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!

When push comes to shove

The very first challenge for colonists was moving things (and themselves) around the local area. In the very early days, with only a settlement at Sydney Cove and few animals that could be ridden or used to pull vehicles, transport was by foot or by boat. One of the first needs for Sydney was barracks, and bricks and tiles had to be made for these, and timber had to be cut. Then all of these heavy items needed to be transported.

When the First Fleet arrived in Australia, it was very poorly equipped. One problem they faced was the quality of the tools sent out by the government. Back then, cheap tools were made for bartering in the “African trade”, and that was the quality of tool sent to Australia. Governor Phillip was savage when he made this point in a despatch to Evan Nepean’ the Home Secretary, written on 28 September 1788 after eight months’ experience of the shoddy tools.

The tools and articles in the inclosed lists will be much wanted by the time they can be sent out, and I cannot help repeating that most of the tools were as bad as ever were sent out for barter on the coast of Guinea.
— Historical Records of Australia, 1(1) 86.

The First Fleet had sailed in 1787 with enough food for two years, if everything went well, and things might have gone well. They had seeds of many crops, and they also had a good supply of animals, but these were mainly brought for breeding, and many others were, in any case, animals to eat. Any making and building meant things had to be moved, but where were the horses and bullocks to do the hauling? With just seven horses and seven cattle in the colony, the hauling would at first have to be done by humans, and the mills would need to be turned by wind power or water power.

To convey these materials from the brickfield to the barrack-ground, a distance of about three-quarters of a mile, three brick-carts were employed, each drawn by twelve men, under the direction of one overseer. Seven hundred tiles, or three hundred and fifty bricks, were brought by each cart, and every cart in the day brought either five loads of bricks, or four of tiles. To bring in the timber necessary for these and other buildings, four timber-carriages were employed, each being drawn by twenty-four men.
— David Collins, An Account of the English Colony in New South Wales, volume 1, 331.

All too often, we can only spot what was happening by inference from incidental comments. For example, when a convict fisherman named William Bryant was caught selling part of his catch, he had to be punished, but not by being sent to haul a cart:

For this offence he was severely punished, and removed from the hut in which he had been placed; yet as, notwithstanding his villainy, he was too useful a person to part with and send to a brick cart, he was still retained to fish for the settlement; but a very vigilant eye was kept over him, and such steps taken as appeared likely to prevent him from repeating his offence, if the sense of shame and fear of punishment were not of themselves sufficient to deter him.
— David Collins, An Account of the English Colony in New South Wales, volume 1, 1798, 44 – 5.

In May 1793, as the weather got colder and the days shorter, the working hours for labourers were changed:

… the lieutenant-governor directed that the convicts employed in cultivation, those employed under the master bricklayer, and those who worked at the brick carts and timber carriages, should labour from seven in the morning until ten, rest from that time until three in the afternoon, and continue at their work till sunset.
— David Collins, An Account of the English Colony in New South Wales, volume 1, 1798, 239.

Then there came the mills that were driven by humans. David Collins wrote this in 1794:

Wilkinson’s grinding machine was set in motion this month. It was a walking mill, upon a larger construction than that at Parramatta. The diameter of the wheel in which the men walked was twenty-two feet, and it required six people to work it. Those who had been in both mills (this and Buffin’s, which was worked by capstan-bars and nine men) gave the preference to the latter; and in a few days it was found to merit it; for, from the variety and number of the wheels in Wilkinson’s machinery, something was constantly wrong about it. Finding, after a fair trial, that it was imperfect, it was taken to pieces; and Buffin was employed to replace it by another mill upon the same principle as that which he had himself constructed; and Wilkinson returned to Parramatta.
— David Collins, An Account of the English Colony in New South Wales, volume 1, 1798, 305.

In time, though, bullocks became a major source of power, and I may get around to them, one day. 

A bullock cart on Brickfield Hill (near the present Town Hall), August 1896,
from David Collins, An Account of the English Colony in New South Wales.

In the 1820s, a treadmill was installed in Sydney, so that recalcitrant convicts might not only be punished, but return a dividend:

A tread-mill, the first of the kind in the Colony, we are informed, has been set going this week m the Carters’ Barracks. There can be no doubt but that this instrument will prove as salutary in this Country, in the repression of crime, as it has been found efficacious in the Mother Country.
— The Sydney Gazette and New South Wales Advertiser, 21 August 1823, 2, http://trove.nla.gov.au/ndp/del/article/2182128

There was, it seems from an 1824 report, a graduated series of punishments, starting with the Sydney gaol gang (with or without irons: in the picture above, only one gang member is ‘ironed’), then the gang at Emu Plains, which in 1824 would have been the site where roadbuilding was going on, but there were worse fates:

Should this renovating station, however, prove inadequate to the end contemplated, which in some instances, after every care, will be the case, perhaps a visit to the tread-mill may not be considered quite inexpedient. In the event of all those experiments failing, Port Macquarie then becomes the most desirable spot for correction and reflection on this side of Bass’s Straits…

It is fit that men should smart for the abuse of that Royal mercy, which is so repeatedly despised. When Macquarie Harbour fails in deterring obdurate offenders from the commission of vice, then it is high time that the place of execution should sweep them away from the face of that earth, which groans under the weight of the many years’ atrocity that supplicates Heaven’s justice!
— The Sydney Gazette and New South Wales Advertiser, 10 June 1824, 2, http://trove.nla.gov.au/ndp/del/article/2182973

Sometimes, though, humans were willing to take on a load: if there was a promise of gold at the end. Here, in 1851, Rudstone Read describes the scene as people “rushed” over the Blue Mountains:

The whole line of road to the diggings presented an animated appearance: drays, carts, cars, equestrians, pedestrians, and last, though not least to catch the eye, was an old man wheeling his all up in a barrow, his son harnessed in front, and his wife bringing up the rear, with a child on her back, who generally got a cheer from passers-by, or a question, as to how much she’d take for her “swag.” At every public house or coffee-shop on the road, all sorts of people were to be seen, and, excepting Esquimaux or Patagonians, every nation on earth might here be found represented.
— C. Rudstone Read, What I Heard, Saw, and Did at the Australian Gold Fields, 1853, 5 – 7.

Read added as a footnote that this party returned to Sydney at Christmas with £1000.

Then there were those who used human power to drive bicycles, which I dealt with recently.