Of cubes and Smart Lazy

As I grow more into Advanced Middle Age, so I am attending to the leftovers, things like my Playwiths web site. I was amazed to discover last year that this had drawn in 4 million visitors.

OK, it's been going since about 1995, but 170,000 visitors a year isn't shabby for what began as a single page, constructed as a teaching tool for teachers from my feeder schools. As you can see if you look, it grew.

Anyhow, last year, I was on a train to Fairy Meadow to have lunch with some Illawarra schoolkids (it's an annual gig), and it struck me that there was probably a book in it.  On the train I had been ruminating on the fact that I had three more books that I would like to get done, but adding this one to the list made good sense. I got out the notebook and sketched an outline.

Six months later, it's largely done, but this note is about a bit of maths that hit me this morning, that just had to go in. Yes, I'm still fiddling, even as I pitch the book, and the writing, plus the fiddling, explain the dearth of posts here.

First, by way of background, I play with numbers to put myself to sleep or just to relax. In the dentist's chair, I calculate the square or cube root of a prime number (usually 17) in my head. It's just the way I am.

On the side, I know things that are interesting, like the story of Ramanujan and the number 1729, but I posed a question on the site and in the book: how can you use those four numerals in that precise order to create a sum generating each of the integers up to 50 (sums like -17 + 29 = 12, or 1 - 7 - 2 + 9 = 1).  Here are some of the harder answers to the game of 1729.

The original thing that got me interested in 1729 was that Srinivasa Ramanujan famously called it the smallest number that is the sum of two cubes in two ways (1^3 + 10^3 = 9^3 + 10^3), but during the writing of the book, I realised that this is not the case. Here's the proof:

6^3 + (-5)^3 = 4^3 + 3^3 = 91.

Well, that's interesting, but I happen to know that 1729=7x13x19, so it has three factors in arithmetic progression, which is interesting, but 91=7x13, which is Interesting.

With me so far? This morning, as the currawongs shredded the pre-dawn calm this morning, I began thinking about the simple pattern of differences between successive squares. I can illustrate this best with a snippet of spreadsheet:

Now why did I generate that, and how? The why: I am Smart Lazy, which means using available tools. The how: here are the codes I used in Excel, in a smaller snippet:

Entering the code is easy, if you use the Fill Down and Fill Right commands in Excel: look them up.

Well that table of manipulated squares has some nice patterns, but being a fiddler, I moved to cubes. This time, you see the codes first: notice the highlighted cells.

Can clever people see how little I had to change in the spreadsheet? That's what Smart Lazy is about! Look at the codes below to work out what diff(1), diff(2) etc. are about.

Notice that I have highlighted certain cells. The 7, 19 and 91, highlighted in yellow, are INTERESTING, but they may be just coincidence, and those do happen if you push enough numbers around. Here's a short quote from the book

The speed of light in terafurlongs per fortnight is 1.803, close enough for government work to the metric equivalent of a fathom, showing that any measured value can be given almost any number by a judicious choice of units. Any reasoning based on the coincidence of two values needs to be questioned closely to see if the coincidence is just, well, a coincidence—or the work of somebody using peculiar units to get a result.

For example, the number of islands in the Hawaiian island chain is 137, and the ratio 1/137, often referred to as alpha, is the fine structure constant in physics. This value represents the probability that an electron will emit or absorb a photon. It is the square of the charge of the electron divided by the speed of light times Planck’s constant, and it is just a number: there are no dimensions or units involved at all.

The significance of alpha was first spelled out in 1915 by a physicist named Arnold Sommerfeld—at the time, measurement errors made the value closer to 136—and physics ever since has been littered with efforts to explain the number.

Meanly, I have left out all the best bits of that story: you'll have to watch out for the book. No publisher yet, but I am pitching Playwiths at the moment.

Now go back to the other numbers, the ones highlighted in blue, all of which can be expressed in terms of the sums or differences of cubes.

117 = 5^3-2^3

217 = 6^3+1^3

513 = 8^3+1^3

728 = 9^3-1^3

999 = 10^3-1^3

1027 = 10^3+3^3

1736 = 12^3+2^3

Coincidence? I doubt it, but I have no idea why it happens. I’m not saying how I did it, but of the first 2000 integers, 150 of them can be expressed as the sum (or difference) of two cubes. I will admit to using both a spreadsheet and a word processor with a SORT function.

Still, the book is called Playwiths: why not play with it?

PS: there are other patterns there that I have left for others to discover. I didn't miss them: the margins of this page were too narrow for them to all fit in. Try 208 and 1216 in the diff(4) column, for starters...

 The book is called Playwiths. Play with it!