Mr Honner

Resources Teaching

The Surprising Simple Math Behind Puzzling Matchups — Quanta Magazine

My latest column for Quanta Magazine is about one of my all-time favorite mathematical ideas: transitivity. Well, technically it’s about intransitivity, a subtly complex mathematical situation which any sports fan knows all about.

It’s the championship game of the Imaginary Math League, where the Atlanta Algebras will face the Carolina Cross Products. The two teams haven’t played each other this season, but earlier in the year Atlanta defeated the Brooklyn Bisectors by a score of 10 to 5, and Brooklyn defeated Carolina by a score of 7 to 3. Does that give us any insight into who will take the title?

Well, here’s one line of thought. If Atlanta beat Brooklyn, then Atlanta is better than Brooklyn, and if Brooklyn beat Carolina, then Brooklyn is better than Carolina. So, if Atlanta is better than Brooklyn and Brooklyn is better than Carolina, then Atlanta should be better than Carolina and win the championship.

Sports fan knows things are never this simple, and in my column I explore some of the surprising mathematical reasons why it may be the case that A is better than B and B is better than C, but C is better than A. You can read the full column for free here.

By MrHonner, 11 months11 months ago

Geometry Photography

Math Photo: A Most Mathematical Building

Here are some images from Harpa, in Reykjavík, Iceland. Harpa is home to the Iceland Symphony Orchestra and the Icelandic Opera, and is one of the most mathematical buildings I have ever seen.

The face of the building is a solid wall of glass prisms whose faces are hexagons and pentagons.

Here’s a look up through the wall from below.

Different perspectives highlight the different polygons.

Whoever designed this beautiful building certainly knew the theory of pentagonal tilings!

By MrHonner, 11 months11 months ago

Teaching

My Tests are So Hard

Everywhere I’ve taught there have been teachers who brag about how hard their tests are. It’s always a central part of their identity as a teacher, of how they see themselves, and how they want to be seen. They proudly consider themselves more rigorous than their colleagues.

But nothing could be easier than making a test hard. You can just put more questions on it than can be reasonably handled in the allotted time. Or put problems on that haven’t been emphasized in class or practiced enough. Or problems that test edge cases and not core ideas. Or problems from the next unit. Or problems you simply haven’t prepared all students to handle.

I’ve seen teachers do all these things. It’s not rigorous. It’s lazy. You know what’s truly difficult? Writing a test that is fair, representative of core ideas, and appropriately challenging.

By MrHonner, 12 months12 months ago

Teaching

Taught Helplessness

I’m currently reading “The Design of Everyday Things” by Don Norman and it’s interesting to think about what the theory of product design has to say about instructional design.

For example, the author discusses how “learned helplessness” can result from poor design. A product whose functionality isn’t discoverable, and that doesn’t provide good feedback, will be frustrating to use, so users will likely give up after trying and failing a few times.

Just as I was making the connection to teaching math in my mind, the author himself brought up math instruction as a common example of “taught helplessness”: When math is presented as unintuitive, and poor or misguided feedback is given, students are likely to just give up. The problem is amplified by the linear way in math is usually taught. In many classrooms, if you don’t understand what happened yesterday, you will probably struggle to understand what is happening today.

Originally posted on Mastodon.

By MrHonner, 12 months12 months ago

Challenge Numbers

2024 and Differences of Squares — Solution

The new year 2024 is a difference of squares, $2024 = 45^2 - 1^2$ , which got me thinking about a fun little number theory problem:

Is there a largest number that can not be expressed as the difference of squares? If so, find it. If not, prove no such number exists. Good luck, and happy new year!

As promised, here’s my solution.

There are infinitely many numbers that can not be expressed as a difference of squares. In fact, we can completely characterize numbers that can be expressed as a difference of squares and those that can’t. It all starts with factoring.

Differences of squares have a useful structure that can be exposed by factoring:

$a^2 - b^2 = (a + b) \times (a - b)$

We can leverage this structure to answer our question.

Suppose n can be factored as $n = s \times t$ . If n can be expressed as a difference of squares, then we can also write

$n = a^2 - b^2= (a + b) \times (a - b)$

Now set $(a + b) = s$ and $(a - b) = t$ . This gives us the system of equations

$a + b = s$

$a - b = t$

We can solve this system by adding and subtracting the equations. Adding gives us $a = \frac{s+t}{2}$ , and subtracting gives us $b = \frac{s-t}{2}$ . This shows us how to express n as a difference of squares: Just factor n into $s \times t$ , compute $a = \frac{s+t}{2}$ and $b = \frac{s-t}{2}$ , and then $n = (a + b) \times (a - b) = a^2 - b^2$ .

There’s only one thing we have to worry about: a and b must be integers. But as long as s and t have the same parity — that is, s and t are both even or both odd — then the sum and difference of s and t will both be even, and so $\frac{s \pm t}{2}$ will be an integer.

This means that n can be expressed as a difference of squares if and only if we can write $n = s \times t$ where s and t are both odd or both even. This is usually possible, often in multiple ways. But there’s one situation when it isn’t: When n is divisible by 2 exactly once. When this is true, then however you factor n into $s \times t$ , the lone factor of 2 will end up as a part of either s or t, making one of them even and the other odd. Thus, in this case, it’s impossible to factor n so that the two factors have the same parity, and so it’s impossible to express n as a difference of squares.

This gives us a complete answer to our question: A number n is not expressible as a difference of squares if and only if it is divisible by 2 exactly once! In other words, every odd number times 2 is not expressible as a difference of squares, and every other integer is.

As an example, given $n = 105 = 3 \times 5 \times 7$ , we can factor $105 = 7 \times 15$ which gives $a=\frac{15+7}{2}= 11$ and $a=\frac{15-7}{2}=4$ , and sure enough, $105 = 11^2 - 4^2$ . Notice, we could also write $105 = 5 \times 21$ , which gives $a = \frac{21+5}{2}=13$ and $b = \frac{21-5}{2}=8$ , so $105 = 13^2 - 8^2$ .

On the other hand, it isn’t possible to do this at all for 6. There are only two factorizations, $6 = 6 \times 1$ and $6 = 2 \times 3$ , and in both cases the factors have different parity, so the a and b we need won’t be integers.

There’s an interesting resemblance here to Euclid’s Formula for generating Pythagorean triples. There’s also an interesting follow-up question about how many different ways a number can be expressed as a difference of squares. And since the numbers that answer our question are those that are divisible by 2 exactly once, I wonder what properties numbers that are divisible by 3 exactly once have.

Thanks to everyone who contributed on the Mastodon thread! There are some cool ideas there as well, so be sure to check it out.

Related Posts

By MrHonner, 12 months12 months ago

Follow Mr Honner