. Katz’s A History of Mathematics he says that the Mayan documents that were not destroyed don’t show how the Mayans did calculation. (Hmmm. Katz also says the Babylonians used tables, and doesn’t mention those formulas I mentioned earlier.)

(23) This is described as the Prosthaphaeretic Slide Rule in this article (brought to my attention in these comments by Jason Dyer), but it really doesn’t use prosthaphaeresis at all. Rather, it’s a physical item that creates the similar triangles that the Greeks used for multiplication.

(24) Repeated Addition. Simple enough. It only works for integers, but that’s true of several of the other methods as well.

(25) On the fly shortcuts to Repeated Addition. For example, Jason Dyer’s example:

49 x 11 = (50 x 11) – (1 x 11) = 550 – 11 = 539

And we’re at 25 ways! Hooray!

I will instead use 257 x 62 to shorten the process.

Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
_______257
_________26
_______ 4

Hello July! It turns out that the family reunion we’ve been at didn’t have internet access, so instead of taking breaks to write about math we had to spend the entire time relaxing.

More multiplication tomorrow, but for now here’s a picture from the folk at Evil Mad Scientist Laboratories (some rights reserved on the picture). Notice that the Margin of Error is included in the net weight! It makes me want to run out and see if I can find similar examples.