Beth Malmskog

Math etc.

Reposting This Week’s Puzzle: The Trivium’s Checkerboard

This week’s puzzle can be thought of as a missing page from the Phantom Tollbooth, a book very near and dear to my heart.  Here is the hero of our story, Milo, with his faithful companion, Tock the watchdog:


As our puzzle begins, Milo and Tock (and the illustrious Humbug) are caught in the trap of the Terrible Trivium.  The Trivium is a faceless man who aspires to waste the time of those he catches, by smooth talking them into trivial, endless, time-wasting tasks.  Like emptying a well with an eyedropper. Anyway, Milo and his friends have been wasting time for a  while, and are about to make their getaway, when the Trivium gives them one last task–win a simple game involving dominos and a checkerboard.


I’ll skip the whole story–listen to the linked audio to hear the dialogue!  The task that the Trivium sets Milo on is to cover a modified checkerboard with dominos, without any overlap of dominos or any dominos hanging over the side.  You see, the dominos are rectangles that each exactly cover two squares of the checkerboard.  It is easy to cover a standard 8 by 8 checkerboard with dominos by these rules–simply put 4 dominos end to end in each row.  However, the Trivium has changed the checkerboard.  He’s gotten out his saw and cut the upper left and lower right corner squares off of the board.  His challenge to Milo is to cover this new board by the same rules: each domino covers two squares, no dominos should overlap or hang over the edge.  Milo starts in to work…


Milo tries and tries different rearrangements of the dominos.  Each time, he finds that it doesn’t quite work out.  The Trivium is incredibly pleased with this–he knows that there are millions of arrangements of dominos on a checkerboard like this, and that it will take Milo his entire life to try them all.  However, Milo suddenly stops.  He realizes that he’s been had.  Milo has noticed a very simple fact that makes it clear that he will never be able to cover the board in dominos.  Though the Trivium is furious, he can’t argue with Milo’s simple and impeccable logic, and he is forced to let Milo and his friends leave.  Of course, they go on to finish their quest and rescue the Princesses Rhyme and Reason.  All because of Milo’s amazing observation.

The puzzle–What very simple fact did Milo notice, and how did he know, then, that he could never cover the board?

Send your solution to me at!  The author of the best solution that I receive by the middle of next week will win an incredible Math Mostly/Somewhat Science T-shirt, courtesy of the Wesleyan University Department of Mathematics and Computer Science.  Make sure and tune in to WESU 88.1 FM (in Middletown, or stream live at next week, from 2:30-3:00 on Friday afternoon (Eastern time) to hear the solution to the puzzle.  I’m so excited to read people’s solutions!

[Checkerboard picture source–David Hardman via Wikimedia Commons]


Radio delay! And big news.

Ach!  There has been a technical issue with putting together today’s show, so my first puzzle won’t air until next week.  I’ll repost the puzzle next week.  Still, there will be a science show on WESU today at 2:30, and I’ll be listening to it!

The big news–I have accepted a job at Colorado College starting this fall.  It’s not tenure track, but it is a fairly long term visiting appointment.  I am really excited about it!  It was a tough decision to leave Wesleyan, which is really an incredibly great place to work.  My secret dream is to merge the two schools in to one, which would in my dream be conveniently located in a strange hybrid of San Francisco, Wyoming, New England, and Colorado.  I would also ride a horse to work in this dream.  Since that’s not going to happen anytime soon, I am thrilled to be moving closer to my family and to have the chance to work with the amazing faculty and students at CC.  Plus, Pike’s Peak is actually RIGHT THERE.  Seriously.  It’s stunningly beautiful.

Return to radio!

Exciting news for me!  Math, my constant companion, is reuniting me with radio, my old flame.  That’s pretty dorky, I know.  What I’m trying to say is that, starting now, I will be producing a weekly math puzzle radio segment for WESU.

My segment will be part of “Somewhat Science,” a show put together by Wesleyan students with guidance from Professor Suzanne O’Connell.  These students put together great pieces about global, national, and local issues and advances in science.  I’m adding a 7 minute or so math puzzle, which hopefully involves listeners emailing me with their solutions to my puzzles.  The listener who submits the best solution each week will win a prize.  I’m hoping for math t-shirts.  Maybe some other fabulous prize for this first week, since the t-shirts aren’t set up yet.  As I said, I am just thrilled.

In the spirit of alliteration, I think the segment will be called Mostly Math.  Or Much Like Math.  Or Mucho Math.  You get the idea, I’m sure.  In any case, you can hear it on WESU 88.1 FM Middletown, CT (or stream at  Somewhat Science airs 2:30 and 3:00 PM Eastern time, on Fridays starting February 15.  Not sure exactly when Math Mostly will come on.

To answer a puzzle, send an email to me at mathmostly at gmail dot com.  The best answer as of production time for next show will win an incredible prize, as mentioned earlier.  I will be posting the puzzles here so that people can reference them, though hopefully we will get an actual show website put together in the next couple of weeks.


As for the rest of it, I really don’t know where to begin.  A stress and excitement filled chronicle of these winter months of job interviews and decisions?  An account of the incredible (and apparently really nice) students in my classes?  Ah, I don’t have time now anyway.  Suffice to say it’s been weird.  Will post a puzzle soon, but now I have to go to the radio station!   Hooray!

Winter Break

Fall turns to winter in Connecticut

I love you, winter break.  I love to spend my days skiing, picking up my niece and nephew from school, sitting at my mom and dad’s kitchen table, and reading novels.  This is what I always picture when fall semester gets tough.  And wow, it’s really great.

Another thing that I picture, and has actually happened to some degree this break, is getting lots of math work done.  Paper revisions, talk writing, email, check.  Planning for next semester, semi-check.  Brand new math, not so much.  Maybe tomorrow?  Not likely.  Hmmm … kind of worried about when this part will happen.

Now that I voiced that anxious thought, I will get back to enjoying myself.  Time to pick up the kids.  Will write soon about my trip to the Joint Math Meetings in Boston last week.  In the meantime, here is a picture preview:

The donkey! One of the many statues I saw in Boston. The only statue I rode in Boston.

Sorry, shell method, we just weren’t meant to be. Maybe in another time, another calculus class…

How does the end of the semester, so long in the distance, so suddenly rear its head with such an anxiety-inducing roar?  I am not pleased.  There is just not going to be time for everything I want to do.  I’ve been through the stages of grief now:

1) Denial–“I’m totally going to get through everything.  Of course we’ll have time to talk about finding volume by the shell method!  And, even though the days don’t add up at all, I think I’ll plan on doing some fun applications of linear transformations at the end of the semester.”

2) Anger–“I hate you, snow storm!!  You stole my extra time!  Jerk!”

3) Bargaining–“If I prepare LaTex notes for all my classes, then I can talk really fast because my students won’t have to write, then can I get through the material?  If I stay up all night and make precision lesson plans, with minute by minute breakdowns, then will I have time to talk about the cross product?”

4) Depression–“My heart is broken.  I might as well quit teaching right now.  This is it.  Why bother?”

5) Acceptance–“Actually, we’ve gotten through a lot this semester and my students are going to know a ton.  They’re doing really well, and they be able to do the things that are really important for the future.  If we missed something, it’s not the end of the world.  They’ll totally figure it out.”

So, much like the e-collar that I finally accepted as part of my life with Arlo the dog, and the dark and cold that I accepted when the power was out, I have accepted the fact that we sure as hell aren’t going to get to calculating volumes in my Calc I class.  And we are not going to have time to spend on cool visualizations of linear transformations in Vectors and Matrices.  That’s it.  The best I can do is to talk about u-substitution and eigenvectors.  Everything extra is out the window now.  All I can do is give 3 more good lectures.

Other news: Our paper was accepted to the Proceedings of the AMS!  Way to go, team!


And some pictures from a Connecticut fall.



Same dog, same place, more snow.

Yellow and blue.



Red, yellow, and blue.


Orange and blue.

The Final Countdown

Late afternoon, rainy day in November.  My students just finished taking their second midterm exam.  On my desk: the usual Mason jar (my to go mug of choice) clutter, one knitting needle, two bulging file folders of ungraded tests, and a long list of application materials, e-mails, and travel arrangements to sort out.  But it’s Thursday, and tomorrow is my research day, so this is my big chance to write a little bit without it causing me to actually miss another deadline.  So–here’s the update.  We’re in the Final Countdown (cue lick from Europe song, (the ultimate late night math conference singalong)).  There are 5 more class sessions left in the fall semester and I am simultaneously relieved to see the end of the semester coming, sad that soon I won’t get to see these students any more, and really anxious to cover all of the important stuff.

I remember how this time of year felt when I was an undergraduate.  Final projects looming all over the place.  The idea that all of these projects and overdue homework assignments will be easily handled during Thanksgiving break.  The reality that I never started any of them until the Sunday after Thanksgiving.  Then I would stay up late for the next two weeks to work on everything.  Then I would get a bad cold, just like everyone else on campus.

The plague has already taken hold here at Wesleyan.  Students are e-mailing to report that they’ll be missing class, homework will be late, they are too sick to move.  I would suspect these illnesses as being perhaps too convenient but, after seeing some of these students face to face, I just hope they don’t die.  Seriously, there were some miserable people taking the test today.  And what a goldmine for the viruses involved.  42 stress-weakened students in one tiny little room, stuck there for 80 minutes, too distracted by the task at hand to defend themselves.  I feel like some kind of procurer.  You’re welcome, cold viruses.  I’m sorry, students.  Hope you feel better soon.

Lights on!

Me and my beloved headlamp, earlier this week.

Last weekend’s early snowstorm here in Connecticut reminded me of home (seems like there was always snow by Halloween in Laramie, WY), but my reminiscing was cut short when the lights went out at my apartment on Saturday night.  They stayed out, along with the heat and hot water, for several days.  I was a bit concerned because I had a talk scheduled for Tuesday at the University of Connecticut and I needed electricity to work on my slides.  However, hundreds of thousands of other people in the state were without power and had bigger problems, so I don’t want to complain too much.  I stayed warm enough and had great company.  I must say, though, I am really happy to report that the lights are back on!

The power also went out on Wesleyan campus on Saturday night.  It wasn’t until Tuesday evening that the main part of campus had steady power.  Classes were cancelled for two days.  The dorms and other student houses were dark and cold during that time, too, so many of the students also had a really tough week.  My classes were well attended on Thursday, though, which was great.  My students are awesome!

Despite the preparation darkness, the talk at UCONN went well.  I was pleasantly surprised to see some familiar faces in the audience–I hadn’t realized that Milena and Arendt, who I met as friends and visitors of CSU’s Renzo Cavalierri, were at the University of Connecticut.  I feel like Renzo possibly knows more people than anyone else on on earth.  Really nice to see them again.  I spoke about my recent work with Rachel Pries and Bob Guralnick on the automorphism groups off a family of maximal curves.  This was my first math talk since June, so I was worried I would be rusty, and of course imagining all of the incredibly hard questions that I would never have time to prepare to answer…  In fact the audience was engaged but friendly and I had a great time. It was great to talk about that work again and remind myself why I have been so crazy about that problem since 2007.

The week of talks continued with an algebra seminar here at Wesleyan on Friday, this time about Ihara zeta functions of graphs.  Had a good discussion afterwards with Chris Rasmussen, trying to think about how these graph primes decompose.  One more talk to go–on Tuesday the 8th I’ll be at UMASS Amherst for the 5 Colleges Number Theory Seminar, speaking about the automorphism groups again.  Can’t wait!  Neither can my Linear Algebra class, since they’re getting out early that day.  When I told them why, they were shockingly very interested in hearing about my work.  Seriously, my students are really cool.

In other news, we have a new paper on the arxiv:

Six weeks of teaching = much better than six weeks in an e-collar

Teaching report:

After six weeks of teaching, I’m still having a great time.  In fact, it is only getting better!  My students are great.  The material is getting more interesting.  Everything is beautiful.

My linear algebra class is studying determinants.  The first section on determinants in the book that I use is strictly on the method of cofactor expansion, and gives no properties of the determinant outside of how to calculate it.  I assigned the True-False problems from book without giving careful thought to the tools available in each section.  So of course I’m leading up to how I assigned a problem that I couldn’t solve with those tools.  Here it is. True or False: If A is a square matrix in which all minors have the same value, then det(A)=0.

Okay, so the answer is true, which you can see in several ways.  The easiest way is to consider the adjoint formula for the matrix inverse.  The adjoint is the transpose of the  matrix whose entries are the cofactors of the original matrix.  A inverse is 1/det(A) times the adjoint matrix.  If all the minors have the same value, then this supposed A inverse is not invertible, so A is not invertible, so A has determinant 0.  But wait!  The book has not made any connection between invertibility and the value of the determinant.  And the adjoint formula doesn’t come until 2 sections later.  So how could you solve this problem without either of these pieces?

The excellent Billy Chan, my next-door office neighbor here, found a very nice proof that doesn’t require the adjoint formula, but does require invertibility… and oh, hooray!  I just figured out how to skip the invertibility step.  Here’s the idea–create B by replacing the second row of A by another copy of the first row.  The determinant of B will equal the determinant of A because if we do cofactor expansion along the first row of B, the minors we encounter are just the second row minors from A, and all these have the same value, so they are in fact the same as the first row minors from A.  In any case, B has two identical rows, so is not invertible and has determinant 0.  But we don’t need to know that connection if we just write the cofactor expansion of B along the first two rows.  The two expressions for the determinant are identical but have opposite signs, so the determinant must be 0.

Whew!  So that took two PhDs.  Still, I bet that at least one of my students figured it out.

In other good news:

I finished revisions on a paper about the global zeta function of Gauss’ curve (will post soon)!

I get to talk about my maximal curve research at 3 different colleges next month.

Fall in Connecticut is looking like it might be as beautiful as everyone claims.

One of my students spent an hour in my office with me this morning teaching me how to knit socks.

In the making good news from bad news section:

After six weeks with a cone on his head, my dog Arlo has forgotten what it was like before he wore one.  Arlo was hit by a car during our first week in Middletown.  He got away with what seemed a miraculously small injury–basically a big laceration on his leg.  He needed a small surgery to stitch it up, but no bones were broken and there were no internal injuries.  However, he has to wear the aforementioned Elizabethan collar to keep him from licking his leg off, which I think that he would do if given half an hour or so.  This collar was, for about 5 weeks, the most annoying thing ever for both of us.  Arlo kept getting stuck in doorways because he couldn’t comprehend that he needed 3 times the usual clearance to get by something.  He would drag it along the wall when he walked, which is VERY LOUD.  Yes, all caps loud.  Someone should look into recycling e-collars in the form of really annoying instruments!  The semi-rigid plastic makes a great resonator and creates almost a roar as it is dragged along any surface.  Especially in the middle of the night when my poor dog is restless and gets up and walks around the room once an hour.  One day, in the course of dragging the thing around, he got the collar stuck under a chair and managed to get his head out of it.  Of course he pulled all his stitches out and reopened the wound.  Disaster.  The new plan has been to let the wound heal without stitches, with bandage changes at the vet’s office every two days.  I have been pretty frustrated with the whole thing, and getting very impatient for the bandage changes to be done.  At various times, the end has seemed tantalizingly close.  Maybe only another week?  Two weeks?  It seemed possible that it could be almost over!  When?!!  Every tentative deadline has passed, though, and the end has continued to slip back out of sight.

But, remember, this was good news. The good news is that I am just not that worried about it anymore.  As I said earlier, I think that in the last week Arlo has forgotten what life was like before the cone and now seems perfectly content.  And, I mean, really, at least *I* don’t have to wear the cone.  And I get to read in the waiting room.  Things could be worse.