March MATHness – It’s Madness every March!

In 2009, Dr. Chartier along with collaborator Dr. Amy Langville, along with their student collaborators, adapted existing Bowl Championship Series ranking methods to integrate weighting. To test their methods, they applied the ranking methods to March Madness by creating brackets and submitting them to the ESPN Tournament Challenge. All the methods did well with the best bracket beating 97% of the brackets submitted to the online tournament.

Dr. Chartier working with Erich Kreutzer ’10 on ranking methods.

Since that time, Chartier has taught and continued researching ranking methods, advising dozens and dozens of students. Each year a team of students aid Chartier to prepare for the tournament with their work often beginning months in advance. As March approaches, Chartier is contacted by press around the country for Madness insights. He also speaks broadly about the bracketology and sports analytics.

Front page of the New York Post covering Dr. Chartier’s bracketology.

In 2014, with Warren Buffett insuring Quicken Loans’ billion dollar bracket challenge, Chartier’s methods went viral. His work hit the front page of the New York Post and from there he was interviewed about a dozen times a day for several weeks approaching the tournament. He appeared on ESPN, CBS Evening News, Bloomberg News and NPR and was covered by such national press as The New York Times, USA Today, The Atlantic, and TIME, to simply name a few.

Want to give the methods a try? Here is Tim’s quick 12-minute tutorial for the 2022 tournament:

Try the methods yourself at the March MATHness web page! There are a lot of options now so think carefully and have fun with the MATHness of March!

Magically Multiplying with Lines

“Magic is just science that we don’t understand yet.” – Arthur C. Clarke

The same can be said of mathematics. Such enchantments can occur when a professor stands at the front of the room and magically finds the sum of an infinite series, the product of polynomials, or even the product of two numbers. When we stand on the other side of understanding, that which seemed magical can seem quite logical.

Math is a place where we can help each other bridge those gaps and see the magical become understood. To tap into this, let’s see an example of multiplying numbers that simply mystified me when I first saw it back in 2006. While more polished videos exist now, the simplicity of this video has its charm and the silence adds to its mystery.

Before unveiling the math (trick) behind this magic. Let’s diverge to another form of multiplication by using a table to review the process of multiplication. We’ll find the product of 21 and 13. To begin, we form the table as below:


Then each entry of the table is filled with the product of the numbers in its respective row and column as seen below:


Now, to simplify the discussion below, let’s concentrate only on the entries that were products of row and column headers in the table above. This gives us:


Adding the entries of the table gives the desired product to our problem, 273. However, notice for a moment that we could also add along the 3 diagonals that run from the upper right to the lower left. Our first diagonal has only one entry 200. The second diagonal has the numbers 60 and 10; so the sum is 70. Finally, the last diagonal has the entry 3. As you sum along each diagonal, the numbers have the same place value. The first is hundreds, the second tens and the last ones.

Let’s use this knowledge to look at the trick again. Below is my rendering of the trick with the intersection points delineated by color.

Compare this to the table and adding along those diagonals. Notice a similarity? If not, the colors of the dots may help. Each color corresponds to a different place value. Each purple dot is a hundred, each green ten and each orange one. Further, notice how adding the dots, particularly in the middle of this picture, corresponds to adding along the diagonals in the table.

Got it? Try the method of drawing lines to find the product of 32 and 12. When you get that, try 62 times 14. You might want to go back to the table, if you struggle to think through what to do with the sums you get with the points.

Magical? Possibly…but when we bridge that magical gap with understanding, such a trick, at most, is magical for mathematical muggles.

Super Bowl Blink


In our pandemic world, Super Bowl ads were our opportunity to laugh, think and possibly feel some of the Super Bowl party charm.

Here is an impactful half-minute with Drake from State Farm.

And, here is Big Bird with a big ad at a minute in length.

A 30-second 2021 Super Bowl ad was reported to cost $5.5 million. That’s over $180,000 a second. It takes about 0.4 seconds to blink. That blink-ful of content cost just over $120,000. How much money is that? A blink of Super Bowl ad content is enough cash to buy a high-end 2021 Jaguar F-TYPE as seen below!

Be the light in the pandemic war

Light in Tunnel

[Note, this piece was written on January 23, 2021, just four hours before Myron Chartier, father of Tim Chartier died.]

With each vaccination, the light at the end of the pandemic tunnel grows brighter.  We might be brave enough to look to that light.  But, to lean on Amanda Gorman’s penetrating words, we must be brave enough to be the light in this dark journey. 

As my fellow American, be my light as my tunnel is very dark.  My father lies in a bed with labored breathing. He is dying, having tested positive for COVID-19 on his 83rd birthday – the day before he was to receive his first dose of the vaccine. 

I cannot see my father or hold his hand as he dies a mile away and a world apart. My story isn’t unique. We know this.  The news is filled with family after family mourning losses.  As President Biden said, we are in a national emergency.  Our journey through the pandemic is collective.  COVID-19, our national foe, only strengthens when we succumb to our fatigue of this war. 

The day before the inauguration, the death of over 400,000 fathers, mothers, brothers, sisters, husbands, wives, partners, friends, mentors, and leaders were remembered at the reflecting pool at our nation’s capital.  We must remember these deaths to heal.  We must heal to be strong.  We must be strong to win the war. 

My father will take his last breath in coming days, possibly even today.  My father will not survive this pandemic.  So please, be the light in this tunnel.  Social distance.  Wash your hands.  Wear a mask.  It takes we the people of the United States.  Our actions in this time will help form a more perfect Union, establish justice, ensure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity.  And, as Amanda Gorman recited, “We move to what shall be, a country that is bruised, but whole. Benevolent, but bold. Fierce and free.” Be the light. This hill we climb is steep.  War will leave us battered and bruised.  Yet, we can emerge beautiful.

Computing magic

Suits of cards on a laptop screen

A clever use of mathematics can mystify audiences. A variety of resources divulge such secrets for a budding mathematically-inclined magician. Colm Mulcahy’s (Spelman College) column Card Colm on MAA Online reveals the mathematics behind a variety of tricks that involve such topics as Fibonacci numbers and results of Paul Erdös. Mathemagician Art Benjamin (Harvey Mudd College) unveils secrets of rapid mental calculation in his book, Secrets of Mental Math. How magical such tricks appear is correlated to one’s insight into the underlying mathematics.

Many math-based card tricks exist on YouTube. I’m fond of math-based magic. My fondness is proportional to my inability to do any magic that isn’t self-working. Simply put, I like the math but I like that the math makes me look magical. Further, magic can engage audiences of many ages. Here you see a picture of Tanya Chartier engaging a youth with card magic trick at the National Museum of Mathematics’ New York City Math Festival in 2019.

Want to be a mathematical card magician? YouTube is a trove of tricks. Here is a fun (self-working) one (with a short video):

As with any trick I see, my initial reaction is, “Wow! That’s amazing!” I can sense the math but am overtaken with the mystery. An attribute of a self-working trick is only the steps need to be remembered – the magic takes care of itself.

Try the trick in the magic in the video. It works — every time regardless of the order of the deck. And then comes the real trick — after the initial “Wow” comes the desire to understand. Why does the trick work and this is when the magic turns to math. Not surprisingly, trick relies on placing the chosen card ninth from the bottom. There are 52 cards so the chosen card is 43 from the top.

First, imagine capping all four piles of 10 cards. Then, each pile contains 11 cards and the last pile IS capped with the chosen card. However, in every other case, the trick works as outlined in the video. Sum the cards that are face up and count that many cards down. If the sum is 10, for example, the tenth card dealt will be the chosen card. Why does the trick work? I’m going to leave that to your mathematical exploration, mainly due to my repeated inability at writing a clear, succinct answer.

Unless you cap all four piles, you are able to add up the sum of the cards that are face up and count that many cards down to unveil the chosen card. If you cap all four piles, there are no cards face up. In such a case, the “zero” card (rather than the first or thirteenth card, for instance) is the chosen card. Granted, a magician, such as yourself, could finesse unveiling such a “zero card,” which would be the card to cap the fourth pile. In my mind, this is less magical than counting down and finding the chosen card. I found myself wondering, “What is the probability of this even happening?” Feeling quite nerdy but not minding (much), I grabbed a pencil and paper and begin trying to compute such a probability. Soon, it appeared I would need to cover several cases or sit until I perceived another approach. Then, I laughed in the realization, “I’m a mathematician and computer scientist! Let the computer do the computing!”

I quickly coded a program to perform the trick. I created a deck and randomly permuted the cards to serve as my shuffle. Then, I simply performed the trick 100,000 times keeping track of how often I capped all four piles in the trick. The first time I ran the computation, this occurred 1352 times. The second time I (via my program) performed the trick 100,000 times capping all four piles 1336 times.

So, I would have about a 1.3% chance of capping all four decks. Given I rarely perform magic, this is a pretty small chance. If I decided to perform the trick more often, could I live with that probability? Would you? It may partially depend on what you do when the case arises. Frankly, I might just put the chosen card as the 8th card from the bottom and then count down the desired number of cards and the next card is your card. In such a case, if every deck is capped, the next card is your card. The math is easier to perceive though…But, I often perform math-based magic tricks so people can learn and struggle with the math. Yet, maybe that’s precisely why the original trick is compelling! I, for instance, sat and coded a computer program to perform the trick, and am now blogging about it! Were the card the 8th from the bottom, maybe once I saw the trick, I might have even forgotten about it several hours later. I guess, now that I think about it, I can live with that 1.3% chance in the hopes that someone else might enjoy a similar mathematical journey. And then again, maybe I’m the only one who might not be that great at finessing a zeroth card!

What choice would you make?

Chicken or Egg?

Reducing patches of a wireframe

A trove of 3D printing models are available at Thingiverse. Warning: be sure you have available time if it’s your first time to look! If you have access to a 3D printer, you can print your very own models from such Makers as Henry Segerman and Laura Taalman.

A year ago, I taught a seminar on linear algebra in computer graphics and data science based on my MAA book, When Life is Linear: from computer graphics to bracketology.

We used 3D models to motivate rotation matrices. Students downloaded their model of choice and then used 3D rotation matrices to place the wireframe in a desired position on the screen. We used Matlab in the class making the use of an STL File Reader quite helpful.

Some wireframes are quite refined and cumbersome to visualize and rotate. So, we also made use of the reducepatch command in Matlab. This command reduces the number of polygons in a wireframe as seen at the top of this post.

Soon, the students were downloading wireframes, using rotation matrices to place them and then repeatedly executing the reducepatch command. Soon, the process became a game. Who could most closely predict the lowest level wireframe? For instance, what would we get after repeatedly reducing the wireframe of the model of the Peanuts character Woodstock available on Thingiverse posted by Masterdid? I’m not sure if the chicken or the egg came first…but if Woodstock comes first, would the end result be an egg? Make your guess and then click the image below to see what results!

Animated GIF of Peanuts Woodstock character

Zoom Bloopers

Zoom fail image

As we move into our new semesters, remote learning keeps us zooming through our content. Even with so many hours on Zoom, bloopers arise and are ever-present! It can help to acknowledge them. It’s also helpful for us all to share the journey together and even laugh as we ascend that learning curve.

I worked with artist Olivia Weaver (@_ollie_oop on Instagram) to create a comic on Zoom bloopers, which I used in my class last term. The students offers fun responses, many of which evolved into tips for each other on learning in our remote world.

A comic of Zoom bloopers

Zoomable Success

Online Learning tips graphic

While 2020 has passed, aspects of 2021, at least for now, look very similar to the year gone by. For one, many of our classes will be taught remotely at least in the first part of the year. We will share with our faces and writing framed by a monitor screen. While we all continue to adapt and increase in our proficiency to online learning, there is still significant challenges.

As we begin teaching in 2021, we should talk about what can lead to success in online learning. To this end, I worked with artist Olivia Weaver (@_ollie_oop on Instagram) to create a comic on dangers of remote learning. Below you see the graphic, suitable for use in your classes, just be sure to credit Olivia.

What other tips would your students offer? Which of the tips ring true for them? Might some of us even want to make post-New-Years online learning resolutions?

tips on online learning, created by Olivia Weaver

Ant-Man’s Superhero Bone Structure

A large Ant-Man peering down on deck of a ship.

Ant-Man and the Wasp has been released into theaters.  Of their various super powers, mathematics underscores a huge one for Ant-Man, his bone structure!  When you see the superhero towering over the boat in the scene below, taken from <a href=”” target=”new”></a>.

A large Ant-Man peering down on deck of a ship.

We should cheer as if he’s the man of steel, really more than steel!  To see why, we become cubists, of a kind.  In particular, consider the following cube.

A cube

Suppose you double its size, meaning its length, width and height all double.  Then the cube becomes:

A cube

Suppose you decided to paint the cube and it took one bottle.  Then, it would take four to paint the enlarged cube.  However, what if we fill the original cube with one cup of water. The larger cube would take eight cups.

What does this mean for Ant-Man?  From my quick estimate, Ant-Man is approximately 10 times bigger as he peers down at the deck of the ship.  Therefore, he weights 1,000 times as much.

The problem? Ant-Man’s bones scale relative to the rate at which the cross-section of his bones scales, which is only 100.  Ant-Man should be crushed under his own weight.  He isn’t and, as such, mathematics gives us another superhero power to cheer as we watch the movie!

Adapted from course notes entitled “A Tour of Mathematics – Exploring the Borders” by Dr. Tim Pennings

Welcome to my blog!

While my title indicates I’m a professor of Mathematics and Computer Science, it’s important (and will become clear as you read through these pages and posts) that I’m also an artist.  I’ve trained in mime and puppetry and performed internationally in both.  As such, you’ll see posts involving my art and math – together or separately.

Mathematically, I specialize in sports analytics having worked with teams in the NBA, NFL, and NASCAR. I’ve also fielded analytics questions from national media such as the New York Times, ESPN, and the Wall Street Journal.  I also direct and work with a team of student researchers (70 in total in the 2017-18 academic year) to provide analytics to Davidson College sports teams.

So, search around and feel free to connect as I’m easy to find on the Internet.