## Seatings and Sums of Squares

At Math for America events, guests are traditionally invited to find their seats in some math-y way. (This is just one of the many ways MfA uses math creatively at their functions). Here is my seating card from a recent dinner at the MoMA.

The keynote speaker, Dr. Eric Lander, is, among many other things, a mathematician-turned biologist who has been working on the human genome project. Dr. Lander gave a remarkably clear explanation of the inherent mathematics of genetics. And, as a lover of number theory, Dr Lander expressed some appreciation for the seating assignments.

Dr. Lander pointed out that the mathematical fact on display here is that **every integer can be expressed as a sum of four squares**. This is commonly known as the Lagrange Four Square Theorem.

Here are a few examples of the phenomenon:

Dr. Lander, a former **International Mathematics Olympiad** competitor, said that it is fairly easy to show that you need **at least** four squares to express every number as such a sum, but it’s much harder to show that you need **at most** four squares to get the job done.

Of course, the words “easy” and “hard” probably have unique meaning to someone cracking the human genome for a living!

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Reminds me of proving any number in base 10 can be written in any base [insert positive integer] form. I question whether if base [insert positive fraction] could pull this off. Probably would lead to just writing up summations instead, then getting distracted by an interesting summation.