Sunday, December 27, 2009

Pythagorean Triples

I got interested in this problem last summer at the Math Circle Institute, held at Notre Dame. I have a bad memory and am having fun reconstructing what we figured out together there.

Introduction to Pythagorean Triples
The Pythagorean Theorem tells us that a right triangle with legs a and b and hypotenuse c will always have the relationship a2+b2 = c2. (Do you know how to prove that?) When all three sides are whole numbers, we have a Pythagorean triple. The most famous of these is 32+42 = 52, often referred to in this context as (3,4,5). The 3-4-5 triangle was used in Egypt to help make perpendicular sides for their magnificent buildings. A loop of rope with 12 equally spaced knots (3+4+5 = 12) was pulled taut at knots 0, 3 and 7 to make a precise right angle.

If the three sides don't all have a factor in common, then they make a primitive Pythagorean triple (PPT). 62+82 = 102 is not a PPT because all three sides have a factor of 2.

Starting to Explore

When I'm making up problems for students, I often want another Pythagorean triple, so I have one more sitting in my brain, 52+122 = 132. Are there others? Are there infinitely many PPT's? How would we find more?

One approach to exploring these involves thinking about parity. (Parity refers to whether a number is odd or even.) I see that both of the PPT's above, (3,4,5) and (5,12,13), are odd+even = odd. Can we have odd+odd = even? What about even+even = even? If we have odd+even = odd, does the odd leg have to be the shorter one?

Do other questions occur to you?

Recommendation: stop reading and start playing as soon as you have a thought about how you might proceed.

I knew there were more PPT's, but couldn't remember any others. I wanted to find a few more, so I could see any obvious patterns. So I made a list of the first 25 perfect squares and looked for pairs that would add to equal another perfect square, or subtract to equal another one. I found (8,15,17) and (7,24,25). Well, that's one question answered: the odd leg does not have to be the shortest side. I see that all 4 hypotenuses are odd. So I want to address the question of whether odd+odd = even is possible.

Question1: Is odd+odd = even possible?
Here's how I start: Suppose we have two odd legs. We can let a = 2n+1 and b=2m+1. Then a2+b2 = ... Can we come to a contradiction? [See the hint at the end for a bit more direction.]

Question2: If a is an odd number, can I find a PPT for it?
I notice that all 3 triples in which the odd leg is the short one include consecutive numbers for the other leg and the hypotenuse. Hmm. If a is the short leg, then b+c = a2 in all 3 of those cases. Is that important? What if I write a2+b2 = c2 as c2-b2 = a2? Oh! A square minus a square can factor. So I'd have (c-b)(c+b) = a2. What does that get me? [I found a way to get a PPT for every odd number. Can you?]

Question3: Must the even side be a multiple of 4?
I wanted more triples, in case it would help me see more patterns. So I set up a spreadsheet with column a holding 1 through 100, row 1 holding 1 through 254 (first time I've ever used all available rows!). Column b and row 2 had the squares of these numbers. The rest of the spreadsheet showed a 0 if the square root of the sum of these squares was not a whole number, and otherwise showed the number. [Here's the formula in cell c3: =IF(INT(SQRT($B3+C$2))=SQRT($B3+C$2),SQRT($B3+C$2),0)]

For most multiples of 4, I found a PPT. And I didn't find any for the other even numbers. So I wanted to know whether the even side had to be a multiple of 4. If the even side is a, then b and c are odd, and c2-b2, with b=2m+1 and c=2n+1, can be explored.

Questions 4 and 5: Multiples of 3, 4 and 5
This reminded me that I had read (whose blog was that on?) that in every PPT, 3 will be a factor of one side, 4 will be a factor of one side, and 5 will be a factor of one side. (As in (5,12,13), one side may contain more than one of these factors.) I realized I'd already proved it for 4. I started trying to prove it for 3.

I mentioned parity earlier. Even numbers can be expressed as a=2m, and odd numbers can be expressed as b=2n+1. Similarly, if we want to think about whether side c will be a multiple of 3, we can look at three cases: c=3m, c=3m+1, or c=3m+2. Using this, I started with the question of whether c would be a multiple of 3. (It wasn't in any of the triples I'd found.) If it is, then neither a nor b can be. (Why?) Once I solved that problem, I wanted to prove that one of the legs would be a multiple of 3. Suppose b is not a multiple of 3 and consider c2-b2. There will be 4 cases, each of b and c can either be 3x+1 or 3x+2. What does this make a?

I tried to think about 5 in this way but got nowhere. I'm writing this blog post in hopes that explaining my thinking will help me get further on some of my dead ends.

I have a few other questions I haven't answered:
  • Given a multiple of 4, how can I come up with a PPT?
  • Can the same number show up in 3 different PPT's?
Let me know if you have fun playing with this. Maybe the directions your thinking takes will be different than mine...

Hint: c2 is a multiple of 4. (Why?) What about a2+b2?

Friday, December 18, 2009

Math Teachers at Play #21

Welcome once again ...
to the Math Teachers At Play blog carnival. Puzzlers, riddlers, thinkers, doers, novices, experts, come one
, come all!

[photo by santarosa]

First off, in honor of the number 21, is a puzzle, fresh from the oven.

The Numberland News
runs personal ads. 21 was looking for a new friend and put an ad in.
Two-digit, semi-prime, triangular, Fibonacci number seeks same. I'm a binary palindrome, what about you?
Will 21 find a friend?


Kendra at Aussie Pumpkin Patch has written about the estimation lesson she did with her sons, which they started by reading Counting on Frank. It sounds like fun!

What is your child's favorite small toy? Will it help them learn division? Ashley at HyperHomeschool headed for the Legos, and here's what happened.

A really good way to understand place value is to work with other number bases. A book I recently discovered, How to Count Like a Martian, by Glory St. John, tells a detective story in which the history of other number systems plays a starring role. The last few chapters discuss place-value-based systems. Want a more hands-on approach? Sol, at Wild About Math!, offers us some math magic with index cards, based on binary numbers.

Megan Wong has written some books she'd like to share with us: Math Power is Fun and Brain Power is Fun are part of her Mind Power series.

Algebra and Geometry
At Let's Play Math, Denise gives another algebra lesson in Pre-Algebra Problem Solving: 4th Grade. I'm thinking this would be a good first step even for adults just learning algebra. I've seen mention of bar diagrams plenty, but this is the best illustration of their use I've seen so far.

Maria Miller offers us 3 videos of her proofs of some basic geometric relationships in Angles in a parallelogram and a triangle.

John Golden at Math Hombre shares his geogebra sketches (available as webpages and geogebra files) at Net Results. Students can use these to create their own prisms and pyramids. Print the nets, fold them up and see the funky solids.

One of my favorite things about MTAP is discovering new blogs. Here's one: Guillermo presents a Tutorial on Geometer's Sketchpad.

Calculus and ...
Pat's Blog has Fun with Parabolas.

Dan at Mathrecreation offers us a curious population model, with directions for exploring the logistics functions in Fathom.

Our Favorite Proofs
Brent, at The Math Less Traveled, presents a proof that pi is irrational. He thinks calculus students should be able to follow it. I've treid to follow this proof in other places and not had the patience for it. So far, his explanation is right up my alley.

The Count, at Discrete Ideas, gives us Discretely Simple, on his two favorite proofs.

On Teaching
Simple things can make such a difference. “What’s a question that someone else might get wrong?” So simple, and such a good way to get students thinking. Here's JD's post on it.

Riley Lark, at Point of Inflection, offers us his index cards. Well, the students get the cards, and some quickie interaction.

Then they can review with Trashketball. Post by Dan Greene at The Exponential Curve.


The Holiday Connection
When mathematicians hear about the gifts "my true love gave to me" on the 12 days of Christmas, they start counting. How many gifts would that be altogether? Sol at Wild About Math! wrote this post a few years back. And John at The Endeavor wrote this post more recently. One of the commenters on John's post wondered: "Funny that the 12 days of Christmas turn out to have just short of one gift per day for a full year. A coincidence?"

Remember the Soma Cube? Rachel, at Minds in Bloom, gives directions for making it here, and thinks a home-made puzzle would make a great gift. If you'd like to do that for this holiday season you may not have time to wait for the cubes to be shipped. But if you can wait, the cubes are pretty inexpensive online: 100 1" plastic cubes on Amazon for about $15, 100 1" wooden cubes for about $10 here, or 1000 centimeter cubes for $25.

Between Maria and JD and a few others around here, I've started to think that creating puzzles (or authoring math, as Maria would say) is something I too can do. So last week I made a logic puzzle, Holiday Logic. I hope you'll enjoy it.

This edition of MTAP was composed in Richmond, California and Chicago, Illinois. It's coming out late in the day because leaving home and flying here yesterday, even though uneventful, did take up my whole day. May your holidays be peaceful. May peace spread exponentially from our hearts through our actions to the world around us.

Thursday, December 17, 2009

Holiday Logic Puzzles

I'd like to share two puzzles here. One is by Mike Shenk, who has a site called puzzability, and is interviewed here. It's called Oh Deer!, and it's a killer. The other I created last week. I wanted a logic puzzle with a holiday theme for my math salon, and I wanted it to be much easier than Oh Deer! When I searched online, I got tired of seeing so much about buying presents. I also wanted to include other holidays besides xmas.

I've always loved doing these puzzles, but this is my first attempt at creating one. I believe it has a unique answer. Please let me know if I goofed.

I wish you all peace and joy during these winter holidays.

Holiday Logic
by Sue VanHattum

1. The Green girl’s favorite Christmas tradition is singing carols.
2. The Brown boy celebrates Kwanzaa indoors.
3. DJ and Jordon joined their friend in her candle lighting ceremony.
4. Layla and Amani joined their friend for his annual walk through the woods.
5. The Gold girl came to Jordon’s house to join his family in their feast.
6. The Fox family celebrate the Yuletide, and Amani comes to their party.
7. Amani couldn’t make it to the Gold family’s Hanukkah celebration.

[Edited for clarity.] Each child celebrates just one holiday with one special activity as a tradition in their family, though they do join in the fun with their friends this year. Your mission: Decide who celebrates each holiday, and what they do to celebrate.

Oh Deer! A logic problem by Mike Shenk
(first published in Games Magazine, December 1992)

Twas the night before Christmas, and at the North Pole
The last-minute planning was taking its toll.
As Santa was hastily making a scheme
For the placement of deer in his sleigh-pulling team,
The good Mrs. Claus was crocheting bright bows
To be worn by these reindeer (four bucks and four does).

The ribbons were colored in eight festive hues:
One ocher, one rose, one cerise, one chartreuse,
One maroon, one magenta, one white, and one blue.
(These ribbons helped Santa keep track of who's who.)
The deer pulled the toy-laden sleigh in four rows,
Arranged so no row held two bucks or does.

The order of pullers was changed year by year,
For Santa was thoroughly fair with his deer.
He summoned the elves and instructed them thus:
"Let's hitch up the reindeer with minimum fuss.
The bow on the buck behind Dasher is white,
While Blitzen, a doe, sees cerise to her right.

The blue bow is nearer my sleigh than is Dancer,
But nearer the front of my team than is Prancer.
The doe in chartreuse gets a front-of-team honor,
But not on the same side as Cupid or Donner.
Now Comet stands two spots ahead of the rose.
And three deer of four on the right side are does.

The cerise bow is worn two in back of maroon,
One of which is beside the bright ocher festoon.
Oh-Cupid's in front of a buck, by the way.
Well, that's how they line up for pulling my sleigh.
I trust that you elves, being clever, now know
Each reindeer's position and color of bow."

In no time each colorful ribbon was tied
And the team was hitched up for the transglobal ride.
Can you ascertain where each member fits in?
Who's Comet? Who's Cupid" Where's Donner? And Blitzen?
Who's Dasher? Who's Dancer" Where's Vixen? And Prancer?
With logical thought, you'll determine the answer
And write down the color and place for each deer.
Happy Christmas to all, and to all much good cheer!

Thursday, December 10, 2009

Hannah, Divided, by Adele Griffin

Over the past year I've learned a new term: 2e, or twice exceptional, is a term used by advocates of kids who are exceptionally smart, along with having exceptional learning differences. (I'm paraphrasing Tiffani, who blogs beautifully at Child's Play about 2e issues. Another blog I've enjoyed on the subject is Life Among the Gifted.)

Hannah, Divided is the sweet story of a girl who would be designated 2e nowadays. Growing up during the depression on a farm, she's not too worried about her struggles with reading, and the comfort she takes in numbers is very personal. She doesn't much expect either her learning trouble or her gift to take her away from milking cows and sharing the chores with her family. But they do.

Her teacher, Miss Cascade, has prepared the one-room schoolhouse and all its students for a visit from a possible benefactor. On the day Mrs. Sweet arrives, she takes an interest in Hannah's math abilities and quizzes her after school. On the way home, Hannah is so fired up, she just has to run, and count.
Finally, she took this year, 1934, and divided it by two. Over and over, skipping the decimal point like a checkers piece until it stood at the front of the line.

Granddad McNaughton encouraged her mathematics. Sunday afternoons, they passed gleeful hours inventing games with figures and sums, making up riddles and puzzles to solve.
She's given a chance to go to Philadelphia to study math, and takes it. Leaving home is difficult for her, and she leans on her need to pace her room 32 times, get each item in exactly the right place, and tap her paper 32 times. As hard as it is, the math she's able to learn from her tutor at Ottley Friends' School makes all the hardship worthwhile.

As I read this simple story, I kept thinking of how moving it might be for my young friend Artemis, and other kids like Hannah. I hope some of you have a chance to enjoy it soon.

Wednesday, December 9, 2009

Math Teachers at Play coming up...

Submissions are due in a week, on Wednesday, December 16. Please send your favorite posts that haven't been in MTAP yet, new or old. Send your own; send links to other people's posts that you like; send math questions that have you puzzled.

Monday, December 7, 2009

On Theorems and Proofs

There's a good discussion, over at f(t), on "What's your favorite theorem?" (It started on Twitter, but I like blogs better.)

And Brent (of The Math Less Traveled) has begun a lesson I've been longing for. π (that's pi) is irrational. I knew that. But I knew it in a way that doesn't count in math. I took it on authority. I've tried to look up the proof, and didn't have the patience for following what I saw. I'm confident Brent will walk us through it gently. I'm looking forward to this. Maybe this will be my favorite theorem, once I learn it. ;^)

What's my favorite theorem? Hmm, I like:
  • Why the square root of 2 is irrational,*
  • Rationals are countable and reals aren't (that's the one Kate explained so well at f(t)),
  • Pythagorean Theorem,
  • Fundamental Theorem of calculus,
  • Infinity of primes,
  • Angles in a triangle add to 180 degrees,
  • The ones in linear algebra that all go together.
It's hard to pick just one. The angles one is nice, because you can show it by ripping off the corners of a triangle. I know, that's not a proof. But the proof parallels the torn paper demonstration nicely.

* I'm having a bad internet day. I couldn't find the site that makes pretty equations.

Saturday, December 5, 2009

Liars, Truthtellers, and Octopuses*

Tanya Khovanova has translated some puzzles from Russian and created some herself. Now I want to try it.

Here's the intro and the easiest of the problems:
... our characters are genetically engineered octopuses. The ones with an even number of arms always tell the truth; the ones with an odd number of arms always lie. ... Not only do octopuses lie or tell the truth according to the parity of the number of their arms, it turns out that the underwater world is so discriminatory that only octopuses with six, seven or eight arms are allowed to serve Neptune. ... our octopuses who worked as guards at Neptune’s palace were conversing:
  • The blue one said, “All together we have 28 arms.”
  • The green one said, “All together we have 27 arms.”
  • The yellow one said, “All together we have 26 arms.”
  • The red one said, “All together we have 25 arms.”

How many arms does each of them have?

If you enjoyed that, go visit Tanya's blog and try the others.

Now it's my turn. Can I do it? My 4 all work at Neptune's also.
  • Aqua says, "Turquoise has 6 arms."
  • Turquoise says, "Blue has 7 arms."
  • Blue says, "Green has 7 arms."
  • Green says, "Aqua has 7 arms."
  • Aqua says, "The truthtellers have the same number of arms as each other."
Not as elegant, but I think it works. (I'm up in the middle of the night, so my brain may not be functioning well enough to do this.) Your turn...

* Wikipedia explains why it's not 'octopi'.

Friday, December 4, 2009

Gifts for math lovers

I posted back in June about my favorite math books. Any of those would make a great gift. But I'm excited about a few books I've read recently, and wanted to share them here for those of you who like to give books as gifts. Both are biographies of mathematicians.

Carry On, Mr. Bowditch, by Jean Lee Latham (1955) is written for younger readers, but will charm many adults too. It's a fictionalized account of the life of Nathaniel Bowditch, who loved math, but had to leave school when his family needed his help. He was indentured to a ship chandlery for 9 years, which dashed his hopes of someday going to Harvard to study math. But he spent his spare time learning everything he could on his own - he learned Latin so he could read Newton's Principia Mathematica, and then learned French so he could read another book recommended to him.

After his indenture ended, he sailed with a merchant ship, and became interested in the mathematics of navigation. He was incensed at the errors in the book of tables used for navigation, and began the laborious work of correcting them.
"You don't 'cast your eye' over navigation tables!" Nat barked. "When I checked that one table of Maskelyne's, I worked every figure three times, just to be sure I was right!"
"Three times? Every figure? But why in ..."
"Why not? Nat roared. "Mathematics is nothing if it isn't correct! Men's lives depend on those figures!" (page 161)

He also taught the crews how to "do a lunar", a startlingly egalitarian action back then. In this passage he's talking to a young woman who later becomes his wife, but this is much like the lessons he gave the crews he sailed with:
Nat said, "That's the North Star. If you think of the North Star as the middle of your clock face, and the line from it through those other stars as the hour hand, you can tell time."
"It says about one o'clock. Is that right?"
"No, this clock runs backwards."
"Is it eleven o'clock?"
"No, there's one other difference. It takes twenty-four hours for the Big Dipper to swing around the North Star. So every hour space on the clock face stands for two hours." (page 85)
Bowditch eventually decided to write his own book, which he hoped would be error-free, and would also include navigation lessons and general information needed by sailors. His book, the American Practical Navigator, published in 1902, is still carried on every U.S. naval vessel (according to Wikipedia).

Bowditch was born in 1773 in Salem, Massachusetts. I enjoyed reading about the early days of the U.S. as an independent nation from his perspective. My only concern with a book like this is that I'll mix up what's fiction and what's true. The astronomy lesson I've quoted is a bit oversimplified, but apparently close.

The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel, would be unbelievable if it were fiction or even slightly fictionalized. A friend of mine, who works with gifted kids, describes their learning style as sitting in the eye of a hurricane and grabbing at the ideas whirling by. As I read about Ramanujan's mathematical discoveries, I keep coming back to that image. Other mathematicians who worked with him were astounded by his process. Bruce Berndt noted that, although Ramanujan's proofs were often full of holes, his results were almost always correct, and suggested, "We might allow our thoughts to occasionally escape from the chains of rigor, and, in their freedom, to discover new pathways through the forest." (page 183)

Ramanujan grew up in South India and attended school sporadically. (In his younger years, he preferred learning on his own. Later, he couldn't deal with exams in subjects outside mathematics, and was kicked out of university.) It took him years of working as a clerk to support himself before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. He went from just scraping by with a job he had little interest in, to a paid position as a research student in mathematics at Presidency College in Madras, India. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others.

I'm only halfway through, but have learned much already about India, mathematics at Cambridge in the late 1800's, and the history of mathematics. Kanigel does a good job of giving us enough background so that we have some chance of understanding different cultures and different times. (At times, his own bias shows, but subtly.) His description of Hardy's writing (readable, clear, cogent, almost suspenseful) makes me want to go to my local math library and borrow his 1908 Course in Pure Mathematics.

One of the delights for me is the sweet trivia I'm picking up. Here are two bits I enjoyed.
  • When young, Ramanujan played Goats and Tigers, a traditional game in India in which:
    Three "tigers" sought to kill fifteen "goats" by jumping them, as in checkers, while the goats tried to encircle the tigers, immobilizing them. (page 18)

  • It is so hot traveling through the Red Sea that cabins on the east side of the boat, which can cool down away from the heat of the afternoon sun, are quite a bit nicer than those on the west side. Hence the acronym POSH for Port Outward, Starboard Homeward (page 199).

I've bought both these books to give to my young friend Artemis. (If you prefer giving games as gifts, I highly recommend Blink (under $10), Set (under $15), and Blokus.) I'm always searching for ways to enjoy the holiday spirit and still consume less. Used books are part of my solution to that conundrum. And I like getting them from Better World Books because of their interest in global literacy and taking care that old books don't end up in landfills.

May your holidays be peaceful.

Thursday, December 3, 2009

Blog Heaven

  • Would you like your best blog post ever (BBPE) to be in a book?
  • Will you still be blogging in a year or two, when people are reading your BBPE in that book?
If you write about teaching or learning math, or about how people learn, and you said yes to those questions, then send me a link to your BBPE.

I'm working on a book with the (way too long) current working title of Learning Math Outside the Classroom and In: Stories from Math Circles, Homeschoolers, and the Internet. I don't have a publisher yet, but about 15 authors are working with me, each writing a chapter about the cool stuff they're doing. It's moving a bit more slowly than I'd expected (not surprising), but it keeps looking better all the time!

The book will have 5 sections:
• Math Circles, Clubs, Centers, Festivals, and Salons
• Homeschoolers and Unschoolers Do Math
• The Internet is Changing Our Lives: Teaching, Learning, and Doing Math with New Resources
• Classrooms
• Issues: Race, Gender, Gifted Kids, Public Policy (etc)

The internet section will have a few in-depth chapters, but I recently realized that 'chapters' won't do it justice. We need a compilation of BBPE. I'll be going through old editions of Math Teachers at Play, but you can save me the effort of digging out your gems by sending them in. If you're on my blogroll, you probably have a post that would work. Please don't be shy.

If you'd like to point me to anything else that ought to be in this book, I'd be delighted. If you'd like to write a chapter, that may still be possible. (I do not have enough material for the classrooms section yet.)

Email me at mathanthologyeditor on gmail etc. with links, comments, questions, and ideas.

[The fine print: A few authors have asked who gets the money. All authors of full chapters, including me, will equitably split any profits, but profits are minimal for books like this. No one is expecting to make much money, if any. It's glory for math that we're after.]
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