Arthur Benjamin, the man, the maths, the magician. Part 1.

Recently I got the chance to meet up with Arthur Benjamin, a mental maths extraordinaire. If you’ve never heard of him I urge you to at least go watch his TED talk. His ability to manipulate numbers with speed and skill is astounding. Whilst interviewing Art I found that his enthusiasm for expositing mathematics could not be contained and is extremely infectious. Thus, what follows is a greatly reduced interview. I’ve cut out two large chunks on patterns in the Fibonacci numbers and continued fractions that deserve a whole post to themselves. Moreover, I've split the interview into two halves. This week I try to get to the mathematician behind the mathemagician.

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Arthur Benjamin enjoying a sunny Oxford day.

Thank you for allowing to be interviewed for my website. The first question, of course, has to be: how are you doing?

I’m doing well thank you. I’m enjoying my time in Oxford.

What have you been up to on your sabbatical year in Oxford?

I was kept pretty busy here. I gave a TEDX talk in November and a public lecture organised by the University of Oxford’s physics department, which led to a number of schools contacting me.

What are you currently working on research wise?

The main research that I’ve done has been in the area of combinatorics, specifically, finding combinatorial proofs of interesting identities. To take a simple example we know [1] that

\begin{equation}
\sum^n_{k=0}\left(\begin {array}{c} n\\k\end {array}\right)=2^n.
\end{equation}

Now, one can prove that algebraically very easily, but combinatorially you can look at this as an identity where you have a question that gets answered in two ways: how many subsets, 1 through $n$ are there? Well we know there are
\begin{equation}
\left(\begin {array}{c} n\\k\end {array}\right)\nonumber
\end{equation}
subsets of size $k$ and $k$ can run from 1 to $n$, which we can add up, which is the left-hand side of the identity. On the other hand the subsets of can be created in $2^n$ ways because each element is either in or out of the subset.

Each side of the equality tells you something different about the numbers. You can think of it as the same story being told from two different perspectives, which leads to two expressions meaning the same thing.

I assume that, unlike your given example, you do not always know that the identity is true to begin with.

Often you do. It isn’t often that I find brand new identities; rather, I find simpler explanations for known ones. As nothing in mathematics is simpler than counting if you can prove something without heavy algebra or analysis then it is much more satisfying. Sometimes these insights lead to new patterns and identities.

Are your sequences mathematically motivated, or do they stem from a specific application?

To be honest I don’t mind. I just prefer sequences that produce pretty patterns and results.

Which came first maths or magic? And did your love of both lead you on to choosing your route into combinatorics?

If you mean real maths like we’re discussing here then magic certainly came first. If you mean maths as a love of arithmetic then that’s a much harder question as they occurred at a much closer time. I’ve always loved playing with numbers, since I was a kid. I always saw it as a game. I particularly like pulling problems apart and solving them in different ways. 

I’ve always been something of a show off; a ham, as we would say in the US. I grew up in a rather theatrical family, so I’ve always been encouraged to find things that I was passionate about. While I was in High School I would entertain children’s parties with the traditional rabbit out of a hat type of tricks, anything to make them laugh.

As I got older I took an interest in magic of the mind and my dad said,

“Hey, why don’t you do some of that maths stuff in there? What you can do with numbers is much more impressive than the fake mind reading stuff.”

Well, I tried it and, because it is so real, it got the best reaction. That made me think wow! If they like that imagine what would happen if I did bigger problems.

Of course you’re well known for your mental arithmetic abilities, but do you ever entertain people with your research?

If I am talking to a group of mathematical students, then I could talk to them for hours without multiplying any numbers. The general public though are much more interested in the speed mental skills. I’ve always viewed myself as an entertainer; when I’m in the classroom, or when I’m writing my papers, I want them to be enjoyed. That’s not to say that they need to contain corny jokes, just an interesting story.

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Part 2 will be up in two weeks.

[1] Now, of course you may not known this identity. An analytic proof can be found here. However, the mathematics is not the point of interest here. It is simply that each side of the equality means something different.