Additional Math Pages & Resources

Tuesday, August 31, 2010

Wearing many hats, Part II

Continuing from yesterday, we are looking at our many hats. How can math help us understand what we have accumulated?

I think we might want to sort through the hats and classify (divide) them into sets today. So my wife sorted through the ball caps and here's what we found. We have this many CAPS of these COLORS.

Is the visual representation clear enough for you? If so, then that's enough of ball caps; we'll move on.

We have hats with these different FUNCTIONS - 5 mesh, 6 rain-proof, 6 winter, 3 driving, 2 helmets.

Helmets? Are helmets hats? We say yes; helmets cover the head. They go in the hat count.

I also noticed a visor or two. Are visors hats? We say no; they shade the eyes but don't cover the head. They don't get counted.

Are caps hats? They shade the eyes AND cover the head. We say yes. Why not?

Do you see where this is taking us? Suddenly we need a new term for these head things. Let's call them head gear. We need the right vocabulary to CLASSIFY and SORT head gear. Here's one way we could do it, with graphics:

Sadly, it gets a little difficult to show how many hats have light blue fabric and are intended for winter use, and contrast them with caps that are dark blue and aimed at summer use. What if we made a grid that would allow us to see two sets of characteristics at once? Then we could classify our head gear a bit more conveniently.

This could easily be turned into a pretty graph. But I think you get the idea. Math is a language that allows us to count, to classify and sort information.

This is how we teach it to kids (and adults). We introduce words that precisely describe a set of items. We distinguish one set from another. We find ways to clearly share our findings with others. We look at the data and try to learn from it. What can you learn from the chart above?

First, I don't need any more dark gray or black caps! Second, I need to decide how to spell gray (grey).

Monday, August 30, 2010

Wearing many hats, Part I

All of us play a variety of roles in life - we are students, employees, parents, children, hobbyists, athletes, etc. When we do lots of different things, we use a phrase "wearing many hats." We're not talking about this phrase today. We're talking about real hats. HATS.

I read that the Queen of England has worn more than 5000 hats in her reign. Even though she's been Queen for nearly 60 years, that's a lot of hats! More than Dr. Seuss and his book "The 500 hats of Bartholomew Cubbins".

We have a veritable boatload of hats in our house. We live in Southern California where you normally don't need to have a hat to stay warm, you need it to stay cool.  But we have winter caps as well as summer sun hats.  My wife guessed 80 when I asked her how many we had. Then we got them all out and counted.

 We found:
  • 43 baseball caps
  • 6 winter caps
  • 3 dress hats
  • 5 woven sun hats
  • 4 bucket-type hats
  • 6 rain hats
  • 3 driving caps 
  • 2 bicycle helmets

Tomorrow we will look at these hats in other ways, and see how elementary math, statistics, graphs and charts can help us understand the census of hats.

Friday, August 27, 2010

Trying to be a farmer

It's harvest time ... plants are producing far more than we can eat ... come over and take some home!

Have you been lucky enough to hear that kind of talk? Some of us who live in the city don't hear it often, although we might have family with squash, or tomatoes, or (in our area) citrus or avocados.

Today I want to celebrate tomatoes, because this summer we're trying to be farmers. Even if we do have to plant in bins to protect our crops from the gophers, we are enjoying our bounty. We bought seeds from a seed library with heritage varieties from 50-100 years ago. Here are some on my cutting board.

Here are some more (not ours).

and more

and more.

This photo shows I'm in the midst of making Jamie Oliver's Mothership Tomato Salad Recipe. I used to think tomatoes were boring but now I love them. We are very fortunate to be living where we can grow them OR if we don't grow them, the prices aren't too high.

Here comes the token Friday blog math:

There are between 7500-10,000 varieties of tomatoes.
The heaviest tomato weighed almost 8 lbs.
The largest tomato plant was at Epcot Center in Florida and produced 32,000 tomatoes in 13 months!
The amount you can spend buying seeds is infinite!

Take a look here... and if you have 2 minutes you can watch a video about the place we got our seeds. Or buy some seeds from them. And have a nice salad!

Thursday, August 26, 2010

Double-butted when you spoke? Part III

Please bear with me for another post on bicycle wheels, and the math that goes along with them.
The most mathematically-complex issue in building bike wheels is getting the spoke length right. Why? How is it done? What are the options?
  1. You can figure out which spoke you need by trial and error, IF you have a large number of different-sized spokes and you know what you are doing, and you don't mind taking a lot of time to build and rebuild a wheel until you get it right.
  2. You can use moderately-long spokes and chop off the ends that would otherwise perforate your tire. Then take one out and measure it. There's your length.
  3. You can use a spoke length calculator. Some of them require lots of precise measurements of your components but they do the math for you. Here's one calculator and here's another one
  4. You can use a book that has lengths already recorded in chart form. This works IF you use components that were measured by the author of the chart.
  5. You can do the math yourself. But not with what you learn in Excel Math. The calculation is a bit harder-than-average 6th grade math. 
Here's a diagram of a rear hub and the dimension you measure on a hub. Overall width across the locknuts (where the fork fits), the width between the flanges, and the widths of either side outside the flanges. You measure the same things on front hubs; they are less complex because they don't have gearing.

For some charts you need the effective rim diameter (ERD) which is from side to side inside the rim where the spoke enters the nipple. Plus the length of 2 nipples. ERD =  A + (B x 2). Other methods use the individual measurements rather than ERD>

And then add some geometry and math. This isn't a perfect diagram but it shows where the spoke fits  between the rim and the hub. We are calculating the length of an angled line between points on two circles. With a tolerance of a millimeter or two.

I was going to spare you the actual formula, but what the heck. It's not too long, too complex, and too messy, is it?

Yes, it is too long and messy. And once you get a result then you still have to build a wheel!

Making good wheels is difficult. It's very much a "craft" with some science mixed in. You can try it at home, but be forewarned, it's like rocket science. It's safer to send your wheel needs to an expert.

Wednesday, August 25, 2010

Double-butted when you spoke? Part II

We're continuing to investigate the math of bicycle spokes today.

In the following discussion, I am assuming standard-sized (700c or 27 inch) wheels on road bicycles. Let's say we want to get some new wheels for our bicycle.

To gather data, I relied on several bikes I own, the website of DT Swiss (a major spoke manufacturer) and a great website operated by Sheldon Brown. Sheldon has been into bicycles for a long, long time. When I started working at a bicycle shop in 1972, Sheldon was already an authority in the business.

Here are the numbers we will start with:
  • Spoke thickness can range from 1.6-2.3 mm diameter
  • Spoke length can range from 215-310 mm from the bend to the tip of the threaded end
  • Spoke weights - about 4-7 grams per spoke
  • Spoke nipple weights - range from 0.3-1.1 grams per spoke
  • Front Hub weights range from 70-230 grams
  • Rear Hub weights range from 200-600 grams
  • Quick Release skewers (hold hub to frame) weigh 50-200 grams per pair
  • Front hubs widths range from 125-135 mm
  • Rear hub widths range from 125-140 mm
  • Wheel rims weigh 300-600 grams each (per rim)
  • Rim tape (covers the spoke tips and nipples) weighs 30 grams per wheel
  • Inner tubes weigh 50-110 grams
  • Tires weigh from 160-400 grams per tire
OK, now it's time for some math. Let's just focus on weight today.

Q1. Using these numbers, calculate the weight of an imaginary pair of wheels of your choice.

Ignore the gears on the rear wheel and any hub-mounted brakes, generators or reflectors. Otherwise your weight calculation must be complete with skewers, hubs, spokes, rims, nipples, tape, inner tubes and tires. Include both wheels. Round off any sums or products to reasonable numbers. Do the work in your head for each component, show your work, and add it all up without a calculator. Convert to pounds at the end.

A1. Here's how I would weigh my set of wheels:

140 g = Quick Release; one pair
200 g = Front hub
300 g = Rear hub
360 g = Spokes @ 36 front, 36 back = 72 x 5 g
70 g = Nipples @ 72 x 1 g, rounded  off
800 g = Rims; one pair 2 x 400 g
60 g = Rim tape; one pair 2 x 30 g
160 g = Tubes; one pair 2 x 80 g
600 g = Tires; one pair 2 x 300 g
2690 g = 2.7 kilos x 2.2 lbs/kilo = 2.2 + 2.2 + 1.5 = 6 lbs rounded off
2690 / 454 g/lb = 2700 g / 450 g = 6 lbs

Click here to see your wheel spinning

Tuesday, August 24, 2010

Double-butted when you spoke? Part I

  I apologize for the pun in the title of this blog. We are going to look at bicycle spokes today.
  • spokes are wires which are woven between a hub and a rim to create a wheel
  • spokes usually have a bent head that keeps the wire from coming out of the hub
  • spokes have a threaded end which goes through the rim and engages a nipple
  • spokes can be straight or single-butted, double-butted or triple-butted
  • spokes can be have oval or flattened shapes in the middle of their length
  • spokes should be rust-resistant, strong, light and have some flexibility
Here are some spokes and nipples. This handful is about the number you need for a single bicycle wheel. And here are the side views of each end of two kinds of spokes - the flattened aerodynamic type and the normal round style - and some nipples.
A butted spoke is thick on the ends and thinner in the middle. This saves weight, reduces drag and adds some flexibility to the wheel.
  • single-butted spokes tend to be thick at the hub end, thinner in the middle, and maintain that thickness to the rim (threaded) end.
  • A double-butted spoke is thick at the hub end, thinner in the middle, then returns to the original thickness at the rim end.
  • A triple-butted spoke is thick at the hub, thin in the center, and a third (thicker) diameter at the rim end.
The aero spoke is flattened to reduce wind resistance. If it's too flat, it won't fit through the holes in the hub, so there's a limit to the flattening.

OK, I think that's enough background information. Now it's time to ask, "Do we need math to make a bicycle wheel?" Probably not, right? I haven't provided any numbers.

No, it turns out that you need a surprisingly large amount of math. Wheel (we will) look at the issues more closely tomorrow.

PS - I guess it's unfair to have NO numbers at all, so here - you've heard the phrase "let's not re-invent the wheel." Why not? Because everyone else is. I did a patent search and found 4668 patents on bicycle wheels!

Monday, August 23, 2010

Math Tests

How do you assess / measure / evaluate how well someone has learned a subject? Alas, in the education business it often means you must give them a test / quiz / assessment.

I tried to look up the question, "What is a math test" and I get no useful answers from Google.

I found free tests, copies of tests, reports about tests, rantings about unfair tests, free test preparation hints, offers of paid test tutoring, and sites that offer everything but rational discussion. I read papers about how badly we (in the USA) perform compared to other countries. And so on.

I found this statement: "valuing what we measure rather than measuring what we value.”

Which is a nicely-stated philosophical quandry but not very helpful in a discipline like math - which is all about measuring things whether we value them or not.

I got distracted and played with the Toy Theatre Site for awhile. I tested myself on all sorts of elementary math (of course it's fun when you can solve all the problems correctly). I especially liked the water pull.

In Excel Math, most of our curriculum grade levels provide 24 tests of various sorts. We have placement tests, weekly tests, quarterly tests, year-end tests, and a few others. We offer questions that require calculations, multiple-choice, match-the-answers, fill-in-the-blanks, bubble-in-the-oval, write-us-an-essay, and charting or graphing.

They are not designed to make kids' lives miserable, I assure you. Nor are they designed to hold them back or to propel them forward in life. They are problems from the curriculum itself, which offer a chance to demonstrate mastery of the topics taught throughout a math education.

Luckily, once we leave school there aren't too many tests that we need to take. There's occasional one at the DMV for a driver's license. If you want to be an attorney or a realtor you take some tests. There are forms to fill out, and a few "competitions" for a job. But in general, adult life is free of scary formal written assessments of math competency.

Unfortunately we also use the word test to describe the results of medical research as well as for learning assessment.

And we are asked the same kind of questions "How did you do on your test?" or "did you get the results of your test?" as if it were something you could study for. Co-mingling those concepts isn't the best way to free ourselves from the fear of testing.

Score poorly - maybe I flunk school. Score poorly - maybe I die. It's no wonder people fear testing.

Friday, August 20, 2010

Broken, orphaned and outside, Part II

Today we continue on from yesterday's blog about website maintenance (yawn). No, don't leave, it's not that bad. Remember we are talking about how to use elementary math we learned long ago.

After being horrified by yesterday's status report, I did some cleaning. The new website status report shows:
  • Total files     575
  • HTML files  153
  • Orphans        260
  • All Links     3946
  • OK links      3748
  • Broken links       0
  • External links 198
The HTML files are where our content is located. They contain the text about our products. They also have links to other files:
  1. other HTML pages that continue on the same or related subjects
  2. interactive graphics that appear in the midst of the page and "know" they are there
  3. some graphics, videos, or downloadable files are considered orphans on our site, because they don't "know" they are being referenced
In addition to these links, we also have "external links" which go to other places besides our website:
  1. Mailto: links reach out to a mail application; customers use these to send us questions
  2. some links take people to other sites, such as our webstore, or to Adobe to get Acrobat Reader
  3. other links go to interesting sites with articles about math education
After fixing all these file links, and deleting hundreds of unnecessary files on my local machine, I uploaded all the fixes and ran another report on the site. No change!! What happened, I wondered.

Then I remembered that all the old bad files were still on the host (server in the sky somewhere). Although I had relocated the new, clean-up files to the server, I had not done a cleaning to remove all the old unlinked files. So now there were even more broken links and orphans. Sigh.

So I got up my courage, erased the whole site and uploaded the changed version from my local machine. Five minutes of sweating, and whew! it's all up there and working again.

Math questions for the day:

Q1. How many files got removed?
A1. 1496 - 575 = 921 files were removed.

Q2. How many total links were removed?
A2.  5218 - 3946 = 1272 links were removed.

Q3. How many orphans have found new homes?
A3. 1146 - 260 = 886 orphans are no longer lost and alone.

    Thursday, August 19, 2010

    Broken, orphaned and outside, Part I

    Broken and orphaned and outside.

    Sounds sad doesn't it? In this case I am not talking about lonely orphan children, but rather LINKS from one page to another. They are classified as:
    • links between various pages within our website
    • links to other websites outside our own
    Those links can be good (intact), the links can be bad (broken), or pages can be completely without any links at all (orphans).

    In a way this is a sad state of affairs, because it means work! I maintain the Excel Math website. While I'm not a programmer, I've done HTML, XML and other kinds of coding in my publishing career.

    Our website software checks how pages are linked together. The starting report looked like this:
    • Total files     1496
    • HTML files    187
    • Orphans       1146
    • All Links      5218
    • OK links      4825
    • Broken            62
    • External        331
    When our site was first built, our developer was also creating some other sites. We agreed it would be worthwhile to share some design elements, saving each of us time and money. Today I realized that many links had been set up to one of the other websites. It closed down (was rebuilt) a few months ago. Those links were to an obscure little utility file used to align columns of tables. Now they were broken.

    I hunted and researched and tested and erased. After a couple hours the stats were better. Sometimes higher numbers are better. But not always. With taxes, blood pressure, or broken and orphaned files, lower numbers are better.

    Tune in tomorrow for a full report!

    Wednesday, August 18, 2010

    Ping Pong, Hong Kong, Ding Dong

    The blog today is NOT about table tennis, a large city in Asia, or door bells. It's about time and math. However, the title was chosen because I was trying to find words to rhyme with the blog titles from the last few days - How Strong, How Long and How Wrong.

    In my search I found Ping Pong, Hong Kong, Viet Cong, Mahjong, Sing Song, Ding Dong, and so on. Ding Dong caught my eye because I was adjusting some clocks yesterday. Take a look:

    These two 7" brass Seth Thomas clocks are more than 50 years old. The left clock has an internal bell and the right clock an external bell. The movements inside are otherwise the same.
    • Each clock requires 12 twists clockwise of the key on the left-hand shaft (near the 8) to wind the strike spring.
    • Each requires 15 counterclockwise twists of the key on the right-hand shaft (near the 4) to wind the main spring.
    These marine clocks strike ship's bells. That's a shipboard tradition dating back at least 500 years. Visit the US Navy's Historical Site to read about how bells play an important role on ships even today.

    Ship's bells communicate to a group of people standing watch on a ship; people whose eyes should be on the horizon and not on the clock!

    A watch is a 4-hour shift doing a specific job. A bell is rung every 30 minutes to indicate the passing of time. Let's say you and your pals start on watch at 12 noon.

    12:30   1 bell    ding
    13:00   2 bells  ding, dong
    13:30   3 bells  ding, dong, ding
    14:00   4 bells  etc.
    14:30   5 bells
    15:00   6 bells
    15:30   7 bells
    16:00   8 bells! Your shift is over and a new one begins

    16:30   1 bell
    17:00   2 bells, etc.

    At some point the term 8 Bells became slang or shorthand for dead (your life's shift is over).

    The good things about this system are:
    • Everyone stays on the same time
    • You can tell the time in the dark
    • You don't need a clock for every person
    • You don't have to count too high
    • You get a reminder every 30 minutes
    • You can't easily fall asleep with the loud bell ringing
    The bad thing about these clocks? You can't easily fall asleep with the loud bell ringing. You have to throw a couple thick towels over it because the strike spring doesn't run down for about 3 days!

    Here's one clock striking 6:30 - it's the clock with the external bell. Be sure to turn on your sound first.

    The two clocks striking 7 over each other. The later one has the internal bell; its ring lasts much longer.

    Tuesday, August 17, 2010

    How Wrong?

    In a math class, after discussion of a topic, investigation of its relationships to things they already know, and some practice, we then ask students to find the answers to a question or problem.

    Sometimes students succeed and sometimes they fail. Sometimes they are close and sometimes they miss by a country mile. They can right on, or flat wrong.

    But what is wrong? Does it mean "I blew it but I was sincere - so I should receive credit anyway?"
    • the original sources of wrong meant twisted, crooked, bent, uneven, sour
    • wrongness usually refers to a state of incorrectness, inaccuracy, error or miscalculation
    • a wrong answer refers to an error of calculation or judgment
    • wrong can mean inaccurate, incorrect, false, untrue, mistaken, improper, amiss, askew, at fault, awry, defective, erring, erroneous, fallacious, false, faulty, fluffed, goofed, erred, miscalculated, misconstrued, misguided, mishandled, mistake, off-target, off-track, out of line, out of order, perverse, specious, spurious, unsatisfactory, unsound, untrue, or unsuitable
    A clever tool called the Visual Thesaurus constructs diagrams to show you relationships between words and meanings. They chart the word wrong like this:

    We are most interested in the branch at the bottom that's labeled incorrect. Generally speaking (and thankfully) answers to math questions can be wrong without being immoral.

    The dictionary tells me that we can say wrong, more wrong and most wrong, but we can't
    • use wronger or wrongest or wrongified
    • say he wronged the problem 
    • wrong'd is incorrect as well
    • finally, wring and wrung aren't even related to wrong

    Monday, August 16, 2010

    How Long?

    How long has multiple mathematical and social meanings.

    • Distance - How long are your legs, oh giant giraffe?
    • Future Prediction - How long can you keep up barking, you ornery dog?
    • Past Remembrance - How long have we been going together, honey?
    • Both past and future - How long was Bill Murray stuck in the Groundhog's Day movie?

    Believe it or not there is a blog dedicated to Groundhog Day movie. You remember the story about the groundhog who didn't see his shadow ... well I won't spoil the movie for you. Go see it for yourself.

    A blogger worked out how many days the Murray character was stuck there. About 8 years, was his guess. Director/Screenwriter Harold Ramis said he reckons it was more like 30-40 years.


    I once worked on a curriculum study guide to help readers learn more about the CS Lewis series The Lion, The Witch and the Wardrobe. During the production of this study guide, one of our artists developed a map of Narnia. She actually called a US Forestry Service office and asked,

    "How long a walk could three small children take through a snowy wood if they walked from lunch time to tea time? And how long would that journey be if they had a beaver to guide them, etc. etc."

    If I recall correctly, it was not a very long conversation. Click. Call another office. Repeat. Finally she came up with enough suggestions to calculate the distance from one end of Narnia to another.

    There have been lots of maps done as people have imagined Narnia - and for me to avoid stepping on anyone's copyrighted toes, you can click here  to see a long list of some of those maps.


    How Long Has This Been Going On is Van Morrison's twenty-fourth album. I have about 20 of his albums. My wife gets a bit tired of me listening to them, especially when he does a song called Take Me Back 

    Take me back, take me back, take me back
    Take me way back, take me way back, take me way back
    Take me way back, take me way back, take me way back
    Take me way back, take me way back, ah!
    Take me way, way, way, way, way, way, way back, huh!

    Ah, ah, take me back, take me back, take me back, take me back
    Take me back, take me back, take me back, take me back
    Take me back (woah) to when the world made more sense
    Way, way back, way back
    Take me back, there, take me way back there
    Take me back, take me back, take me back
    Take me way, way, way back, way back
    Take me back, take me back, take me back
    Take me way, way, way, way, way, way, way
    Oh, ah, take me way back, when, when, when, when, when, when
    When, when, when, when, when, when, when
    Everything felt, so right, and so good
    Everything felt, so right, and so good
    Everything felt, so right, and so good, ah
    Everything felt, so right, and so good
    Everything felt, so right, and so good, so good
    In the eternal now, in the eternal moment
    In the eternal now, in the eternal moment
    In the eternal now
    Everything felt so good, so good, so good, so good, so good

    When you lived in the light

    OK, I admit I left out some of the words, but didn't you find yourself thinking "How long is this going to go on?"

    Friday, August 13, 2010

    How Strong?

    Since this is an elementary math-oriented blog I am NOT going to ask how much weight you can press. I'm going to ask how many passwords you have and how strong they are (resistant to attack).

    We all have multiple requests to create passwords, PIN numbers and combinations for our school lockers (oops, obsolete!). We have to remember birthdays and anniversaries. Thus our memories are full and fallible. When prompted to create a new PIN or combination, we usually choose something easy to remember (but also easy for a hacker to guess).

    The numbers around this state of affairs are the subject of today's blog.

    Have you ever had a padlock like this? I bought mine the day we got our first house, to lock the garage. Since I could set the padlock's combination myself, I chose my wedding anniversary date, so I could recall two numbers with one allotment of memory cells. Because I opened this lock so many times, I can still remember my anniversary, even though it was 12,760 days ago.

    This simple combination lock only has 4 numbers, yet there are 10,000 combinations (10 x 10 x 10 x 10). Anyone trying to break it has to spin the rings around manually, while kneeling down in front of my garage door, so it's fairly secure.

    I have a lock on a cabinet in my house that only has two tumblers. But each has letters instead of numbers so the sum ( 26 x 26 ) gives 676 possible combinations.

    Computer passwords have to be a lot more secure than these locks, as hackers can sit back and let special programs do the work for them. They don't have to twist dials around and worry about you coming home and finding them fiddling with your locks.

    A good password is complicated but not impossible to remember. We've got one password at work that's so long we have printed it on an 8.5x11" piece of paper. It's extremely secure, but that's too much for most of us to manage every time we want to get into a website.

    Here's a nifty site that does an evaluation of sample passwords just to see how strong they are.


    If you are going to check a password there, I suggest you check lots of random ones as well as your own, because almost any site can be analyzed like this Password Meter. Here's a typical evaluation:

    There are 270,000 sites with a better three-month global traffic rank than this site. It is ranked #140,000 in India, where 20% of its visitors are located, and is popular in New Zealand, where it is ranked #15,000. About 90% of visitors view one page. Visitors tend to be childless, lower-income men aged under 35 and over 55 who browse from a computer at work.

    Are you a typical visitor of the right age and gender? Did any of your passwords measure up? If not, an article I read by an IBM security specialist suggested a pattern easy for you to remember and hard for others to decipher:

    Odd-Name   Special-symbol   4-Numeral

    Combinations assembled in this way are good and secure. Adding upper and lower case letters are even better. For example:

    Boeing#747 is easy and got a 89% rating.

    RatRod&1953 did much better at 100%.

    Have fun with all your new passwords!

    Thursday, August 12, 2010

    The Poincaré Conjecture

    This blog is usually about math taught in elementary school and how we might use it later in life. But we have an esoteric subject today.

    Jules Henri Poincaré (1854-1912) was a great French scientists of the last century. He did creative research on mathematics, physics, and celestial-astronomical subjects. The Poincaré Conjecture has to do with spheres. He proposed that a three-dimensional sphere is the only "bounded space" with no holes in it. Proving this assertion was an extremely difficult problem that remained unresolved until 2003 when Russian mathematician Grigory Perelman published a proof (and turned down a million-dollar price for doing so).

    The surface of a perfect globe is called a two-dimensional sphere. The globe has the property that a lasso of string (or rubber band, etc.) encircling it can be pulled tight to one spot. In the process, the lasso can slip along the surface and does not leave the surface at any time (for example, if it was going over a valley), nor cut off any part of the object (like a mountain). A Wikipedia illustration shows this:

    On the surface of a doughnut, by contrast, a lasso passing through the hole in the center, or around the outer edge, cannot be shrunk to a single point without cutting through part of the doughnut or leaving the surface of the object. Another Wikipedia illustration I have enhanced shows two sample lines:

    I'm not sure I fully understand this, but I'll try to build up a foundation for the Poincaré Conjecture:

    A 1-dimensional sphere is a circle;
     all points (x, y) that satisfy the equation x2 + y2 = radius2

    A 2-dimensional sphere is the surface of a globe;
     all points (x, y, z) that satisfy the equation x2 + y2 + z2 = radius2

    (now we leave elementary geometry behind and boldly go where few have gone before ...)

    A 3-dimensional sphere is the set of points in four dimensions;
     all points (x, y, z, w), that satisfy the equation x2 + y2 + z2 + w2 = radius2

    Poincaré not only thought up this sort of thing, he surmised logically how these objects ought to behave. Grigory Perelman actually understood what he meant, and proved it to be true.

    Speaking for the rest of us, it's hard to imagine these concepts. I would probably say something like "imaginary shapes could behave like tied balloons - they can be a sphere, then become a sausage, then a cute dog." Not very mathematical, am I? Some art by Michael Floyd easily illustrates my concept:

    Here is a real-word summary of the Poincaré Conjecture by a noted post-doctoral researcher :

    From here on out we'll stick to elementary subjects, ok?

    Wednesday, August 11, 2010

    Would you turn down a million dollars?

    One of the premiere mathematicians in the world recently turned down a prize of $1,000,000.

    Since 2003, Russian mathematician Grigory Perelman has offered complete proofs of a one-hundred-year-old problem called the Poincaré conjecture (we'll talk about it tomorrow). He derived his proofs using some concepts offered decades earlier by an American mathematician named Richard Hamilton.

    Perelman's proofs were published, studied, verified and recognized. Perelman was named a winner of the Field Prize ($10,000) and the Clay Prize ($1,000,000). A special awards ceremony was scheduled by the Clay Mathematics Institute this July in Paris, but Perelman refused to attend or take the money.

    It's not that he doesn't need it or come up with a way to spend it. Perelman lives with his mom in an apartment in St. Petersburg. He doesn't work on mathematics anymore, and the family has no money. But the city he lives in, and his friends, have offered him lots of suggestions on how they could spend his money for him. If he took all their suggestions, they would take all his money. He avoided this dilemma by not taking the cash.


    Sometimes people have complex issues with wealth. Some of us are laden with it (remember the generous billionaires from a few days ago) and some of us have none. It's rough when you have no money, but life isn't a piece of cake when you are rich either. Most people have a moderate amount of wealth and a moderate amount of trouble in their lives.

    Perelman says he doesn't want the money because he doesn't respect the process of chosing a winner, and he says he didn't do anything unique on his own - that his work was based on and equal to Hamilton's effort, and the prize should be shared.

    This might be the equivalent of taking a test and refusing to accept an A grade because your study partner once showed you how to solve the type of problems that appeared on the test.

    Or it might be that his extra-ordinarily perceptive thinking which found the solution to an incredibly complex problem also enables him to see conflicts of interest and dreams of glory in the institutions (and his neighbors in St. Petersburg). He wants to stay clear of all this.

    Isn't math interesting?

    Tuesday, August 10, 2010

    Color Full

    How full of color does something have to be before it's called colorful?

    (Is this a nonsense question? No.)

    Here's a cat with color. There's a bit of black and a bit of white, along with lots of blues and greens.

    Here's a bit more color, on a cow. More orange and green and more black.

    Here are a few more colors, and more brightness. Less green, more black. This is a goat, not a dog.

    Here is a raven - there seems to be a bit "more color" and fewer colors. If you see what I mean.

    How many colors are in these paintings? One way to count is to put a grid over it, and decide which is the primary color in each square. Make the squares small enough, then count the colors.

    Of course you would need a lot more squares than this, and lots of time and good eyes.

    Here's a mathematician's viewpoint on how to do it with a computer:

    Let P be a set of n points in Rd, so that each point is colored by one of C given colors. We present algorithms for preprocessing P into a data structure that efficiently supports queries of the form: 

    Given an axis-parallel box Q, count the number of distinct colors of the points of P ∩ Q. 

    We present a general and relatively simple solution that has polylogarithmic query time and worst-case storage about O(nd). We then present several techniques for achieving space-time tradeoffs. In R2, the most efficient solution uses fast matrix multiplication in the preprocessing stage. We give a reduction from matrix multiplication, which shows that in R2 our time-space tradeoffs are close to optimal in the sense that improving them substantially would improve the best exponent of matrix multiplication. 

    Finally, we present a generalized matrix multiplication problem and show its intimate relation to counting colors in boxes in any dimension.

    We DO NOT cover this in Excel Math, just in case you were wondering. And I didn't make up any of the words - that's how some mathematicians communicate ...

    FYI - The pictures are by Joe Nyiri. He teaches art to kids at schools and at the San Diego Zoo.

    Monday, August 9, 2010

    Divide Evenly, Again

    A few months ago I did a blog on dividing evenly. The subject was cutting a equal-sized piece of pie for each person. A few days ago I did another and showed how to evenly-space 3 pictures on the wall.

    In both cases the object was to get the same size for each (person or picture). Today I have a more interesting challenge from of a new book called Numbers Rule by George Szipiro.

    George the mathematician/journalist says that it's impossible to evenly or fairly divide the 435 representatives in our Federal government among the 50 states. Some states are always under-represented; some states are over-represented.

    Why? Because we don't have a way to impose fractional voting shares on representatives. Since we have to allocate, for example, 1 or 2 delegates to each state rather than 1.5 delegates, we have to give more to one state and take some away from another. This has been studied for hundreds of  years, using dozens of different methods of dividing and rounding. You can read all about it in his book, and in another book called Fair Representation.

    There is another problem with counting votes which is related to small parties. If your country has more than two major political parties, it's impossible to allocate representatives evenly in proportion to the votes cast for them.

    For example, in one district, the elections results look like this:

    • Green party 1000 votes
    • Yellow party 600 votes
    • Orange party 599 votes

    If there's only one seat, then Green is represented and the other two parties are not. If there are 22 seats, then we can give 10, 6 and 6 seats.

    In another district, it's like this:

    • Green  400 votes
    • Yellow 801 votes
    • Orange 799 votes

    If there's only one seat, then Yellow is represented and the other two parties are not. If there are 20 seats, then we can give 4, 8, and 8 seats and all are represented.

    If we only have the one seat per district, then Green gets one and Yellow gets one. The elections are over and Orange gets left out completely.

    But if you add the two district votes, you find that the votes were split with one-third to each party.

    • Green 1400
    • Yellow 1401
    • Orange 1398

    Green and Yellow are now evenly represented in the Assembly and Orange has no representation at all despite being only 2 or 3 votes down out of 4200 votes. This is unfair to one-third of the electorate.

    You might say "Too bad, tough luck" but come on, it's still unfair. This situation would not meet anyone's hope of having proportional representation for all viewpoints - one of the goals of a democratic government. In many places (including here) people get very angry. What are the alternatives?
    1. Let cooperative parties pool their fractional (or excess) votes across districts to get more representatives? That's what Switzerland, Israel and some other countries do.
    2. Rank the acceptable candidates then choose the one with the overall highest rating? This method is used in Australia, New Zealand, Scotland, Ireland, Malta.
    3. Put a check next to everyone we could live with, and no check next to those we dislike? Then the ballots could be tallied and the overall winner chosen. This is used by the United Nations and a lot of professional societies.
    4. Evaluate candidates on excellent, good, average, poor and unacceptable, then select the best people and have another go at it if necessary? That's how the latest French presidential elections were counted.
    5. Have a dictatorship. This is mathematically acceptable. One viewpoint is fully represented.
    Alas, mathematicians say there are no perfect solutions and there is apparently no perfect democracy. All are subject to paradoxes, inconsistencies, and manipulation behind the scenes.

    If you want to see fireworks, forget the 4th of July. Wait until the 2010 Census figures are in and the next reallocation of our 435 state representatives begins!

    Friday, August 6, 2010

    Why Learn Math? I have a Calculator. It says 250

    This is my 250th blog posting, in exactly one calendar year. The first post was dated August 7, 2009. It was entitled:

    Why Learn Math? I have a calculator

    On that day I started to write about using math in daily activities of life. Easy math. The kind of math we learn(ed) in elementary school. Math we teach with our Excel Math curriculum. I want to demonstrate how to think mathematically, which a calculator will not do for you.

    In that time, we've had nearly 10 thousand visitors, from at least 126 countries around the globe.

    Note: if you are an English-speaker outside the USA, substitute the term "maths" where appropriate.

    I've written lots of words and posted 1000 images, most of which I have created here at my desk, or taken with my very own cameras. The volume of words is harder to count, but I can tell you the total bits and bytes -  21,067,290 bytes for 1,116 items.

    The blog has been here rain or shine, with just a few days off for short vacations and weekends. We've made it through brush fires and "the coldest summer in nearly 100 years" (which in San Diego is no big deal) and a half-dozen Internet outages.

    We have had 2 rattlesnakes inside the office during the last 12 months!

    Being near Miramar Air Base, we have been strafed by Blue Angels and stealth bombers

    We have not "monetized" the blog, nor have we attempted to cover outrageous topics to provoke emotional outbursts from readers. I have received only one complaint (from a professor that I used his picture without permission; though I did credit him and link back to his site).

    Why do we do it?
    • We want people to visit our website, buy Excel Math curriculum, and teach math to kids
    • We want to de-mystify math - it's not rocket science (usually) and it's not hard (frequently)
    • We like learning new things and understanding how and why the world works as it does
    • We hope to present a investigative but charitable and humorous look at life, not the contentious, argumentative and cynical viewpoint so prevalent today
    In conclusion, I present this Speed Bump cartoon, which Dave Coverly graciously gave us permission to use. Last year I had hoped to use as part of a promotional campaign for our math books. We had to take it off the website because a few people complained that they couldn't calculate curriculum purchase prices ... sigh ... they should have worked harder when they were in school to cultivate a sense of humor!

    I hope you enjoy it.

    Thursday, August 5, 2010

    A dollar is not always worth a dollar

    Of course, you say, the dollar's value can appreciate, depreciate, etc and so it doesn't always have the same purchasing power.

    That's not what I meant. I was just repeating a statement in a fascinating new book called Numbers Rule. 

    The statement meant that money is not equally valuable to people. For example:
    • If you have 3 dollars, one more dollar increases your wealth dramatically (is it 25% increased? or 33%?)
    • If you have 300,000 dollars, one additional dollar makes very little difference to your wealth or purchasing options.
    The more money you have, the less each of those units should mean to you (assuming you're not a maniac or miser). With lots of mathematicians weighing in on this, the odds have been calculated and we've determined that
    • if you have only one dollar, the next dollar is worth 0.3, which I think means it increases your purchasing options about a third more than just having $1.00
    • if you have a million dollars, the next dollar is worth 0.0000004, meaning it does virtually nothing for your purchasing options
    This whole subject is known as the St Petersburg Paradox. It evolved from some probability questions posed by Swiss cousins, Nikolaus and Daniel Bernoulli. Daniel is famous for lots of other things, including the Bernoulli Principle which has to do with the flow of liquids and gases over solid objects. This plays out in the lift of an airplane wing, the flow of air and fuel through a carburetor, calculating speed of a boat or airplane, etc.

    OK. Enough history. Taking this subject, and looking back at yesterday's blog on Billionaires donating their fortunes to charity, does this dollar is not always worth a dollar principle mean that they don't feel the pain of giving up their dough?

    I'm not sure that it does. Perhaps if you learn to give when you have a little (the working poor), you find it easy to keep on giving when you have a lot. And alternatively, perhaps if you begrudge giving when you have a little, you will rarely want to give a large amount, even if you have a large bank balance. My grandfather was a wealthy man but he definitely didn't like sharing ... or donating.

    I'm sure the mathematics of generousity have been well studied and we won't add to this today with our elementary math blog.

    And we don't have to get into a lot of math to know that the dollar goes up and down against other currencies. Here's the graph of the dollar against the euro for the past 6 months.

    The higher the bars, the more purchasing power the dollar has compared to the euro. Our company's accountant and his family are off on a European Vacation today. He's bemoaning the depreciation of the dollar and its effects on his wallet.

    Q1. During what month on the chart would the accountant's dollars had the most value compared to the euro?

    Wednesday, August 4, 2010

    How Much is More Than Enough

    How much is more than enough?

    An excellent question that math can help calculate, but not decide. Today's blog is about people who have decided they have more than enough. One honorable approach is to give the excess away. But where do you draw the line?  

    How much is more than enough?

    Bill Gates and Warren Buffett are encouraging billionaires to pledge the majority of their fortunes to charity. As of today, 40 billionaires have agreed. Here's an abbreviated version of their campaign:
    The Giving Pledge invites the wealthiest individuals and families in America to give the majority of their wealth to philanthropic causes and charitable organizations during their lifetime or after their death.
    Each person who chooses to pledge makes a statement publicly with a letter explaining their decision ... the Pledge is a moral commitment, not a legal contract. It does not involve pooling money or supporting causes or organizations.

    The Giving Pledge focuses on billionaires, but is inspired by millions of Americans of all financial means and backgrounds who give generously to make the world a better place.
    Just for fun, take a few moments and read some of their personal thoughts. The Giving Pledge website  tells you about the hearts of these wealthy folks.

    How much is more than enough?

    I first learned about exceptional giving 40 years ago, when I read about R. G. LeTourneau. First a laborer, then mechanic, he joked "I was such a small-time operator that I couldn't power a treadmill in a flea circus".

    LeTourneau came to my attention because a huge shopping center was being built in the canyon behind my house, and the fields were full of giant tractors from his company. On weekends kids climbed all over them - we were the fleas on the elephants (so to speak).

    LeTourneau became an earth-moving contractor, then a builder of road grading equipment, an inventor of hundreds of patents, and a very very rich man. The company he founded makes the largest earth-moving equipment in the world. Here's a loader that can lift 53 cubic yards of soil (72 tons) per scoop.

    In addition to running his company and continuous inventing, LeTourneau started a university, was president of Gideons (hotel bibles), helped to fund the development of the nation of Liberia, etc. He said he gave not just 10% of his income to charity, but kept 10% and gave away 90%!

    How much is more than enough?

    I was stunned by his biography. How could anyone work hard to get money yet be so cheerful about giving it away? Is it just billionaires who do this? No, on the contrary billionaires aren't alone in their generosity - they are joined by the working poor. Both rich and poor groups give away twice as much (by percentage of income) as middle-class folks do.

    The average annual donation in the USA is about $1650 per year, or 2.2% of household income. Average annual giving by religious families is $2,210, compared to $642 from secular families. The Hoover Institution says "Houses of worship teach their congregants the religious duty to give, and the physical and spiritual needs of the poor ... people may be more likely to learn charity inside a church, synagogue, or mosque than outside."

    This matches what my accountant brother-in-law said to me one day, "We are trained in school to count money, grow the nest egg, and legally avoid taxes, NOT how to give money away."

    How much is more than enough?

    Tuesday, August 3, 2010

    Math for Hanging Pictures

    This blog is about using elementary math in our daily lives as adults. Here's a practical example.

    I bought three old icons and I want to hang them above a small fireplace mantle. Unlike many of my simple hanging projects I can't just bang a nail in the middle of the wall then move it around to get things straight. Here are the issues:
    • The wall is made of brick so holes have to be drilled for hangers
    • I want my 3 items to hang at the same height and to be evenly spaced
    • The three pictures are slightly differing sizes and are not "square"
    • The pictures already have hangers on their backs which I don't want to move
    • The hangers are not in exactly the same position on the backs
    For you visual learners, here is an illustration of the situation:

    • I need to know the sizes of the 3 icons (they are almost the same, I learned).
    • I need to know the width of the mantle - is it wide enough that the three will fit with appropriate spacing between them? (yes)
    • I need to decide on the height of the icons above the mantle. I think an inch or two is okay.
    • I need to know the drop from the top of the icon to the wire, to know where to drill into the wall.

    Let's call the icons A, B and C and the mantle M and the hook locations HA, HB, HC.
    We'll worry about the horizontal spacing first.

    Is   Mwidth - (Awidth + Bwidth + Cwidth) greater than 0?   Yes. The answer is 7 inches, therefore

        A + 3.5 + B + 3.5 + C = M           

    This means the horizontal hook spacing will be, starting from the left edge of the mantle,

        HA = Awidth ÷2

    and starting from the right edge of the mantle,

        HC = Cwidth ÷2

    and finally, a hook for the last icon is
        HB = HA + (Awidth ÷2) + 3.5" + (Bwidth ÷2)

    or to be safer, just put it in the middle of the other two hangers

        HB = (distance between HA to HC ÷ 2)

    The vertical position is calculated like this -

    The icons are 12" tall, subtract 2" inches drop to get 10 inches then add 1" for clearance at the bottom.

    If I put the hangers 11" above the mantle the icons should be at the right height.

    Of course, on a brick wall it's difficult because I'd prefer to drill into the mortar rather than the bricks. I had to put the icons slightly lower, but not touching the shelf, so my holes went into mortar.

    Here's how it came out:

    Monday, August 2, 2010

    Math without numbers

    A friend of mine said he was looking at his daughter's college math textbook, and there were very few numbers in it. How can you have math without numbers! he exclaimed. I was sympathizing until he went on to say "it was all word problems."

    Now that's another story. I like word problems, or story problems as some people call them. I've written thousands of them. Sorry kids, the buck stops with Janice Raymond (she started Excel Math) and me. And despite what my friend said, there are numbers in story problems.

    Here are some common FAQs on story problems.

    Q1. Do you write story problems deliberately to torment us?

    A1. No. In fact I often simplify what has a useful problem for years, just because people sometimes don't solve them as easily nowadays.

    When I simplify problems, that doesn't mean the calculations get easier. It means I pay close attention to the presentation of the problem on the page. As a life-long editor and typesetter, I think a nicer presentation makes a difference in comprehension.

    Q2. What do you mean? Can you give me an example of "presentation simplification"?

    A2. Sure.

    Stan had $31.42 to spend on
    school supplies, so he bought 4 
    76¢ pens with black ink and 3
    books that cost $4.87 apiece. How
    much money does he have left?

    Stan had $31.42 to spend on supplies. 
    He bought 4 pens that cost 76¢ each.
    He bought 3 books that cost $4.87 each. 
    How much money does he have left?

    The second presentation has a separate clause on each line. The order of presentation is the same on each line, starting with the buying, then the quantity of items and the amount per item. The math is the same. The reading is easier. Some of the extraneous information is left out. 

    If you like to suffer while you solve story problems, maybe this is too soft for you.

    Q3. Where do you get your ideas for story problems?

    A3. From real life. From you - once in awhile we talk to parents and kids to get new ideas for story problems. We have either done these things ourselves, or know someone who did, or we wish we could do them. I don’t like to write stories about talking animals and stuff like that.

    Q4. What other factors make you change a story problem?

    A4. Changes in society. We are taking out all the candy, cookies, cakes and sodas from our books, even though candy and cookies are a convenient individual unit that kids are interested in baking (addition), eating (subtraction) and even sometimes in sharing (division).

    We took out trikes and put in scooters. We took out skates and put in skateboards. We're adding various activities and items that are relevant to kids today - always mindful that we don't mention brand names or commercial products.

    Q5. Do you have word counts or readability for different grade levels? 

    A5. Yes and no. We use a larger font on lower grades, which restricts the words that will fit, and that limits how much we can write. We have limited space on the page and so we tend to use nouns with just a few letters. 

    If you think we discriminate against Ascension who raises artichokes, you're right. We would  prefer Anna grows beets.

    My sister Kathy is a big wheel in reading education. She says write good stuff and kids will figure out how to read it. Write bad stuff and they won’t bother to try. We don’t do a formal readability analysis on each story, but we do change and clarify stories in response to comments from our users.

    Q6. Will you send me an answer key so my mom can help me study?

    A6. No. I didn't just fall off the turnip truck   razor scooter   hybrid bicycle, you know! Solve the problem yourself and confirm it with the Checkanswer.

    (Turnip, in the summer time, at Excel Math)