Prelude#
In A mathematician’s apology (Hardy, 1940), G.H. Hardy calls chess problems a mere exercise in pure mathematics. However, unlike the usual worksheet exercises, chess problems are extremely accessible to the general public; No mathematical education is needed to understand and solve them. They also bear a strong sense of aesthetic: Chess problems are intriguing, beautiful, and competitively addictive. (Desjarlais, 2011) and have been the thorough fascination of mathematicians (Watkins, 2004).
However, even if beautiful, Hardy argues, chess problems are essentially unimportant. They lack substance and “seriousness”. For him, the mathematical ideas underneath a chess problem have very little significance as they can’t “be connected, in a natural and illumating way, with a large complex of other mathematical ideas”. In Hardy’s conception of mathematics (where mathematics exists in reality and we unpick it theorem by theorem), a serious, important, relevant problem should have all these qualities. For example, Euclid’s proof of the existence of inifnitely many prime numbers would be a first-rate ‘real’ theorem: It is an essential part of the structure of arithmetic, of mathematics as we know it.
Nevertheless, Hardy wrote his apology in a very different world than ours: Mathematicians were just starting to figure out the applications of number theory in encryption (which we now use on every single contactless payment), and graph theory wasn’t much of a thing either. In the UK, there are more people learning and doing maths than ever before (amsp, 2024).
Whether G.H.Hardy likes it or not, mathematics is now mainstreamed into popular culture and, as so, why should we be gatekeeping any mathematical knowledge at all? What is the need of creating a hierarchy of problems, puzzles, and mathematical contraptions? Is Cantor’s theorem of the non-enumerability of the real numbers, which requires extensive mathematical familiarity both to understand its proof and meaning, more important than the hidden permutation problem on a chessboard which brings such joy to solve and does not require hours of further study?
This week’s puzzle(s) are, essentially, unimportant. Hardy explicitly calls them “uneappealing to a mathematician”. I thoroughly disagree. Yes, they didn’t tell me anything about the deep intricacies of the world of mathematics, but they amused me and made me think of what we consider as important in this beloved field of us. And who has a saying on that anyway?
Problem#
The natural number 9801 is a four-figure number which is an integer multiple of its reversal: \( 9801 = 9 \times 1089 \) What is the other four-figure number with this property?
There are just four (after 1) natural numbers which are the sums of the cubes of their digits:
\( 370 = 3^3 + 7^3 + 0^3 \)
\( 407 = 4^3 + 0^3 + 7^3 \)
What are the other two?
Problem submitted by Pablo Ortuño