Today we will look at the sixth and final problem from the 2018 South African Mathematics Olympiad:

Let be a positive integer, and let be distinct positive integers with . Construct an table where the entries of the -th row are for . Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.

1. Prove that at least three entries remain at the end of the procedure.
2. Prove that there are infinitely many possible choices for and such that only three entries remain,

There are some heuristics that are often helpful when solving a problem, such as

• Looking at small cases:

  This helps us to understand the problem and how the various pieces in the problem relate to each other.

From Princeton University Press.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…