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Saturday, 26 July 2014

The Great Speewah Flood

Now let's see, I was going to tell you about working dogs on the Speewah, wasn't I? As I recall, I was going to explain how the dogs out there really need to be as versatile as a shearers' cook when the food runs out.

Which reminds me: Crooked Mick demonstrated that sort of versatility more than once, during the Great Speewah Flood, the one that lasted two years without a single break in the rain.

I think I've already told you about that: if I did, I would have explained how the flood was so bad that all the fish drowned, and afterwards they had to import new fish to stock all the waterways. Well whatever I said before, that's the way it really happened.

Anyhow, the rain got on everybody's nerves, especially the shearers. So being a rough bunch, they took it out on the cook, who was sensitive, as cooks go. In the end, what with the shearers not being able to shear, and the hard time they gave the cook, the cook went, all the way to one of the outer sheds, where there was a large barrel of dust, kept for emergencies like long floods.

Climbing up on a high beam, he weighted himself down with scrap iron, jumped into the barrel of dust, and drowned. The boss was so sorry he stopped keeping a dust barrel after that. He said it might give the place a bad name, even if it was a long-standing tradition.

Well as you know, Mick was a marvellous cook, so after the barrel of dust had been dumped into the water, the station boss came down to the shearers' quarters, and put the hard word on Mick to take over as cook. There was no way they could get a new cook in, he said, not while the flood was on, and Mick was the only man up to the task.

Mick agreed that he was the best choice, but he mentioned that he was getting quite a good classical education from The Professor while the rain was coming down, and he'd miss all that if he had to cook for the whole shed. He was happy just to sit there, soak up an education, and wait for the rain to end.

I should explain that Mick had left school early, but he could see that a bit of education could be really useful when you were digging post holes, and didn't have anybody else to consult about dangling participles and things. You never know when one of them will dangle into the hole you've just dug, if you don't know what to look for.

So in the end, the boss agreed to pay The Professor a bonus to sit in the cook house and talk to Mick while he was cooking, as well as pay whatever the boss had offered Mick, and we all ate very well for the next eighteen months. Well, within reason we ate very well, but there were a few worrying moments when the food ran short, once or twice.

As you might realise, with the rain keeping on and on, the water levels got higher, and Mick and The Professor had to climb up a gum tree and build a new cook house, in a low fork of the tree. Then, as the water level kept rising, they kept moving the cook house up the tree, higher and higher.

Now that would have been all right, except that the two of them used to go up, right to the top of the tree to collect firewood. If you think about it, they were moving the cook house up the tree, while the top of the tree was slowly being lowered, and in the end they had to build a combined boat and cook house, and sail off to a tree that still had its top branches, all the while keeping up a supply of tucker for everybody.

At the same time, the rest of us had built a whole range of barges and pontoons to hold the stock, most of the local wildlife, the sheds, and other farm gear. That was how the great Speewah Ironbark Forest became the Speewah Plain that you can find in the atlas today. And if you look in the middle of the Speewah back paddock, you may even see a small rise that's forever labelled "Mount Ararat".

We couldn't have done it, though, if Mick hadn't put in an hour a day felling the trees for us, because he was the only one able to dive to the base of a Speewah Ironbark and cut through it on one lungful of air. It was hard on the forest, but we saved the stock and the wildlife.

Then there was the problem of the food. Until the fish drowned, they were able to get some of those, and the dogs kept the sheep herded in the upper branches of the taller trees until the floats were made. Remind me to tell you about the dogs some time.

Luckily we managed to save two of the Speewah rabbits, and we were able to live on their progeny for about twelve months. We ate two a night, one for Mick and one for the rest of us, and those parent rabbits just kept on breeding at the same old rate, the whole of the flood time.

Anyhow, the real problem came when we ran out of salt. We tried to work out how to get some brought in, but we were completely cut off. Even Mick wouldn't have been able to paddle or swim back against the raging torrents that were running off the Speewah in all directions.

So Mick ended up getting into a hollow gum tree, finding his way down to ground level, and sinking a hole down to the sub-artesian water, deep under the ground. Then he collected this water by the bucket, and carried it back up to the surface, where we evaporated it over a small fire, so we could have salt with our food again.

Later, when the whales came, it was Crooked Mick who fitted special stabilisers to the cook house, in case a whale bumped into it during the night, and he caught the two smaller whales with a hand line.

Not that we ate the whales, of course, but he used the oil to power his stove when we ran out of firewood, as the last of the remaining trees disappeared below the surface, just before the rain stopped. It was touch and go, there, whether we'd have to start burning some of the boats and rafts, and people were beginning to look meaningfully at Greasy Smith, wondering how much oil they could get off of him.

Yes, well I know I was going to tell you about the dogs, but I have to go now, so maybe next time, if nobody interrupts me, I can tell you all about them. Make sure you remind me . . .

* * * * *

Note: there is a whole book of these stories, which I am currently pitching to publishers, but they will probably appear in an e-book.

There will be quite a number of these on the blog, all with the tags Speewah and Crooked Mick.

Thursday, 24 July 2014

Crooked Mick Builds a Railway

This came out of the bottom drawer.

* * * * *

I reckon the hardest work Mick ever did was when he laid the last ten miles of the track on the Speewah Hills spur line. It wasn't just the lay of the land that was against him, though, it was the geology itself. It was all jumbled up, with sandstone and shale all mixed in together, and outcrops of quartz sticking up in all the worst places.

So Mick had the problem of digging through really hard rock in the cuttings, but even though there was soft rock in places, it was almost useless for fill. That didn't help any.

Mick was left to do it all on his own, on account of a ghost that had turned up. It wasn't the ghost as such that made all the other workers pack up and leave. They didn't mind ghosts: it was the way it kept whining and grizzling about what a rough after-life it was having. It was  a real whinger, and just wouldn't shut up, no matter what, and so Mick was there on his own, and that didn't help any.

But there was worse: there was Royalty coming out to open up the line, and there were still ten miles left to do, with just nine days to do it in. That's what governments are like: slow to react and then demanding everything in a rush, and that put Mick under a lot of pressure, which didn't help any, either.

Of course, Mick couldn't have done it without his dog, but the dog was in one of its scatter-brained and lazy moods that it sometimes got into, and that didn't help any, which is why it turned into the hardest piece of work that Crooked Mick ever did.

Anyhow, the dog went out and surveyed the rest of the route and pegged it, but being lazy, it didn't drive the pegs in too far, so a few got knocked out by passing kangaroos, and some passing galahs took a couple more for dessert, which meant Mick actually went the wrong way a couple of times, and that didn't help any.

So after that, Mick gave the dog a good talking-to and sends it out with a team of horses, a team of bullocks, and two scrapers, so they can do the rough work on grading the route. Seeing the dog was being a bit absent-minded, Mick didn't want to overload its intellect. Personally, I reckon the dog knew exactly what it was doing, and it played stupid to get an easy life. Whatever it was, Mick had to do more of the work, and that didn't help any at all.

Still, with the dog supervising the horses and the bullocks, as well as acting as billy-boy, they could've made it, easy as pie. As it was, they got the whole of the way cleared and graded in five days, and that was when Mick found there were no sleepers. So he had to give half a day and all that night to cutting the hardwood sleepers to go under ten miles of track, and here he was lucky. See, there was a full moon, and that helped a bit.

He tried tossing the sleepers up ahead as he cut them, but any that he threw further than a mile just splintered when they landed, and the dog couldn't carry more than four at a time, so Mick got the dog to round up a couple of hundred Speewah bull ants. He tied two sleepers on each ant, one each side, and the dog marched them off, biting through the strings from time to time, to release the sleepers.

After a while, the dog got the idea of undoing the knot, and it learned from that about how the knot was tied. The next morning, it took over the job of tying the loads on, as well as dropping them, and it increased the work force to four hundred ants and had them marching four abreast. That helped a bit.

The only thing is, the sleeper-cutting hadn't been allowed for in Mick's timetable, and he was now behind schedule, so he worked his way along, a hammer in each hand, driving the spikes into the sleepers, which went faster when the dog was there to hold the spikes on the left-hand side, but it was mostly still at work transporting the sleepers.

That was when Mick made a bad mistake: he overworked, and broke both his hammers, with two miles of line still to be laid. That didn't help any, but he still had his fists, and he kept going and got it down to just half a mile to go, when he ran out of spikes. Now that definitely didn't help any at all.

So he sent the dog out, early the next morning, to bite the tips off as many mosquito stingers as possible, getting only the youngest ones, so the tips hadn't hardened yet. When the mosquitoes first leave the water, their stingers are no harder than half-seasoned ironbark, but half an hour later, they get really tough, so the dog had to get in quickly. That didn't help any, because the dog had to pick and choose among the mosquitoes, dodging the ones that were too old.

But in the end, the dog collected enough of them. As a matter of fact, if you go out there today, the rails have rusted away, and the sleepers have all fallen to the termites, but you can still see those mosquito stingers, marching away across the landscape in two parallel rows.

I suppose you're wondering how Mick transported the rails. He didn't have to do much, because they were loaded on railroad trucks when he took over the work, so he just hooked them up, and pulled them along behind him. That helped quite a lot, having them all loaded up like that.

And I suppose you want to know how he fixed the ghost: well, that was the easiest part of all. He collected the sap that oozes out of the gum trees, the gum that gives them their name, and he heated it up in an old billy. Then, when the ghost came round, he grabbed it by the throat, and poured the hot gum down its throat, and shut it up for good.

And I know what you're thinking now: how could the hot sap stick to a ghost? Well that was where Mick used his brains. He collected all the sap from ghost gums. That helped a lot.

But apart from that, it was the hardest bit of work that Crooked Mick ever did. It would've been easier if the dog hadn't been so lazy.


* * * * *

Note: there is a whole book of these stories, which I am currently pitching to publishers, but they will probably appear in an e-book.

There will be quite a number of these on the blog, all with the tags Speewah and Crooked Mick.

Monday, 21 July 2014

Ant lions rule!



This is another retread from a nursery news letter, but I have added more information at the end about catching, keeping and managing these cute little carnivores. Ant lion has a different meaning in other parts of the world, but I gather that there are strong similarities.

Aged seven, I was given a book called Beetles Ahoy! and read about ant lions there and fell in love with them. Family Myrmeleontidae (Neuroptera) to entomologists, these are the larval stage of lacewings. They dig neat holes.

The name is a misnomer: they aren't lions, as anybody can see. More importantly, they don't always eat ants—I have seen one catch and presumably eat, a small weevil.

Any loose material like sand has a natural angle of rest. This is the steepest angle it can hold without tumbling down. Sand dunes, sand heaps and sand banks are all limited by this angle. So are wells dug in creek beds.

This angle shows up in sandstone cliffs which contain fossilised sand banks, and you can see these all over North Head. The best view is from the lookout off the Fairfax Track.

So sand has a position of maximum stability. Ant lions rely on this. They dig conical pits in the sand by burrowing into the sand, and flicking sand up and away with their heads so the sides settle at the angle of rest. Then the predator sits hidden at the bottom, waiting for something to fall in.

Anything going over the edge dislodges sand and tumbles down. As it tries to scramble out the ant lion flicks the fallen sand out. This undermines the side which start to slide down, while some of the flicked sand knocks them down. The prey slides down as well.

Once the unlucky animal reaches the bottom of the slope, the ant lion seizes it in its pincers and starts sucking it dry. In the end, it flicks the empty husk of the prey out of the pit.

They are all over North Head, but you have to know to look for a small conical pit, 1–3 cm across in dry sandy soil. The soil may be close to one of the gum trees that kill grass, inside a hollow tree, along the edge of a building or under a rocky overhang. Sometimes, you can even see ant lion pits, right out in the open.

At times, and for assorted reasons, I keep some as pets. Here is all you need: just add ants—or weevils.

Catching ant lions: if you chase them they can burrow fast.  I use and old cup and scoop up the pit and everything for about 3 cm below the base, and I tip this sand into a jar.

Once i have several of the animals, I transfer the jar's contents to a tray containing about 3 cm of dry, clean sand, sprinkling the sand from the jar over the surface. The ant lions lie very still for a while, so you may not see them. Then they move backwards across the sand before backing down into it. You can see a trail going from left to right in the photo above, half-way up.

They often wait for a day or so before making a pit, so be patient.

Catching ants: Do not use a pooter! Ants release formuc acid when they are handled, and this burns the throat. Depending on the species, put a sheet of paper with a spot of Vegemite 
 or a scrap of meat (for meat ants) or honey (other ants). When enough ants are on the sheet, pick the paper up and shake the ants into a jar.

The rig on the right shows a neat way to stop the food ants escaping.  There is water in the larger tub. In use, the handle on the inner tub is upright, so there is no escape for the ants.

Water: Ant lions live in sandy soil that is somewhat shaded and also protected from direct rain, so they probably don't like full sunlight or damp conditions. I imagine they get all the fluid they need from their prey, but I usually keep one corner of the tray clear of sand, and add small amounts of water in that corner, enough to saturate the lowest millimetre or two of the sand.

When you have finished with your ant lions, release them back where you caught them, or keep feeding them until they mature, change into lacewings and fly away. If you have a covered tray, you may be able to see the adults when they emerge, but check it every day, and don't open it inside.

The photos are all mine, the drawings are from my 1985 book, Exploring the Environment.

Incidentally, it struck me that maybe I was repeating myself, and indeed I am: there are even some pictures in common, but the approaches are a bit different.  Now I wonder: am I getting better or worse?  You decide: the first version is here.

Friday, 18 July 2014

About leaves

In my spare time, I am a volunteer in the Nursery group at a local sanctuary on North Head. We concentrate on raising plants and restoring damaged bush, and from time to time, I contribute a piece to their newsletter.  I don't think they are very accessible there, so I will pop them in here as well.



Lomatia sp.

Johann Wolfgang von Goethe had a bit of a thing about leaves. He wrote a poem about the leaf of the Ginkgo, and probably saw the leaf as a symbol of love. Goethe was many things, and also a curious botanist—some might say a peculiar botanist. He thought the leaf was the basic unit of the plant: "from top to bottom a plant is all leaf…".

Banksia sp.
I thought of this when I sighted a Lomatia along one of the tracks a few weeks back. At least, I think it was a Lomatia, but now I have my doubts, because of where it was growing. I'll need to visit it later in the year to check the flower, but Lomatia is one of those once-seen-never-forgotten leaves.

That started me thinking about distinctive leaves, like Canada's maple leaf, the serrated leaves of the Banksia and the gracefully curved leaves of some gums. Again, once seen, never forgotten, though I'll bet that somewhere out there, some other plant has taken on a similar design.

Allocasuarina, or she-oak.
That's why botanists, both before and after Goethe, used flower parts for identification, despite Goethe's ideas. Still, leaves help in identification, and they are certainly worth attention.

A leaf is just a plant's way of catching sunlight, while hopefully not losing too much water. Most Australian plants have tricks to hang onto their water. She-oak needles are really branches with the leaves tightly attached, all except for little scales sticking out.

Every walk brings me "leaves" to admire, but some are fake leaves like those on Bossiaea which are really cladodes, flattened stems. The leaves of wattles are often phyllodes, flattened petioles or leaf stalks, and in each case, the change is designed to save the plant from losing water.
Bossiaea sp.

Another way to avoid losing water is to discourage animals from eating the leaves. Biting a leaf opens wounds that the plant "bleeds" from, and what is eaten represents a loss as well.

That explains the rainforest leaf below, which I saw on the Dorrigo Plateau.

It has remarkably nasty spines to keep larger browsers away, though as you can see from the picture on the right, below, small animals just dodge around the spines and much away.


One of the things that changes the form of leaves, that shapes them, is what Charles Darwin called the struggle for existence against predators, though another aspect is the fight against other plants to get a place in the sun.
 
Listen, young Goethe, forget about plants as symbols of love.

Even the leaves remind us there's a war on out there. Some leaves are even mined!

Unnamed leaf which has been attacked by a leaf miner.

Sunday, 13 July 2014

Gold for free



Well, this is post 250, it seems, so it's time to revert to something a bit more closely related to my day job, which is writing. That, after all, is what I started my blog for.

As it happens, I have the signed contract for Not Your Usual Gold Stories from Five Mile Press, and I am doing last-minute intensive revision at the moment. These snippets won't be in the book, so I thought I might post them here. So they are bits of the gold book for free, but they happen to be about people who claimed they had a sure-fire way of getting free gold.
The Grape that can with Logic absolute
The Two-and-Seventy jarring Sects confute:
The sovereign Alchemist that in a trice
Life's leaden metal into Gold transmute.
Edward Fitzgerald, The Rubaiyat of Omar Khayyam, (5th edition) #59.
There are three effective ways of winning gold. You can find it, you can trade for it, or you can steal it. All of these are either hard or risky in some degree. There is another method which has attracted people: the use of alchemy to transmute base metals into gold. Over the years there has been a great deal of talk and perhaps even some hard work to this end, but if any alchemist ever succeeded in "making gold", there is no historical record of any success.

Some of the alchemists seem to have really believed that gold could be "made" by treating some cheaper metal in the right way. Modern chemistry tells us this is impossible, but they knew nothing of that. The honest men among the alchemists may have been thin upon the ground—but while they found victims, there were others who saw right through them

An Italian poet named Augurelli presented a work in hexameter to Pope Leo X in 1518. It was in hexameter and dealt with "Chrysopoeia," the true art of making gold. The canny pope is said to have made the poet a gift of an empty purse, as the possessor of the secret of the Philosopher's Stone lacked nothing but a purse, in which to store his artificially prepared metal.

Georgius Agricola knew all about the alchemists' tricks as well, and believed the crooks should be executed. More to the point, he described their methods, something like 470 years ago.
… these throw into a crucible a small piece of gold or silver hidden in a coal, and after mixing therewith fluxes which have the power of extracting it, pretend to be making gold from orpiment, or silver from tin and like substances.
— Agricola, De Re Metallica, preface xxix
Their game might have been easier than digging or stealing gold, but it could also carry risks. After he died, there were those who maintained that Alexander Seton had really succeeded in making gold, which probably just means he was cleverer at sleight-of-hand than his audience, though his history suggests that he wasn't all that clever.

His yarn was that a Dutch vessel was wrecked near his Scottish home in 1601, and he rescued the crew, put them up, and paid their passages back to the Netherlands. What follows is not trustworthy, but it is a widely believed popular myth.

In 1602, he visited one of the Dutchmen, and demonstrated his new-found skill of making gold. The Dutchmen told his neighbours, showing off the piece of gold that Seton had "made" for him, and the offers and enquiries came flowing in from scholars, and more to the point, from noblemen and monarchs.

Seton travelled around Europe, putting on a show where others under his instructions, placed lead in a crucible, with a powder, and found a mass of gold, equal to the lead. The gold was tested by assayers in Zurich, and declared to be the real thing.

The young Elector of Saxony, Christian II, invited Seton to call, but Seton dispatched another Scot, William Hamilton, who performed the same demonstration, with the same success. Christian issued an invitation that amounted to a command, so Seton presented himself.

This was a bad move, because when he did not reveal his secret, he was imprisoned and tortured with rack, fire and scourge, but still he refused to deliver the secret, claiming that such secrets were not for the profane. He was finally rescued by a Moravian chemist named Michael Sendivogius, who spirited the now weakened Scotsman to Cracow.

Realising that Seton was dying, Sendivogius also tried to get his secret, but the man died. Unabashed, Sendivogius married Seton's widow, in the hope that she knew the secret, but she did not. Still, he had Seton's treatise, which he published in his own name, and a supply of the wonderful powder, but when that ran out, that was the end of him.

Another (and more likely) version has him dying in a prison in Dresden in 1604. At least neither of them perished on golden or gilded gallows like Georg Honauer (Württemburg, c. 1597) or Count Ruggiero (Berlin, 1709).

James Price was elected a Fellow of the Royal Society in 1777. He had a fortune from relatives, and gave his life over to chemical experiments. Then, in 1783, he repeatedly demonstrated the transmutation of mercury to silver and gold, using white and red powers. Assayers tested the gold, and declared it genuine.

The Royal Society expected Price to publish his method, but he refused. He also declined their invitation to repeat his experiments before the Society. The labour had weakened him, he said, but the President of the Society insisted that he perform, "for the honour of the society". He agreed, and so he came undone.

Even back then, the Royal Society had more than it share of subtle minds, and one of them spotted the false bottoms in Price's crucibles, which hid pellets of silver and gold. Caught out, Price drank "laurel water" (prussic or hydrocyanic acid) and died.

Jernegan's wheeze

This is the third in a series. You can go back and read them first, if you like: they are Gold for free and More modern gold makers, but this is a stand-alone tale.
In the cases mentioned in the last two entries, "gold making" was just a device to defraud, but one ingenious case went a lot further—and unlike most gold swindles, it had an Australian element. Professor Liversidge, professor of chemistry in Sydney University, had his efforts reported in both The Lancet and also in the journal Science, when he began studies on the prevalence of gold in New South Wales coastal seawater.

The concentration was a mere of ½ to 1 grain per ton, but that converted, said the breathless newspapers, to somewhere between 130 and 260 tons of gold in every cubic mile of seawater. The news also reached the USA, and that is probably important in what follows. Here is the source: 'Scientific Notes and News', Science, OCTOBER 23, 1896, 615. (This erroneously gave the yield as "230 to 260 tons per cubic mile").

Liversidge, a highly respected scientist, gave learned papers on his work, and these also were reported both in Australia and overseas. You can read a further report in Science, November 6 1896, 685-6.
… this would be in round numbers about 200 tons of gold per cubic mile, and if the volume of the ocean be considered 300,000,000 cubic miles, a total amount of gold in sea water of sixty billion tons. Yet this amount is probably insignificant in comparison with the amount of gold disseminated in crystalline and sedimentary rocks apart from gold in veins and other deposits. Experiments seem to indicate that sea water contains about the same amount of silver and gold.
Nobody got too excited by this. In a philosophical piece, the Sydney Morning Herald (Saturday 12 December 1896, p. 15, http://trove.nla.gov.au/ndp/del/article/14079389) considered that if a gold magnet were invented, the hundred billion tons of gold that might be in the hundred billion cubic miles of the oceans might be accessible, but that, said the paper, was unlikely.

Liversidge's work wasn't even new, except that he had shown that the gold concentration estimated by Edward Sonstadt in 1872 was too high, but given that his results were published in 1896, there must surely be a link between that and the approach made to a jeweller by one Prescott Ford Jernegan, described as the pastor of a Baptist church in Middletown, Connecticut.

Jernegan claimed to have the equivalent of a gold magnet. It was a wooden box with holes, wires that connected to a battery, and inside, there was mercury and a secret ingredient. Together, these would draw gold from the sea. He invited the jeweller, Arthur Ryan, to test his invention, and if it worked, to join him in forming a company to profit from the company.

Now Ryan would have known how to test gold, so he was safe from fakery there. He must have heard of Liversidge's work, or maybe Jernegan had a clipping to show him. Later, the inventor explained how the idea had come to him while recovering from typhoid fever and after reading a news article headed "Gold in the Sea".
A note to would-be hoodwinkers: it is these idle sidelights and mundane addenda that add sparkle to a bald and otherwise unconvincing narrative. He asked Ryan to set up the accumulator (as the device was known), to ensure that there was no trickery—another ploy of the stage magician and the con-man alike. The test went ahead in February 1897, and lo and behold! The next morning, there was gold in the mercury.
 (For more, see The Inquirer & Commercial News, Friday 18 February 1898, 5, http://trove.nla.gov.au/ndp/del/article/66692015 — and no, that is neither an idle sidelight, nor a mundane addendum.)

The gold passed assay, and was valued at $4.50, but it was a start, and with just two or three boxes, a man could "make wages", as gold diggers used to say. They founded a company to install 1000 accumulators, which would process 4000 tons of water each day.

Larger-scale trials were soon bringing in $145 a day—and a stream of investors. Assays continued to prove that the gold was the real thing, and the share price rose from $33 to $150. Then in July 1898, Jernegan and his assistant, a man named Fisher, both disappeared. That would not have mattered, because they had left the gold-making equipment, but the accumulators stopped working.

The secret to their success was that Fisher was a trained diver, and while potential investors sat on a pier to ensure that nobody came along and interfered with anything, the dark of the night and the chop of the waters stopped them detecting the diver who came stealthily, bearing gold to feed to the boxes.

Nobody seems to know what happened to Fisher, but there is some trace of Jernegan. He debarked from a ship at Le Havre in France on August 1, but a lack of the proper paperwork made it impossible for the police there to arrest him. He started for Paris by train, and while he was reported in October to be willing to return to America and face charges, he never did. On the other hand, he refunded $75,000 of the estimated $200,000 he had collected, and in April 1899, he was in Belgium

Tradition has it that Jernegan fought off an extradition case in Vienna, but lost all his money when he invested in a gold-from-seawater scheme, but this claim seems to lack evidence — I suspect that it is wishful thinking. He seems to have later gone to the Philippines and published books there on geography between 1905 and 1914, and at the age of 61 in 1927, on religion in Palo Alto, California.

See also two earlier entries: Gold for Free and More modern gold makers.

Saturday, 12 July 2014

More modern gold makers



This follows on from Gold for free, and it may help to start there,

Over time, the gold makers found safer ways to operate. In the 1880s, an American named Wise ran a scam in Paris. In the Wise process, an ounce of gold, a little silver and some base metal was added to a crucible. Vile-smelling chemicals were placed in there, but these were so vile, that everybody was forced out of the room. On their return, the crucible contained one and a half ounces of gold, and some silver and other metals. In short, there appeared to be a 50% profit.

Two French aristocrats, Prince Benjamin de Rohan and the Comte de Sparre, provided 10,000 francs for working gold, and Wise pocketed this and returned to America, a rich man. He would not have worried about the sentence of two years' imprisonment recorded against him in his absence.

If the mysterious Mr. Wise was not either an admirer of, or even one and the same as Edward Pinter who faced the Old Bailey in July, 1891, I will transmute my hat into gold. Pinter entered a plea of guilty to a charge of unlawfully attempting to obtain £40,000 from Edwin William Streeter, a Bond Street jeweller, by false pretences with intention to defraud, and was imprisoned for three months.

It seems that Pinter had been pulling a similar fraud in Paris in 1888, going on what appears in Science, August 28, 1891, 114. In this case, the promised gold return was three-fold, but the process took three weeks. The furnace was left operating in a locked room, with $90,000 in gold in the furnace. When the "alchemist" was nowhere to be found, the room was opened, and the gold was found to have been transmuted to stones and scrap iron.

How might it be done? After his retirement as head of Scotland Yard's CID, Sir Robert Anderson reminisced on a case where the swindler insisted that he be thoroughly searched as he left the room, and that nobody else was to enter. He was searched, and none of the 20,000 sovereigns were found on him, yet they all "walked". Anderson claimed that the man's gold-headed cane, one each occasion, was packed with sovereigns.

Two German did well out of gold-making in the 1930s, but lacked the sense to quit while they were ahead. Perhaps a psychologist somewhere can explain this in terms of the rise of the Nazis, the trauma of losing the Great War and the massive inflation that followed, but clearly things were different in Germany.

One of them, Heinrich Kurschildgen, otherwise "the gold maker of Hilden," even put a scare into the world's economists briefly, when he claimed to be able to make enough gold from sand to settle Germany's reparations bill, the damages that Germany was forced to pay for starting the Great War. He must have been convincing, because financiers and even academics fell for his clever talk, but in late 1930, he was sentenced to serve 18 months in gaol after being found guilty on 15 counts of fraud.

A month later, "Baron" Charles Tausend, a former plumber (or tinsmith: reports vary on this) who had been held for 20 months, went on trial on a similar charge. Tausend had apparently netted £125,000 from assorted wealthy Germans, including aristocrats and General Ludendorff. He had magnificent cars, a castle in the Tyrol and funded a "Hitlerite" (as it was called back then) newspaper.

He had come undone when the authorities forced him to repeat his successful "experiments" under the close scrutiny of analytical chemists. He was caught out, dropping a cigarette end into his "gold making machine", and when this was retrieved and examined, it had a small piece of gold in it. He got 44 months, and his laboratory and gold were confiscated.

Just a year later, and as if to prove that Germans weren't the only gullible nation, Professor John Dunikowski was on trial in Paris. A Pole, Dunikowski claimed to use radio-activity to release the gold trapped in ordinary soil. His backers, a group of English bankers, pointed out that they were too bright to fall for "gold making", but this chap seemed to be on to something.

The bankers even had an unnamed "famous scientist" who had studied Dunikowski's operations and could not fault them. Perhaps if they had employed a competent stage magician, they might have obtained better advice. Magicians know that the stirring rod, introduced into a crucible, is likely to have gold attached to it, covered in black wax that melts and burns away, leaving the gold behind.

During the 1940s, as science learned more about nuclear reactions, a number of scientists were quizzed about the possibility of "making gold", but their answer was always the same. For example, Columbia's Harold C. Urey said it was " … possible, but not commercially practical, to change platinum into gold."

Two of his colleagues explained that the tiny yield could only be detected by extremely sensitive apparatus.

In 1947, Walter Zinn, director of the US Argonne national laboratory explained that, while gold might be made artificially, doing so would cost more than digging it out of the ground. In 1948, a member of the American Atomic Energy Commission explained that they could make mercury from gold, or gold from platinum, but stressed that it would be unprofitable.

It didn't matter. If people can make gold, no matter how much it costs, a part of that message can be used to make, if not gold, at least a healthy profit. In late 1949, there was a panic on the Paris stock exchange when the same old news about gold by nuclear science was trotted out as "new".

There was a thriving black market in gold at that time, and in countries where gold was hoarded, the price of gold plummeted with the news that anybody could make gold. The damage ran on to New York, London, Mexico, Cairo, Athens, Tangier and Beirut.

These days, we may be too sophisticated to fall for that sort of "scare", but some people might still be caught. In October 2011, A 30-year-old Belfast man, Paul Moran, was gaoled for three months for arson and endangering the lives of others, after he placed his faeces on an electric heater, in an attempt to make gold. *
 

There's one born every minute, and for each one, there is a queue of villains waiting to (golden) fleece them. I will come back to a bit of that in the next entry.

See also: an earlier entry, Gold for Free and the next one: Jernegan'swheeze.

-------------------------
* It would be tempting to compare this with the Australian notion of a "shicer", probably an anglicised form of a German word, but I would never stoop so low, outside of a footnote, because nobody reads those.

Thursday, 3 July 2014

Travel memories

I suppose the hotel chain that spawned Paris Hilton would have to be tacky, but when I found that the Hilton in Quito (Peru) featured a Café Colon, I thought they had excelled.

That was before I found the bowl of popcorn on the Café Colon dinner buffet. Naturally, I added some to my plate, just because I could, but when my dining companions asked me why I had it, I thought quickly.

"To dunk in the red wine," I said, deadpan, "It's an old Peruvian custom."

To prove my point (one must risk suffering for one's art), I dunked some and ate it, and it tasted quite nice. When the popcorn was also on the breakfast buffet, I asked, and found that it is there for people making ceviche. But it's good in red wine, as well, especially carminere. Unsweetened and unsalted popcorn, of course: I'm not a barbarian.

Now about the Café Colon: this is probably the most hideous eatery seen during a month's travelling. The problem is one of management, because the front-of-house staff are delightfully helpful, but the Café Colon seems to be run by Basil Fawlty on a bad day. They played crap pop music in the feeding area for both of the meals we had there. VERY loudly. It may have been Celine Dion, or Barry Manilow in tight trousers, but it fitted the décor.  It most certainly did not help the digestion.

Then there was the giggling idiot who attempted to draw the cork on my wine, and managed to leave part of it behind. This can happen to amateurs like me, but it should not happen to professionals.  Still, the worst was yet to come.

I have no objection at all to pushing the rest of the cork into the bottle, but you need to know that a knife, wider than the bottle neck won't work.  Nor does a fork.  It took him ten minutes to realise that a teaspoon was needed, and then the handle was his second choice.

Then in spite of my giving him our room number THREE times, he turned up at the table after we had retired and demanded that one of our dining companions pay up.

The management aspect also emerged in the totally disorganised layout of the dining area. It was plain incompetent, or perhaps deliberately malicious. Related foods were placed at opposite ends of the serving area, and so on and so forth.

There sre some very nice small hotels in Quito, so avoid the Hilton.

Saturday, 28 June 2014

The meaning of miniature


The first miniatures were created back in the days of hand-written books, when miniaturisation and illumination went together. Of course, these days, those illuminated manuscripts which have survived are given low levels of illumination, because that word has changed meaning.

When they were first illuminated, the manuscripts had light added to them, when the monks in the scriptorium used colour to make the manuscript more interesting and more beautiful.

If they were illuminated now, the manuscripts would slowly be damaged by the light, until they faded away, so illuminated manuscripts are never illuminated now. We keep them in the dark to preserve the illuminations in the original sense, but illumination is not the only word that has changed.

The word 'manuscript' has changed as well. Originally, it meant 'hand-written', but now an author's work, fresh off the laser printer hooked up to the computer, is also referred to as a manuscript, but back to our monks, scrivening away on vellum, and their occasional miniatures.

Sometimes, the monks might be making an almanac, featuring red letter days, special feast day that were indicated in red. In the Church of England, red letter days came to be those days on which the Book of Common Prayer includes a collect, an epistle and a gospel for that day — but by then, the Book of Common Prayer was printed, not written by hand.

The use of red for special matters was nothing new: to the Romans, an ordinance or law was called a rubric, because it was written with vermilion (rubrica), unlike Rome's praetorian edicts and rules of court. In England, the term 'rubric' came to mean those liturgical directions and titles printed in red, and we find Milton writing in Paradise Regained:
No date prefix'd
Directs me in the starry rubric set.
Once again, though, we have strayed into the era of printing. Back in the time before the printing press and movable type, the monks used a special pigment called minium to make the red letters of the rubrics. This word actually had two meanings: it could be vermilion, mercuric sulfide, or it could be red lead, an oxide of lead written Pb3O4 by the chemists. They also used minium to add small drawings to the manuscript.

So when a monk miniated a manuscript, he wrote on it or painted around its borders with minium, and the result was a miniature. Today, we think a miniature is just a small version of something, perhaps because it sounded like other words, including minor and minimum, but by 1716, painting 'in miniature' meant creating a very small painting, often on ivory or vellum, and by the time Napoleon was defeated, the term meant on a small scale — very much the meaning we would understand today.

You could also miniature something when you embellished it with miniature portraits, but sadly, the day of the miniator was closing, for once photography took off, a major part of the miniator's business went west.

Now people could pop into a studio, sit uncomfortably still for a few minutes, and then order as many prints as they wanted.

Oddly enough, there is another mini word with a link to lead. The Minié ball, a type of bullet which expanded in a rifle barrel to make a tighter fit, and named after a captain of that name in the French army in 1853. The bullet went faster as a result, and made a hole in the target which was far from miniature in size, though it would undoubtedly have been miniature in colour.

Thursday, 19 June 2014

The caddy


I have been away (NOT playing golf, but messing with reptiles, swimming with hammerheads and stuff like that, as will be revealed in the weeks to come). Anyhow, service has been resumed with this piece from my bottom drawer.

The caddie or caddy, today, has one meaning only for most people, and that refers to the offsider of a golfer, the person who carries the clubs, and who, if the caddie is any good, provides local advice to the golfer.

It is a Scots word, brought into English when golf was given to the English, apparently as a revenge for Culloden. The English passed golf on to the Americans in revenge for certain events in the 1770s, and the Americans later used it as an object of retribution against the Japanese over certain events in December 1941 and thereafter, and the Japanese are still looking for somebody they can drop it on.

Along the way, even though a number of golfing terms travelled with the game, some of them lost their other contexts. The caddie was originally any kind of lowly servant, and the caddies of Scotland included errand-boys, odd-job men, and chairmen.

Of course, these chairmen were not the grand lords of the board, the rulers of the roost that chairmen are today: these were just the chaps who carried the sedan chairs when people wanted to get from one place to another without exertion, and without getting muddy or wet.

The caddie, in fact was just a Scots form of the English word that could be either a cadee or a cadet, and generally, this implied a young person, but became the name given to all sorts of servants. The original cadet was a younger son, and the word came through Provençal, where it was a capdet, with this in turn coming from capitello, a diminutive form of the Latin caput, a head. So a cadet was a minor head of the family, but then the term was corrupted and downgraded.

Further east, a cadi in Arabic-speaking countries was an important person, a civil judge, and further east still, the cadet turned up, either as a young trainee in the East India Company, or as a young man who enters the army without a commission to learn the arts of war.

Just a little bit further east, the speakers of English came upon an entirely different form of caddy, as is often indicated by the spelling. The Malays used the kati, usually described as a weight of one and a third English pounds, or as 625 grams, which does not sit well with the first definition, nor with the alternative translation of 1 lb, 5 oz, 2 dr. Still, whatever the equivalence may be, the kati was widely used in Asia, and commonly divided into 16 of a smaller unit, the tahil, and if opium was being sold, this unit was divided into 10 chi.

The English, of course, did not partake of opium themselves, or kept quiet about it if they did, but economics had forced them to deal in opium, to avoid a nasty balance of payments problem, because the English were absolutely besotted with tea.  They had it in the morning, they had it in the afternoon, they had it with meals, and it was costing England a small fortune, all of which was ending up in Chinese coffers, so the English started trading opium for tea.

In return, they courteously kept on buying tea by the boat load, rushing it back to England in clippers, the fastest and most beautiful sailing ships ever, where it would be sold in the little wooden boxes that it had been placed in, each holding a kati of tea.

And so we got the tea caddy, a small container which in Australia and England, usually was made of tinplate or thick crockery, held a bit more than a pound of tea, and usually featured some royal occasion or other that was deemed worth celebrating.

And as if to remind themselves of the origins of it all, in some unconscious way, the English lower classes still speak of "a cup of cha", where the Chinese name for tea is cha, and appropriately, we all call the crockery 'china'.

There is another sort of caddy that we hardly ever hear of any more: 'caddy' is also another name for a ghost or bugbear, but it would be hard to say quite why. It just is.

Friday, 30 May 2014

Postcards from South America

Herewith some random scenes...

Jorge Luis Borges and I had a meeting today at Cafe Tortoni in Buenos Aires. He waxed on, but I was sincere, in the etymological sense. Work it out.

Frutilla is a little-known monster which emerges at night from the Tigre Delta to hunt down vegans. As the vegans have learned to fight back, dealing it a deadly blow with a steak through the heart, Frutilla now has its infective viral particles smuggled onto breakfast tables in jars that we must not open. At least, that is what an old Indio lady told me for a small fee.
















A cayman in the river above the Iguazu falls.








A nearby catfish.













Butterflies dito. I don't think the cayman or caiman got fat on THEM.








A coatimundi, a raccoon relative, same area. People are warned not to feed thrm.






Me near the falls.

















Chris ditto. It looks as though she is in front of a backdrop because I used flash-fill.










The falls themselves...
















The Parque das Aves. We could not find the Parque das Avenots





More sunsets and stuff later...





















































Saturday, 24 May 2014

Triangular numbers

Background

When I can't sleep, I do calculations in my head: always have done, always will.  I might try to get the cube root of 17, or the successive powers of three, or other number patterns, like sets of three primes in arithmetic progression.

This story has its beginnings with a boring plane trip when I was playing with the triangular numbers, 1, 3, 6, 10, 15, 21 . . . which are generated by 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, 1+2+3+4+5+6...

These are called triangular numbers because you can make them up into neat triangles like this:
                        *
              *        **                                                  
      *      **      ***
*    **    ***    ****
1    3     6         10  and so on                                                                                                                     
There are several really cute things about these numbers that were known, way back in the days of Diophantus, an Ancient Greek who liked playing with really big numbers. Every perfect square is the sum of two consecutive triangular numbers, as you can see if you plot the square numbers as asterisks or dots like the diagram above. With a bit of effort, you can use a similar method to prove that every odd perfect square is of the form 8T + 1, where T is a triangular number.

The original problem

It occurred to me that the 8th term in the series of the numbers in the series was 36, which is a perfect square, and then I wondered if there are any other numbers in the series which are perfect squares, and if so, how you could generate them. It was late at night, I mistakenly calculated the value of T(15) as 121 (it is actually 120), and so it looked worth trying, but I went to sleep before I found any more.

Next day, I realised my error, and since I suspected that there would not be many numbers fitting the pattern, I created a spreadsheet that would generate them, knowing that term number T in the series is given by

T(n) = n * (n + 1) / 2, which means that term 11, for example, is 11 * 12/2 = 66

The Spreadsheet


Diophantus would have killed for even a simple spreadsheet program. Here is my spreadsheet solution to test for numbers that fit:

A1 is the starting number of the terms to be explored (in the first run, that will be 1).

A2 =A1 + 1

B1 =(A1*A1+A1)/2

C1 =IF(SQRT(B1)=INT(SQRT(B1)),1,"")

B2 and C2 are filled down from B1 and C1, which has similar formulas. Column A is copied down from A2.

I then fill down columns A, B and C to row 50000, and insert the value 1 in cell A1 to test the first fifty thousand terms. To get the next 50,000 terms, I type 50001 in cell A1, and so one.

A "hit" is indicated by the value 1 appearing in column C in the same line as the hit.

I found it a nuisance searching for the hits, so I created 20 cells in column D to help me find them:

D1 =sum(c1:c2500)

D2 =sum(c2501:c5000)

and so on to D20 -- this told me roughly where the hits were.

Then to eyeball for ANY hits in a block (they get fewer and fewer) I created E1 =sum(d1:d20), then all I had to do was insert my starting values 50001, 100001, 150001 etc in A1 and trawl through the terms, zeroing in on a specific 2500 rows to search for the hit when one was indicated in cell E1 and column D. Once I learned to calculate the approximate value, I could jump almost to the correct row, but that calculation is jumping ahead of the story.

The Results

What I had now was a curious pattern, where the terms that were perfect squares went like this:

The Mysterious Constant

About this time, I started to see some patterns. I found that these term numbers appear to be in a geometric progression with the ratios of two successive term numbers converging. Leaving out the first couple of terms of the converging series, I found these ratios between the key numbers:

5.87755102040816, 5.83680555555556, 5.82986317668055, 5.82867346938776, 5.82846938954150, 5.82843437620146, 5.82842836889813, 5.82842733820888, 5.82842716137060, 5.82842713102994, 5.82842712582431 and 5.82842712536459

So I wrote to an email list with  a few number-players in it, drawing attention to the curious patterns that were emerging, and asking for help. I ended by noting that the ratios of the square roots of the successive terms that are perfect squares rapidly converge and stabilise on the value 5.828427124746190, which I thought might give a hint about the value being converged upon above!

It was not long after that Ben Morphett told me that my mystery number was explicable:

"Your number 5.828427124746190 is equal to (3 + 2.sqrt(2)), and is the solution to the quadratic x^2 - 6x + 1 = 0"

I still don't know why, but I suspect that this relationship is just a coincidence, because somebody else managed to explain a lot of it for me in a different way — but that may just prove that I haven't looked hard enough.

But before we get to that, though, I commented to the list that the square roots of the perfect squares I had generated appeared relatively uninteresting, but Bruce Harris saw otherwise. My factorisation missed the point, he suggested:

" . . . if instead you factorise them into just two factors, allowing the factors themselves to be composite numbers, an obvious pattern emerges:

1 = 1 * 1

6 = 2 * 3

35 = 5 * 7

204 = 12 * 17

1189 = 29 * 41

6930 = 70 * 99

In each line, the first of the two factors is the sum of the two factors in the previous line, and the second is this number plus the first of those two factors again. Thus in the fourth line, 29 = 17 + 12, 41 = 29 + 12; in the fifth line, 70 = 41 + 29, 99 = 70 + 29."

He is right in this. Here is a table which extends this — the rest you can do yourself.



So what is the story with that strange factor? Tony Morton explained it to us in full, but I will reveal less than all of his solution, to leave you room to do some discovering yourself. Before I go to that, a few points:

Some hints


A famous mathematician called Gauss once was given, as a young boy, the task of summing the numbers from 1 to 100. He realised that if he added the first and the last terms in the sum, he got a total of 101, and the same if he added the second and second-last, or the third and the third-last: in all, there would be 100/2 instances of 101, making the final result he needed 5050. In fact he had multiplied (n+1) by n/2. Remember that.

    288 may be 2*12*12, but it is also 2*9*16, while 9800 can be 2*70*70 OR 2*49*100. Remember that.

    Every triangular number can be expressed either in the form n/2 * (n+1) OR n * (n+1)/2. Remember that.

    Geometrical proofs are possible for many problems with triangular numbers. Remember that.

You are now equipped to construct a proof which explains why the patterns arise, but you may not be able to explain that factor of 5.828427124746190 from this. Then again, maybe I have missed something :-)

The Morton Solution


So here is part of Tony Morton's solution, enough to get you thinking, and which does explain that factor. I have changed the notation to make it consistent with what has already appeared here.

If we want to know the nth triangular number - the sum of the numbers 1 through n — then we can use the formula T(n) = n (n + 1) / 2.

But suppose we are given a triangular number and want to know which one it is, to calculate n given T. The above formula can be written as n^2 + n - 2T = 0 which we can solve to get

n = (sqrt(8T + 1) - 1) / 2.

So for example, given T = 36, I can apply this formula to deduce that it is the 8th triangular number. So far so good. But if we look at this formula more closely we see it implies that if T is any triangular number, then 8T + 1 is always a perfect square. More explicitly, we can rearrange to get

8T + 1 = (2n + 1)^2.

Now, we are interested specifically in those triangular numbers T which are also perfect squares themselves. If we set T = m^2 for some (undetermined) integer m, then

8m^2 + 1 = (2n + 1)^2.

There is accordingly a one-to-one correspondence between the triangular numbers that are squares and pairs of positive integers (n,m) that satisfy the above equation. That is, given any square triangular number we can find n and m, and given n and m solving the equation we can find the corresponding square triangular number. So the problem of finding square triangular numbers is equivalent to that of solving the above equation.

Now, if n is a positive integer then so is 2n + 1, and if m is a positive integer then so is 2m. So let x = 2n + 1 and y = 2m to obtain

2y^2 + 1 = x^2

or x^2 - 2y^2 = 1.

Any solution in positive integers (x,y) to this equation gives us a solution in positive integers (n,m) to the earlier equation provided x is odd and y is even. It turns out, however, that every solution (x, y) satisfies this property. To see this, note that x^2 has to be odd because it is the sum of an even number (2y^2) and an odd number (1). Since x^2 is odd, x must be odd. But the square of any odd number leaves a remainder 1 when divided by 4, because (2k + 1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1. In mathematical jargon, x^2 is 'congruent to 1 mod 4'. If y were also odd, then y^2 would be congruent to 1 mod 4, and x^2 = 2y^2 + 1 would be congruent to 3 mod 4; a contradiction. Thus x is odd and y is even.

The equation x^2 - 2y^2 = 1 is known as Pell's equation. We thus see that every solution to Pell's equation gives a square triangular number, and conversely every square triangular number is obtained from a solution to Pell's equation. Given (x,y) we can calculate the triangular number T and its index n as

T = (y / 2)^2, n = (x - 1) / 2.

How to solve Pell's equation? Well, it's fairly easy to find the smallest solution: x = 3, y = 2. This corresponds to the trivial case n = 1, T = 1, which is obviously a square. So we'd like to find some larger solutions. It turns out that we can generate larger solutions from the smallest solution by the following trick. Write

1 = x^2 - 2y^2 = (x + y sqrt(2)) (x - y sqrt(2))

and raise this to an arbitrary power k to obtain

1^k = 1 = (x + y sqrt(2))^k (x - y sqrt(2))^k.

Now, given integers x and y, (x + y sqrt(2))^k can be written in the form A + B sqrt(2), where A and B are integers, and similarly (x - y sqrt(2))^k = A - B sqrt(2) by symmetry. We therefore have

1 = (A + B sqrt(2)) (A - B sqrt(2)) = A^2 - 2B^2.

So the numbers A and B, obtained from the kth power of x + y sqrt(2), provide a new solution to Pell's equation. Setting k = 2, 3, 4, 5 and so on through all the positive integers thus generates an infinite number of solutions to Pell's equation, hence an infinite number of square triangular numbers.

Thus, using the smallest solution x = 3, y = 2 we get (using Q to represent sqrt(2))

(3 + 2Q)^2 = 17 + 12Q x = 17, y = 12, n = 8, T = 36

(3 + 2Q)^3 = 99 + 70Q x = 99, y = 70, n = 49, T = 1225

(3 + 2Q)^4 = 577 + 408Q x = 577, y = 408, n = 288, T = 41616

and so on. It turns out that the powers of 3 + 2 sqrt(2) generate _all_ the solutions to Pell's equation in this way. Because, if we have any solution (A,B), we can always find C and D such that

A + B sqrt(2) = (3 + 2 sqrt(2)) (C + D sqrt(2))

and C, D are both positive, with C smaller than A. (Proof left to the interested reader.) If C is larger than 3 we can repeat this process with (C,D), and continue until we arrive at the smallest solution (3,2) (which we must do because C is getting smaller at every step). We thus obtain A + B sqrt(2) as a product of factors 3 + 2 sqrt(2).

***********************

We will leave Tony's proof at that point, but you are well on the way to the full explanation.

Possible enquiries


What is left for the honest enquirer, the interested reader, aside from testing the effects described, or seeking a new proof? Quite a lot, actually!

For a start, you have a method that may be new: using a spreadsheet to identify the interesting cases, and you have a couple of incomplete solutions.

Wondering what else might be explored led me to look at what David Wells had to say on pentagonal numbers in his "Penguin Dictionary of Curious and Interesting Numbers".

As I suspected, my discovery was not new to mathematics: on page 93, Wells notes that "Some numbers are simultaneously square and triangular . . . they are found by using a fact already mentioned . . . the Pell equation: 8x^2 + 1 = y^2" He goes on to give a general formula:

1/32 ((17 + 12*SQRT(2))^n + (17 - 12*SQRT(2))^n - 2)

and he adds that if T(n) is a perfect square, so is T(4n(n+1)) and that there are 40 palindromic numbers below 10^7: 1, 3, 6, 55, 66, 171, 595, 666 and 3003 among them. T(2662) = 3544453, and T(1111) and T(111*111) are also palindromes.

For every triangular number T(n), there is an infinite number of other triangular numbers T(m), such that T(n)*T(m) is a perfect square.

Also (T(n+1))^2 - (T(n))^2 = (n+1)^3, so the sum of the first n cubes is the square of the nth triangular number.

And (T(n))^2 = T(n) + T(n+1)*T(– 1)

And 2 * T(n) * T(– 1) = T(n^2-1)

The sum of the reciprocals of the triangular numbers converges on 2

The triangular numbers 16 and 21 have triangular numbers as both their difference and their sum, and there are others.

The triangular number 6 is said to be the only example under 660 digits whose square is also a triangular number (that sounds like a challenge to find the next one!)

And by the way, 1189 = (204 * 6) - 35

Pentagonal numbers


Pentagonal numbers are generated like this:


The numbers in the series are 1, 5, 12, 22, 35, 51, 70 . . . and there is a general formula which generates the whole set. What is it?

(Interestingly, the first factor in the general formula for triangular, square, pentagonal, hexagonal, heptagonal, octagonal numbers and so on is always 1/2n, and the second factor shows a fascinating pattern, which I leave you to discover when you work out the other formulae, but here are the first few numbers in a few of the polygonal series.)

Polygonal numbers
Polygonal numbers are generated by drawing similar patterns but with more sides, as seen in the figure above. If you draw them correctly, this is what you will get:




    Aside from the trivial cases of 0 and 1, there is another number under 500 which is both triangular and pentagonal, and there is another one under 50,000: are there any others? Can a computer be set the task of looking for these?

    There are three numbers under 2 billion which are at once triangular, pentagonal and hexagonal.

    There is at least one number under 1000 which is both square and heptagonal: are there any others?

    The 49th triangular, square (obviously!) and heptagonal numbers are all perfect squares: is this a record? (Excluding the trivial answers 0 and 1, of course, I have found nothing greater than three so far, but I have found a case where the triangular, square and octagonal numbers are all perfect squares. Maybe I should have tried nonagonal numbers . . .)

    There is an obvious link between triangular numbers and hexagonal numbers. Do any other sets of polygonal numbers have a relationship like this?

    The triangular numbers appear in Pascal's triangle. Do any of the other sets appear there? Why or why not?

    Think up some others yourself by deducing the formula for a set.