Math that made you go whoa!

Saw this tweet from Dan Anderson a few days ago:

I had a 7 hour round trip drive yesterday and spent a little time thinking about the math ideas that really grabbed me in high school. Three really stuck out in my mind:

(A) The Extended law of sines:

We learned in our trigonometry class that for a triangle with sides A, B, and C, and corresponding angles a, b, and c that:

\frac{A}{Sin(a)} = \frac{B}{Sin(b)} = \frac{C}{Sin(c)}

But it turns out that these ratios are equal to 2R where R is the radius of the circumscribed circle. I learned this idea from the wonderful book Geometry Revisited by Coxeter:

Book 2

In fact, the first math movie I uploaded to youtube was about this idea:

This identity made me think that there was a lot more going on in geometry that met the eye. One neat particularly neat thing that the identity shows is that the area of a triangle with side lengths A, B, and C is equal \frac{ABC}{4R}. Beautiful!

(B) 1 + 1/4 + 1/9 + . . . . = \frac{\pi^2}{6}

Mr. Waterman used the idea that the coefficients of a polynomial were symmetric functions of the roots to prove this sum. It blew me away. (yes, this is a non-rigorous proof, but it is what captured my attention)

In general, for a polynomial of degree n, x^n + c_{n-1}x^{n-1} + \ldots + c_1 x + c_0, sum of the reciprocals of the roots is given by -c_1 / c_0.

We know that Sin(x) = x - x^3 / 3! + x^5 / 5! + \ldots. Factoring out an x we are left with a polynomial whose roots are \pm \pi, \pm 2\pi, \pm 3\pi, \ldots, namely:

\frac{Sin(x)}{x} = 1 - x^2 / 3! + x^4 / 5! + \ldots

making the substitution u = x^2, we see that the polynomial

1 - u/3! + u^2 / 5! + \ldots

are \pi^2, 4\pi^2, 9\pi^2 \ldots

By the “sum of the reciprocals of the roots” formula above, we see that

\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \ldots = \frac{1}{6}.

Multiplying both sides by \pi^2 gives us the result.

This result showed me that there was more going on with the integers than I realized! How could they be connected to \pi? A few years later I’d see this identity in a complex analysis class and see that \pi and e were connected in a strange way, too!

Formula

(3) A formula for the Fibonacci numbers

I think it was my sophomore year in high school when a former student, Anita Barnes, came back to lecture to Mr. Waterman’s Enrichment Math class. Her talk showed a way to find closed form solutions for simple recurrence relations like the one for the Fibonacci numbers:

F_{n+1} = F_n + F_{n - 1}

The idea seemed incredibly simple – for the Fibonacci numbers just assume the solution took the form F_n = x^n and solve for x. Solving the recurrence relation for the Fibonacci numbers was reduced to solving the quadratic equation x^2 = x + 1. From there it was not hard at all to show that the Fibonacci numbers were connected to the Golden ratio. If we let \phi = \frac{1 + \sqrt{5}}{2}, then

F_n = ( \phi^n - (-\phi)^{-n}) / \sqrt{5}

That just blew me away – there was a simple formula for the Fibonacci numbers (and any simple recurrence relation). You could calculate the 100th Fibonacci number by just knowing the first 2 plus the recurrence relation. I think this was the first idea from advanced math that totally blew my mind.

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A problem my high school math teacher would have loved

Earlier tonight a friend of mine sent me a link to some old problems from a middle school math contest in California.

Berkeley Math Tournament middle school problems

I skimmed through the 2014 contest for fun and ran across a problem that my high school math teacher – Mr. Waterman – would have loved:

#16: Consider the graph of f(x) = x^3 + x + 2014. A line intersects this cubic at three points, two of which have x-coordinates 20 and 14. Find the x-coordinate of the third intersection point.

I’m a little surprised (to say the least!) to see a problem like this on a middle school test, but it is a really cool problem. Properties of roots of equations was one of Mr. Waterman’s favorite subjects.

ALS and the Ice Bucket Challenge

My high school math teacher, Mr. Waterman, died from ALS last year.  There’s almost no way to explain the influence that he had on me as a kid.  From math, to teaching, to simply understanding how one person can have a tremendous impact on another person’s life, I am a different and better person because of him.   Earlier today  I was tagged by another student of his for the Ice Bucket Challenge.  Here’s my video – I hope the publicity from this challenge raises buckets of money for ALS reserach:

For most of this week the Quanta Magazine piece on Maryam Mirzakhani winning the Fields Medal had me reflecting on my time in high school. Here’s the article, it is absolutely fantastic:

A Tenacious Explorer of Abstract Surfaces

The bit that got my attention was this paragraph about an experience she had in high school:

“Eager to discover what they were capable of in similar competitions, Mirzakhani and Beheshti went to the principal of their school and demanded that she arrange for math problem-solving classes like the ones being taught at the comparable high school for boys. “The principal of the school was a very strong character,” Mirzakhani recalled. “If we really wanted something, she would make it happen.” The principal was undeterred by the fact that Iran’s International Mathematical Olympiad team had never fielded a girl, Mirzakhani said. “Her mindset was very positive and upbeat — that ‘you can do it, even though you’ll be the first one,’ ” Mirzakhani said. “I think that has influenced my life quite a lot.”

So, she asked the school principal to provide a problem solving class and a year or two later won a gold medal at the IMO, and followed that up the next year with a perfect score.   Good gracious.  Even several days later I’m not sure I can wrap my head around that.   That has to be one of the most amazing accomplishments (in any field) that I’ve ever heard of in my life.

That insane accomplishment aside, it also shows the influence that principal and the teachers had.  I was pretty lucky, too, to have a great principal,  Dr. Moller,  along with Mr. Waterman.  Dr. Moller had a effectively infinite trust in Mr. Waterman had gave him enormous room to teach how he wanted.  We didn’t have to beg for a problem solving class, for example, because it already existed!   Linear Algebra and Differential Equations, too.  Really amazing and probably not what you’d expect from a high school in Omaha.  Truthfully, though, it was only many years later that I realized Dr. Moller’s role, but the older I get  the more I realize how important it was.  The article about Mirzakhani probably made me understand the luck I had in this area a little better.

There was a nice article about Mr. Waterman in the Omaha World Herald after he died:

John Carl Waterman inspired as Omaha teacher and coach

And a nice letter to the editor from Dr. Moller, as well:

Waterman was excellent instructor

I was John Waterman’s principal during his career at Omaha Central High School (“Waterman inspired as teacher and coach,” Sept. 29 World-Herald).

He had a rare talent for making weak math students believe in themselves and succeed in learning math, often for the first time, and he inspired strong math students to achieve even more than they ever expected. His math teams’ records still are among the best in Nebraska high school competition. He was among the first in the state to recognize computer capabilities for teaching math, and he implemented computer programs and activities that were cutting edge for the time.

John has former students working all over the world who would testify that he was an exemplary teacher taken from us much too early.

G.E. Moller, Omaha

You never know how someone is going to influence your life, but I certainly was the beneficiary of some good luck in high school.  I sure hope the Ice Bucket promotion raises lots of money for ALS research.

Problem Solving

I was lucky to have my early development in math shaped by an incredible high school math teacher – Mr. Waterman.  He lived and breathed math and got out of bed everyday thinking about how to teaching problem solving to high school kids.   I ate it all up back then and the problem solving techniques that I first learned in his classes remain important pieces of my mathematical tool kit even today.   It is actually pretty fun to look back and see how ideas originating in high school math contests can come into play 25+ years later at work.  Although maybe the fact that I still use these ideas every day explains why I think problem solving is such an important part of math education.

For the last few weeks I’ve been seeing a lot of posts on twitter about problem solving.  Yesterday, for example, I ran across this wonderful post from Fawn Nguyen:

Making Problem Solving Part of the Math Curriculum

While others may not, I find it difficult to talk about problem solving in the abstract.  In talking or learning about problem solving a specific example is much more valuable to me than 1,000 words of abstract discussion.  Two of the most amazing specific examples I’ve seen lately come from the Field’s Medalist Tim Gowers and from the head of Art of Problem Solving, Richard Rusczyk.    Gowers decided to “live blog” his attempt at working through problem 1 in this year’s International Mathematics Olympiad.   It is an all too rare treat to see the thought process of one of the top mathematicians in the world in real time:

Tim Gowers walks through an IMO problem

Richard Rusczyk’s example is also a math contest problem – in this case problem #24 from the 2013 AMC 12.  The problem asks a question about the quadratic function  f(z) = z^2 + i z + 1.  I’m sure that as a kid a quadratic with an imaginary component would have really intimidated me, but Rusczyk’s calm and systematic approach to walking through the problem takes all of that intimidation away.  It is such a great example of problem solving – in fact this blog post has been delayed by about 40 minutes as I’ve watched the video twice because I enjoy it so much!

With these two examples in mind, I thought it would be fun to do my own example, though believe me, it isn’t within a million miles of the quality of either of the examples above.  However, what I think is incredibly important about the Gowers and Rusczyk examples is that they show the problem solving process, and I think that the more examples of that process that are out there the better.  So, neither an IMO problem nor a problem from a big US math contest, but here’s the process that I went through thinking about a fun little problem I saw posted on twitter a week ago:

I was actually up in Boston running around with the family when I saw the original tweet, but for some reason the problem stuck with me and I spent the next few days sort of daydreaming about it.

The first thing I did was think through cases that I hoped would be easy – quadratic equations (and also simplified the problem a tiny bit by assuming that the infinitely many perfect squares arose from positive integers n).  Take a polynomial like x^2 + 2x + 5, for example.  We can rewrite this expression as (x + 1)^2 + 4.   Since the only perfect square than is 4 more than another perfect square is 4, this new expression helps us see that x^2 + 2x + 5 will not be a perfect square for infinitely many integers x.

What if the linear term has an odd coefficient, though – say x^2 + 3x + 1.  Writing this expression as  (x + 1)^2 + x solves the problem in a slightly different way than the prior case.  For x > 1, we see that the new expression is greater (x + 1)^2 but less than (x + 2)^2.  Since  it lies in between two consecutive perfect squares, it cannot be a perfect square and so it will not be a perfect square for infinitely many positive integers x.

Since the coefficient of the linear term will either be even or odd, that takes care of the quadratic case. The only way that a quadratic polynomial can satisfy the conditions of the problem is if it can be written as (x + a)^2 for some integer a.  Great, but how does this fact generalize to higher degree polynomials?

The next example I thought about was the 4th degree polynomial x^4 + 3x^3 + 3x^2 + 2x + 1 = (x^2 + x + 1)^2 + x^3. From the quadratic case I thought that transforming the original polymonial into a perfect square plus a polynomial with a degree of 2n – 1 would be the way to go. Unfortunately I couldn’t see what to do, and certainly couldn’t see anything that would help in the case where the remainder was a general cubic polynomial.  What’s stopping a perfect square plus a number described by a cubic polynomial from being a perfect square just by accident? Hmmmm.

As we ran around Boston the problem stayed in the back of my mind. One day I got the idea to look for a different approach to the quadratic case and that turned out to be the key idea for getting my head around the problem. Writing the polynomial x^2 + 3x + 1 as (x + 3/2)^2 - 5/4 also helps you see that this polynomial cannot be a square for infinitely many integers.  Really for the same reason as above – the value is trapped between (x + 1)^2 and (x + 2)^2. This new expression led me to stumble on a similar statement for higher degree polynomials that was a nice little surprise (to me anyway!).

It turns out that completing the square generalizes in a neat way. For the case we are looking at – a monic polynomial of degree 2n with integer coefficients – the generalization is that you can always find a monic polynomial of degree n with rational coefficients whose square matches the first n + 1 terms of the original polynomial. In math symbols, I mean that we can write:

x^{2n} + a_{2n-1} x^{2n-1} + a_{2n-2} x^{2n-2} + . . . + a_0

as

( x^n + b_{n-1} x^{n-1} + . . . + b_0 )^2 + c_{n-1} x^{n-1} + c_{n-1} x^{n-1} + . . . + c_0

where the a’s are integers, and the b’s and c’s are rational.  The the 4th degree case I was looking at above gives an illustrative example:

x^4 + 3x^3 + 3x^2 + 2x + 1 = (x^2 + (3/2) x + 3/8)^2 + 7x/8 + 55/64.

Convincing yourself that this generalization of completing the square is true isn’t all that difficult.  When you square the polynomial with the b_{i}‘s above, you see that 2b_{n-1} has to be equal to a_{2n-1}, so solving for b_{n-1} is exactly the same exercise you do when you complete the square for a quadratic.  Once you have the value for b_{n-1} you see that a_{2n-2} has to equal  (b_{n-1})^2 + 2b_{n-2}, which give you the value of b_{n-2}.  Similarly, once you have the first k b_{i}‘s, solving for the next one just involves solving a linear equation in b_{n - k - 1}.  Basically, you just end up dividing by 2 a lot.

Although the rational coefficients present a small problem, we can use  ideas similar to the ones  we used in the quadratic case to show that for large values of x the above expression cannot be a perfect square.  In the above example of the 4th degree equation, assume that the expression is equal to z^2 when x and z are integers.  Multiply everything by 64 to arrive at the equation:

(8x^2 + 12x + 3)^2 + 56x + 55 = (8z)^2

For large values of x, the expression on the left hand side lies between the consecutive perfect squares (8x^2 + 12x + 3)^2 and (8x^2 + 12x + 4)^2.   For clarity, note that (8x^2 + 12x + 4)^2 = (8x^2 + 12x + 3)^2 + 2*(8x^2 + 12x + 3) + 1.    As long as x is sufficiently large, 2*(8x^2 + 12x + 3) + 1 will be larger than 56x + 55 since the x^2 term will dominate the x term.  Thus for sufficiently large integers x, the value of (8x^2 + 12x + 3)^2 + 56x + 55 (which we’ve assumed to equal (8z)^2) lies between two perfect squares and cannot be a perfect square itself.  So neither (8z)^2 nor z^2 can be a perfect square for sufficiently large values of x which, in turn, means that the original expression cannot be a perfect square for infinitely many positive integers x.

Essentially the same argument will work for any monic polynomial of degree 2n with integer coefficients, and also essentially the same argument works for showing that the polynomials cannot take on infinitely many perfect square values for negative integer values of x.

So, the generalization of completing the square shows that the only time a monic polynomial in x of degree 2n will take on infinitely many perfect square values for integer values of x is when the polynomial itself is a perfect square.  In that case, all of the values will be perfect squares as desired!

I doubt that my solution is the best or the most elegant, but I had a lot of fun thinking through this problem.  I’m also happy to have stumbled on this generalization for completing the square which I’m surprised to have either never seen previously or (more likely) just forgotten about.

As I said at the beginning of this post, the lessons about problems solving that I learned in the three years I spent in Mr. Waterman’s classes were such an important foundation in my own mathematical development.  Even though I’m not longer in academic math these problem solving strategies play a critical role in my work just about every day.  Away from work I try to communicate those lessons to my kids when we talk about math.

I hope that more mathematicians will follow the lead of Gowers and Rusczyk and give some public examples  of their own problem solving process so that everyone – and especially students – get lots of different looks at the problem solving process.  Seeing that work will help show that mathematical thinking  isn’t  about finding answers instantly and effortlessly, but often involves lots of trial and error, false starts, and most importantly joy at making a little progress.  These are important lessons from math with applications that go far beyond whatever specific problem you happen to be working on at the time.

Another really neat problem from James Tanton

Last week James Tanton posted this interesting problem on Twitter:

It was especially interesting to me because it is closely related to one of my earliest experiences with “advanced” math.

My high school in Omaha had an incredible math department led by an amazingly gifted math teacher, Mr. Waterman. He taught a special class over the lunch hour – Enrichment Math – where kids from ranging from freshmen to seniors would study math that is normally not part of high school curriculum. The year before I came to high school the unquestioned leader of that class, Anita Barnes, had graduated and became the 3rd student from our school in 5 years to receive a full ride math scholarship to Washington University.

During one of the college breaks she came back to give a lecture on recurrence relations in which she showed how to write down an formula for the nth Fibonacci number. It may as well have been magic, but that was one of the first times I saw the incredible power of math first hand.

Tanton’s problem was a happy reminder of the lecture that Anita all those years ago (1987 if you must know!), and I thought it would be really fun to walk through the problem with the boys.

We started by just talking through the problem to make sure that they both understood what was being asked. They both recognized that (1 + \sqrt{2}) was irrational and thought that the irrationality of the number would come into play somehow in this problem.

The next step was to head to the computer to see what powers of (1 + \sqrt{2}) look like when you write them as decimals. It takes a little while for the pattern to emerge, but the boys are able to pick up on the pattern after looking at the first 10 powers.

After getting an idea that we may indeed eventually run into one million 9’s after the decimal point, we go back to the board to talk about why. In this video I introduce the boys to the surprising role that the number (1 - \sqrt{2}) plays in our problem. This video has a little unexpected detour where we talk about multiplying two negative numbers, so that’s why it is a little long.

Now that we’ve understood a little bit about (1 - \sqrt{2}) we head back to the computer to first understand this number a little better and then to see an amazing pattern formed by combining powers of (1 + \sqrt{2}) and (1 - \sqrt{2}). Understanding this pattern provides a neat way to solve Tanton’s problem:

Now that we’ve calculated the first few numbers in the list of (1 + \sqrt{2})^n + (1 - \sqrt{2})^n we head back to the board to see if we can see any patterns at all in this new list. It was fun to see that they did manage to find the pattern by working together.

Having found the pattern, I show them how to write a formula that describes that pattern. If we can prove that our formula is correct (which we do not do, btw) we can understand why our list will only ever have integers, and then why powers of (1 + \sqrt{2} will eventually get as close to an integer as we like. This is the math that Anita explained to our Enrichment Math class in her 1987 guest lecture:

Finally, it would be a shame to have gone through all of this math without mentioning the Fibonacci numbers. We take a few moments to calculate the first few Fibonacci numbers and then I show the boys the formula involving \sqrt{5} that describes this sequence.

All in all a fun way for the boys to see (and participate in) some neat math. Also a nice walk down memory lane for me.

Fawn Nguyen and Bridges

This weekend Fawn Nguyen was speaking at a conference and posted the following problem that she was using as an icebreaker:

I use so much of Fawn’s stuff with my kids that we may as well move to California and have her teach them directly.  This problem seemed like one the boys would love, so we tried it out right away:

I definitely enjoyed the discussion and it was a nice surprise to see how this problem engaged my younger son  In fact, so much so that my older son actually complained after the video that he couldn’t get a word in.  Ha!

After we finished up the video yesterday we went out to join a little neighborhood dog walking group that meets every morning.  One of the people who joins that walk on the weekends is the principal of one of the local high schools.  He and I spend a lot of time talking about fun math activities for kids and I mentioned this problem to him.  He joked (in a good-natured way) that there wasn’t much real world connection to this problem.   Connecting math problems to the “real world” isn’t something that I spend a lot of time worrying about  – we do plenty of problems with real world connections and plenty that are more theoretical.  The reason that Fawn’s problem was attractive to me is the problem solving techniques required to solve it.  The unusual circumstances described in the problem do not bother me at all.

But . . . I couldn’t get the comment out of my head and it occurred to me that there are a few other neat math problems related to bridges that do have “real world” connections.   Two of those problems made for great discussion topics this morning.    The first is a short explanation of why bridges have expansion joints.  The answer to the math problem in the example is really surprising – almost no one guesses right when they see the problem for the first time:

This one turned out to be such a neat problem that my younger son joined in about half way through because he was so interested in what we were doing.

The next problem is the Königsberg Bridge puzzle which is a famous puzzle that was solved by Euler in the early 1700s.  Unfortunately I stated the problem slightly incorrectly – you just have to go over every bridge once, you don’t have to return to where you started.   Since I wasn’t planning on going into any theory here, I decided not to go back and correct the mistake.  The boys really liked playing around with this and spent 10 minutes after the video trying to solve a modified version of the problem (add two bridges – one to each land side – to the island that started with three bridges).  Definitely a fun example for kids:

Finally, I could do a post about bridges without mentioning the book that Mr. Waterman gave to me my sophomore year of high school that was my first introduction to abstract mathematics:

Book Pic

So, thanks to Fawn Nguyen for inspiring one more fun weekend of math!

Geometry, Continued Fractions, and the square root of 2

Of all the interesting and non-standard topics that Mr. Waterman covered with us in high school, continued fractions was definitely my favorite.  I don’t have any great reason, it was just the beauty of the topic really appealed to me.  Each time that I covered fractions with my sons I made sure that we spent a little time talking about continued fractions just for fun.   That exercise with my youngest son happened last week.

I don’t know why continued fractions seem to have vanished from the math ed. landscape.  My working assumption is that for high school the topic was  just one of many casualties of curriculum decisions, and for college the topic was probably viewed as being too elementary.  The actual reasons are probably more nuanced.

The topic was previously held in pretty high regard in mathematics, though.  Take this famous story about Hardy seeing Ramanujan’s work for the first time (from Wikipedia) for example:

“After [Hardy] saw Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy commented that “they [theorems] defeated me completely; I had never seen anything in the least like them before.”

Beyond simply introducing a fun topic, one other reason that I’ve enjoyed playing around with continued fractions with my kids is that  it gives me an easy opportunity to give them a little more practice doing arithmetic with fractions.  The book Mr. Waterman used to teach the topic to us is filled with tons of amazing examples, and it is the same one I use today – C. D. Olds’s “Continued Fractions” from the MAA’s New Mathematical Library.    Here’s a pic of the book along with the only other book in that series that I refer to more frequently (neither quite in mint condition these days) :

Book 2

So, after a brief discussion about the basics of continued fractions on Thursday, on Friday morning we looked at the continued fraction for \sqrt{2}:

Then on Friday evening there was a fun little coincidence on twitter.  First, Steven Strogatz posted a link to a neat proof that the square root of 2 is irrational:

Patrick Honner responded with a link to a second geometric proof:

At the end of this proof was a neat little surprise – the approximation that \sqrt{2} \approx 17 / 12 that we’d found earlier in the day shows up in the geometry.  In fact, the entire series of approximations that you get from the continued fractions could show up.  I thought that the boys would really like to see this particular connection between geometry and arithmetic, so we went through it this morning:

I’m always happy when the neat math that people are sharing on social media sites happens to overlap with the stuff I’m doing with the boys    Given this particular topic already has a special place in my math heart, this little coincidence yesterday was especially nice.