On Monday, July 17, 2017, TV viewers saw Ethan Brown, a young man from Bethel, CT, compete with four other contestants for the $50,000 top prize on Fox’s Superhuman show. He did this with an all-original and never-performed-live-before construction of an 8×8 magic square to meet specifications requested by a panel of judges. Click here to see an edited highlight of Ethan’s performance. The season finale of Superhuman airs Monday, July 31 at 9 PM ET/PT.
Ethan (now 18; he was 17 when the show was taped last year) is no stranger to Gathering 4 Gardner, and has written eloquently of the influence Martin Gardner has had on him. In 2012, at G4G10 in Atlanta, when he was age 13, he amazed us all with his magic squares prowess. He performed on stage as part of an evening bill that included another natural charmer, 93-year-old Ray Smullyan.
That same year, Ethan set a world record by reciting 2,012 digits of Tau from memory. As the video of that performance shows, it took him over an hour; he didn’t rush and risk tripping up, instead focusing on getting the job done correctly. This is the same calm strategy he adopted successfully on the TV show, and on that more recent occasion it must surely have taken nerves of steel under all those bright lights.
We’ve seen him do on-demand 4×4 magic squares for several years now, at Gathering 4 Gardner, MAA MathFest, the MOVES conference at the National Museum of Mathematics, and at other math-related forums. His enthusiasm, skills and very personal mathematical approach to 4×4 magic squares were highlighted to launch Mathematical Awareness Month 2014, marking Martin Gardner’s Centennial.
Typically the common row, sum, and diagonal total is a number selected in advance and out of his control. Furthermore, some numbers in the grids are filled in ahead of time. He then fills in the missing numbers to achieve the desired goal. He’s come a long way since he first did this, getting faster, and more confident, in part thanks to the mentoring of his idol Art Benjamin.
What he pulled off on television recently, a new challenge entirely conceived by him and never before done in public, did indeed makes him appear to a general audience to be superhuman. There is, of course, a rational explanation: mathematics, coupled with single-minded dedication and lots and lots of finely-honed practice. As we learn below, he has a divide-and-conquer strategy that really pays off.
Superhuman is hosted by Kal Penn (of “Designated Survivor” fame) and judged by a panel consisting of Mike Tyson, Christina Milian and Rahul Jandial. The panel picks three of the five contestants to be finalists, and the audience then votes for one of those to be the winner. On the show that aired on Monday, July 17, 2017, Ethan’s assured, flawless performance saw him walk away with the $50,000 top prize. This audience found Ethan’s mathematical tour de force more impressive than the stiff competition provided by four older, much more seasoned contestants with their own diverse and mindboggling skills.
As he prepares to head off to Boston University to start college, Ethan was kind enough to take the time to entertain a dozen questions from us below. His maturity, levelheadedness and humility speak for themselves. We hope that he continues to participate in Gatherings 4 Gardner; who knows, perhaps the skills he picks up in college can be brought to bear in stimulating curiosity and the playful exchange of ideas and critical thinking in recreational mathematics and magic, as Martin Gardner’s 25 years of “Mathematical Games” column in Scientific American did for so many generations of people.
After the interview we add some mathematical comments and share reflections from his father.
1. We’ve been watching you do on-demand 4×4 magic squares for several years now. Was this the first time you’d done it on national TV? What was appearing on Superhuman like? How do you prepare for that kind of “pressure in the spotlight”?
Superhuman was the first time my magic square was highlighted on national television, which was a great honor. I had a lot of fun being a contestant on Superhuman. A big highlight of the experience for me was the experience of seeing and learning more about the operations of a television set. I will be studying film and television in college, and I would like to work in the industry someday, so this opportunity gave me many insights about the process of putting together a production as professional and complicated as Superhuman.
While I hadn’t performed math on stage for a while leading up to the taping of Superhuman, I did have several years of onstage experience under my belt, so I was able to calm my nerves and perform my challenge well. Due to the format of the show, I did not have to worry about saying lines or entertaining an audience (Kal did a fantastic job of that), so all of my practice was just solving magic squares.
2. Please tell us how you moved up from 4×4 to 8×8 magic squares, and what challenges it presents mathematically as well as the sheer scale involved. The one you did on TV asked for a common total of 182.
When I used to perform mental math stage shows, my original 4×4 magic square was my signature bit and often my grand finale. I would have audience members supply a grand total as well as any three numbers to put into any three squares in the grid. Giving the audience the freedom to place numbers anywhere they like is what makes that magic square difficult. A performer cannot memorize a single set of numbers or even a single algorithm that takes into account every one of the 4,480,000 ways an audience can arrange three numbers from 1-20 in a 4×4 grid.
To complete these squares, I had to memorize a set of three algorithms and rotations of those algorithms, which I was able to prove would cover every possible combination. If you fill each quadrant of an 8×8 magic square with a 4×4 magic square adding to 91, you will have an 8×8 magic square adding to 182. That was essentially my challenge: create four 4×4 magic squares, with each one containing three numbers put into any three squares.
3. On TV you had to fill in 52 of 64 numbers. That takes time even if it’s easier for you than for most other people on the planet… Was the show as broadcast edited a bit to make this look faster?
For the show, my priority was to solve the magic square correctly, not quickly. Since the challenge had no time limit, I did spend probably close to ten minutes solving the 8×8 magic square and triple checking my work to make sure I was correct.
4. What are the “flies in the ointment” that can be caused by, (A) the desired total, and (B) the positions or values of the pre-filled numbers the judges selected before you started? For instance, on TV the total was 182, and I noticed that you used some small negative whole numbers in the grid to make it work out. Could you have handled a lower total like 135? Would a large total such as 875 have been harder?
The negative numbers are actually a result of the numbers inputted in the squares by the judges, not the grand total. Since Mike wrote in 1, 2, and 3, I was forced to use some negative numbers in that quadrant to keep each row and column balanced properly. A total between 100 and 150 would actually have made the square look a bit nicer because the negatives would not change and the very large numbers (there were even numbers in the eighties in the square) would become smaller. The positions of the pre-filled numbers can make my job easier or harder, but I do have algorithms for every combination they could have chosen. If you look closely at my grid, you may notice that each of the four quadrants was solved with a different algorithm or algorithm rotation.
5. Do you know of others who can do this, for 8×8 squares? Did somebody help to train you for this particular skill? I noticed that Art Benjamin was in the audience, I know he has mentored you in the past. Was he involved this time?
While I don’t personally know of others who can perform my 8×8 magic square, I do know that it is very easy to learn. With the algorithms memorized, solving magic squares just takes practice, like playing a sport or instrument. I was not born with an ability to do magic squares; it is a skill that I acquired through lots of practice and anyone else can too.
Arthur Benjamin was my inspiration to pursue mental math, and eventually, he became a mentor and family friend. I did consult with him frequently when I created the method for my 4×4 magic square, especially in trying to develop a proof that my algorithms would work for every combination of pre-filled number positions. Expanding this method to an 8×8 grid was pretty simple from there. As he has been with all my endeavors, Dr. Benjamin was very supportive of my appearance on Superhuman, and it was very special to have him in the studio audience with me.
6. Have you used computers in any way to prepare yourself?
I did not use computers to practice my magic square for Superhuman, just whiteboards and notebooks. Since I no longer perform mental math (with the exception of Superhuman), my high school roommate in boarding school would always sneak over to my whiteboard after I had been practicing 8×8 magic squares so he could take pictures of them and send them to our friends, usually captioned “he’s back.” One of my friends referred to my Superhuman appearance as “The Dark Knight Rises.”
7. Given enough time, perhaps more than TV allows, can this be done for even larger squares? Have you explored this yourself? If not, might you do so in the future?
There is a very cool Vedic approach to completing magic squares with an odd number of rows and columns. This method does not allow for the input of numbers into any position on the grid. I elected to use an 8×8 square so I could make the challenge my own, using inputted numbers by the panel and Kal. My 8×8 method theoretically could be expanded to 12×12, 16×16, and so on, though that might get hard to keep track of without a computer. I have not investigated the possibility of using a similar method for 5×5, 6×6, 7×7, 9×9, etc, though I would be very curious to see if someone does take on that challenge.
8. How did you get started on this whole style of challenge? What inspired you to keep at it and raise your performance skills to this level?
My inspiration to begin pursuing mental math was Arthur Benjamin. In fifth grade, I saw his TED performance, and since I thought it was so cool, my dad decided to buy me his book, The Secrets of Mental Math, which teaches many of his mental math techniques. From there, I practiced for several months, and with Dr. Benjamin’s mentorship and support, performed at my fifth grade talent show. After that, things began to snowball, which led to many performances and talks around the world, culminating in my performance on Superhuman.
While I am no longer pursuing mathematics or performing, I am extremely grateful for all the experiences I had as a mental math performer and all the amazing people I got to meet. The most special and memorable moments for me were when audience members would light up not because they were impressed, but because I had taught them some mathematical technique which inspired them to perhaps like math just a little bit more. In my future endeavors in filmmaking, I hope to continue to find ways to inspire people like I used to do on stage.
I would like to be sure to give credit where credit is due. I did do some competitive math in my earlier years of high school, the most prominent competition being ARML (American Regions Mathematics League), a nationwide high school math competition. While I wasn’t skilled enough to make the actual team, I was lucky enough to join my high school’s team as an alternate and support my friends competing. After hours of team rounds and individual rounds, I sat up in what felt like the hundredth row of Penn State’s hockey arena and watched the highest scoring competitors face off in a final competition on the rink, where they were receiving problems I couldn’t even begin to approach and solving them in mere minutes, sometimes even quicker. I will always remember how amazed and humbled I was by those students. When I hear phrases like “one of the smartest young math minds in the country,” I always think of them as opposed to myself.
9. At middle and high school, how has your focus on magic squares impacted the other math you have learned? How did your teachers react to your special interests and skills? Are you interested in math in general? You have published with Art Benjamin, that puts you in a very select group of people. Do you intend to study more math as you head off to college soon?
My middle school was very supportive of my mathematical pursuits, and allowed me to advance two grades in math so I could continue to challenge myself. For high school, I attended Phillips Academy, a boarding school in Andover, Massachusetts. There, I placed into an accelerated precalculus course as a ninth grader, where I had to really struggle to keep up. My high school math classes were some of the most challenging classes I have taken, and some of the lowest grades on my transcript. Despite my high school struggles, my teachers all through my life were always very supportive and enthusiastic about my mathematical endeavors.
I was extremely honored to have coauthored a paper with Dr. Benjamin on my magic square algorithm, which was published in the College Mathematics Journal in 2014 and republished in the Princeton University Press’s Best Writing of Mathematics 2014 in 2015, which gave me an Erdös number of 3. With my astronomy research class in high school, I coauthored a paper (“Finding the Lightcurves and Rotation Periods of 2925 Beatty, 3012 Minsk, and 9060 Toyokawa”) recapping our discoveries of the lightcurves and rotation periods of three main-belt asteroids in the Minor Planet Bulletin in 2016, which, if you are willing to accept the looser Erdös number definition (any paper published in a peer-reviewed scientific journal as opposed to strictly mathematics), gave my teacher and classmates their first Erdös numbers, which they were very excited about.
I will be starting as a freshman at Boston University this fall, where I will be majoring in Film & Television and will potentially do a minor or dual degree in Environmental Analysis & Policy. While I don’t know exactly what I want to do after college, I would love to pursue screenwriting, or potentially another aspect of the film and television industry. I love telling stories and I love making a difference in the world, which I believe I have channelled both through my past interests in math and my current interests in screenwriting and filmmaking.
I am currently writing a blog called The Sweaty Penguin, which explores environmental issues through satirical articles (another way I have been trying to combine these passions).
10. What was it like competing against such a wide range of other talents for Superhuman?
I was blown away by all the other contestants’ amazing skills and fascinating life stories. Some people had trained to perform some obscure skill (like myself), some people were born with their amazing abilities, and some people had just developed such a passion about something that their knowledge base was unparalleled. The diversity in journeys to achieve the skills fascinated me just as much as the diversity in skills themselves.
11. Who did you think might win?
The other contestants on my episode were all incredible, and I don’t think any of us could have predicted the final outcome of the show. I was so humbled when the panelists selected me to the top three and when the audience voted for me as the winner. I have watched every episode of Superhuman and every contestant is deserving of that title. With hard work and dedication, anyone in the world can be a superhuman. Magic squares, and many other challenges on the show, are the result of determination and lots of practice.
12. Do you have plans for your winnings?
I will be using the money to help pay for my college tuition.
The Mathematics of Ethan Brown’s 8×8 Magic Squares
Ethan’s strategy for finding an 8×8 magic square with desired common row and column sums is disarmingly simple, as he mentions above. He just thinks of an 8×8 square as four quadrants, namely, corner 4×4 squares, one each in the top left, top right, bottom left, and bottom right positions.
If the overall target sum is an even number like 182, he simply uses his long-established 4×4 method on each corner square. (His 4×4 magic square methods are discussed at his math website.) As the TV footage shows, Ethan concealed his secret strategy well, by resisting the temptation to fill in the quadrants one by one. This tightrope performance must have taken great presence of mind on his part, switching back and forward between different quadrants and different algorithms, yet never getting flustered or losing his cool.
If the target sum is odd, such as 181, he can find two 4×4 magic squares (for the top left and bottom right positions) with row and column sums 91, and two others (for the top right and bottom left positions) with sums 90. That would indeed yield an 8×8 magic square in which each row and column sum would be 91 + 90 = 181 = 90 + 91 as desired.
On Ethan’s TV appearance, the third judge on the panel (Rahul Jandial) had been asked to randomly pick an even number. This resulted in an even 3-digit number for the target total. A great bonus in such cases is that, since half of 182 is 91, the four 4×4 squares Ethan constructed also had 91 as the sums of their principle diagonals, as well as for other key clusters of 4 entries. As a result, the final 8×8 magic square that Ethan came up with on the TV show had both principle diagonals of eight numbers also sum to 91 + 91 = 182, as highlighted in the broadcast version.
Furthermore, as reported by people who attended the show’s taping, Ethan was able to wow the panelists and audience by showcasing many other diagonals and clusters of eight numbers which also summed to 182. In the interests of broadcasting brevity, TV viewers didn’t see those bonus aspects of Ethan’s 8×8 square; but interested readers can explore this for themselves with the above image of the completed square.
A Proud Dad
Ethan’s father Chris recall how his son got his start: “He was grasping basic mathematical concepts probably from about the time he was starting to talk. When I presented him more and more difficult concepts, he just understood them without much effort, so the knack for math was always there. But when he was in fifth grade, I showed him a video of Arthur Benjamin performing ‘Mathemagics’ and he was immediately hooked. We bought a copy of Art’s book and he just devoured it, practicing the mental math strategies for several hours a day. That led to school talent shows, and more opportunities.”
The Fox TV show was a whole new league of exposure, and Chris describes what it was like for the family to watch Ethan perform for such a large national audience and with such strong competition: “It was really incredible. I had seen Ethan perform in front of some pretty good-sized crowds so I knew he wouldn’t be as phased by this as other 17-year-olds might have been. But even so, this was by far the biggest and highest pressure situation he had been in. We talked about it ahead of time and agreed that whether he won, merely did well, or completely bombed, he would walk away with a great story to tell for the rest of his life. I’m not sure if that helped him keep it in perspective or not, but it was really exciting to be there and filled us with pride.”
The show was taped over a year ago, and Ethan and family were sworn to secrecy about the outcome until the broadcast went out. This led to an interesting situation on Monday, July 17: “On the night of the show, some of Ethan’s grandparents were able to join us at our house. For the past year, we had talked about how fun it was going to be to see him on TV, but sort of downplayed how well he did. They were truly all fooled and none of them thought he had won. So as the show went on and he did really well, you could just feel the excitement building in the room. When Kal announced his name as the winner, the emotions and excitement just exploded. It was really a special moment.”