Oh dear, another long break — but somewhat enforced, perhaps, as I have remained incredibly busy of late. Managing to keep my Listener head just above water, though, and I thought I would surface properly for this one. After all, if a puzzle that appeals to my mathematical background doesn’t prompt me to blog, what will?

Ilver is another of that group of new setters appearing in the last ten years or so, whose first Listener came out just in time for me to have a crack at it (my solving career starting in 2011). Since then, his nine puzzles in all outlets have included themes such as Pig Latin, Doctor Who and Poirot in the Listener, but also a couple of puzzles that have had mathematical elements. As it turned out, this effort belonged to the latter category — the first hint of that emerging when those extra letters revealed something looking like “a very interesting number”. Oh, that quote! I’ve known about it for a while, GH Hardy telling his sick mathematical colleague, the brilliant Srinivasa Ramanujan, about the taxi he had come in (rather than, say, hoping he’s getting better and asking after the family?), and how boring the number 1729 is, and hoping that this isn’t some omen. “But no!” says the young Indian, “for 1729 is the smallest number that can be expressed as the sum of two [positive] cubes in two distinct ways!”

Of course, all numbers are interesting really (there is a ‘proof’ of this, because any smallest uninteresting number would be remarkable for just that fact, and therefore interesting again, and hence there can’t be an uninteresting number!).

At the other end of the puzzle, we are instructed to highlight an identity. And there is some of it, eventually emerging from those odd clue lengths, as “plus one equals zero” emerges below an “i pi”, and all that is missing from Euler’s identity is the initial “e”.

At this point, though, progress came to a juddering halt for a time. 1729 is the key to obtaining a second quote, and then something in the grid that is also related, and fixing the grid will insert that final e in its rightful place. But how to use the key remained a mystery to me (and, it seems, quite a few others) for some time! Perhaps it’s partly because the obviousness of the final highlighting provides something of a distraction. Is there a direct link between Hardy’s work and Euler’s? Much searching of their respective output follows, but isn’t too revealing. Nor does any quote about Euler seem particularly helpful.

But then, of course, this is primarily a word puzzle, rather than a mathematical one, so thoughts should turn to interpreting things a bit more cryptically. It also helps to check other GH Hardy quotes, of which one alone made it into my edition of ODQ. “Beauty is the first test: there is no permanent place in the world for ugly mathematics,” he said as part of a book filled with other gems. Interpreting 1729 as four separate numbers 1,7, 2, 9, and then taking the 1st, 7th, etc letters of across clues revealed the first part of that quote after all. Phew! (after much grid-staring and head-scratching).

To get to the second part, though? Another bit of grid-staring, although at least with the final E in mind thoughts turn to the 1st, 2nd, 7th, 9th columns of the grid (not rows, because we need that e in the second column/ sixth row). Towards the bottom of the 9th column is the Y of “Hardy”, a bit further above is the H, and the rest emerges if you take 1st/7th and 2nd/9th letters in alternating rows. Follow it up and an anagram of “mathematics” is sandwiched between “the world”. And rearrange that to its proper form and you see the e fall in its rightful place, to complete arguably the pinnacle of what is meant by mathematical beauty. “e to the power of i times pi plus one equals zero, where e is the base of natural logarithms, i is the square root of minus one, and pi is the ratio of the circumference of a circle to its diameter”.

What makes it so beautiful is what this identity brings together. In the first place, the numbers above are fundamental to mathematics in a way few other numbers are. And then the operations (exponentiation, multiplication, addition) are also fundamental to all arithmetic, each being used exactly once, as is the equality relation. There’s a certain elegance in everything being used exactly once, too.

But also, it is where these numbers come from that contains the real beauty. pi is a number from the classical Greek mathematics of geometry (and then Trigonometry, its offshoot developed in the Arab world in the 7th-9th Centuries). i has its origins in algebra, developed mainly in the 12th-14th centuries. And the number e emerges naturally from Newton and Leibniz’s development of calculus in the 17th Century. Even 0 is quite special, representing the abstract concept that took a long time to develop, that you can even quantify and work with nothingness. We can thank the Indian mathematicians for that one.

Thus it is, then, that the entire history of mathematics from the Ancient Greeks up to Euler himself can be contained and summarised in just five numbers and four operations. There can never be a purer and more beautiful result.