Wednesday, April 2

Magic multipliers

1, 1, 3, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 9, 1, 1, 3, 1, 1, 1, 9, 3, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 30, 3, 2, 3, 39, 1, 1, 1, 6, 3, 34, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 9, 1, 1, 1, 2, 1, 37, 3, 9, 6, 1, 8, 1, 1, 2, 1, 3, 2, 10, 1, 1, 11, 19, 3, 1, 1, 1, 1, 2, 1, 1, 7, 1, 47, 3, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 8, 3, 1, 24, 1, 3, 10, 1, 1, 1, 1, 9, 3, 3, 13, 1, 6, 1, 21, 10, 9, 7, 3, 1, 1, 1, 7, 2, 3, 19, 1, 1, 1, 6, 1, 1, 2, 37, 1, 14, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 8, 3, 1, 7, 1, 3, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 40, 3, 1, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 8, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 4, 3, 1, 1, 7, 9, 1, 1, 1, 1, 60, 3, 1, 1, 1, 4, 3, 1, 2, 3, 6, 1, 3, 1, 10, 1, 1, 1, 52, 1, 1, 13, 1, 1, 3, 1, 19, 1, 1, 2, 31, 3, 1, 1, 1, 10, 3, 1, 2, 1, 1, 1, 29, 1, 1, 1, 3, 1, 1, 1, 3, 3, 110, ...

This sequence looks superficially like a simple continued-fraction expansion of some constant, but it's not. There are no numbers (I conjecture) ending with the digit 5. How did I arrive at my sequence? They are the magic multipliers (the final number on each line) in the following:

n =  1:  (37+1)/2 =  19*1
n =  2:  (73+1)/2 =  37*1
n =  3: (113+1)/2 =  19*3
n =  4: (149-1)/2 =  37*2
n =  5: (157+1)/2 =  79*1
n =  6: (193+1)/2 =  97*1
n =  7: (269-1)/2 =  67*2
n =  8: (277+1)/2 = 139*1
n =  9: (313+1)/2 = 157*1
n = 10: (353+1)/2 =  59*3
n = 11: (389-1)/2 =  97*2
n = 12: (397+1)/2 = 199*1
n = 13: (457+1)/2 = 229*1
n = 14: (557+1)/2 =  31*9

The initial number on each line is A135952(n). Note that if the magic multiplier is odd, we add 1 before dividing by 2; if it is even, we subtract 1 before dividing by 2. The number immediately after each equal sign is the prime p where A135952(n) divides composite Fibonacci(p). In my sequence of magic multipliers, the first occurrences of the positive integers are at indices:

1, 4, 3, 157, 0, 24, 91, 71, 14, 78, 81, 802, 124, 149, 0, 720, 436, 292, 82, 347, 128, 389, 598, 113, 0, 1245, 454, 1728, 270, 39, 258, ...

Monday, March 24

My Chicks

When Catherine and I first visited in 2007 the grave of her great-uncle George Hennessy (master-at arms on the Mauretania in 1914), I had just some months prior begun my research into her family's genealogy. The header My Chicks was certainly an odd thing to see on a gravestone! In a blog entry written 13 months later, I rationalized it by suggesting that it was biblical metaphor for My Children.

Of course chicks was plural and George wasn't about to be included in the reference — if that's what it was — for another 37 years, during which time he fell in love with, married, and survived another woman, Annie (Harriett Edwards) Hall. It was Annie's second marriage and when she died in 1963, she was buried in a different cemetery with her "devoted husband", William Edward Hall, presumably because that grave was purchased as a double plot, just as George had allowed a spot for himself beside his beloved, Mary (Bennett) Talbot.

Poor George. He had spent easily twice as long with Annie than he had with Mary and there is little doubt that in the years preceding his own demise he dwelled more on her than on his previous partner. But he did now also think about something else. Prior to November 1919, when he and Mary Talbot left Liverpool for Toronto on the S.S. Grampian, he was married to a Hannah Sanwith Gaddes and fathered with her some half-dozen children. His affair with Mary Talbot was unknown to them and they had no idea — after he disappeared — what became of him.

After Annie died, George went back to England to see if he could find the remnants of his family. Unsuccessful, he paid for a memorial headstone to his parents and a brother, before returning back to Canada. Perhaps George was better off not knowing what had happened. His grandson, Derek Hennessy, informed me: "Grandmother Hannah had to put two sons, George and Ralph, into the Fazakerly homes, an orphanage in Liverpool. She took my dad, Billy, Sam, and Richard, the youngest, to live in Sheffield with her sister, as an alternative to the workhouse in Liverpool. George went to Australia as a boy and Ralph, after working in the south of England all his life, finally joined his daughter in New Zealand on retirement."

Last week I discovered at Catherine's father's house a silver cigarette case. On it was inscribed: "From Chicks to George With love XMAS 25/12/18." It appears that Chicks was Mary Talbot's nickname!

Friday, January 31

Mac Pro

An exciting day for me as my Mac Pro arrived! After unpacking, the first challenge was finding that little booklet that accompanies Apple products. I opened up the Pro (the outer casing just slides up) and pulled out of their slots the existing RAM, replacing it with new chips (ordered from OWC two days ago — arrived yesterday). I wasn't sure how I was going to hook it up but, since I had an extra HDMI cable kicking around, I ended up just connecting it to my Samsung TV (for now). I got a ten-year-old Apple extended keyboard and a Kensington trackball out of storage and away I went.

When I got to the desktop, there was no menu bar. I thought this might have been an OSX Mavericks default thing, but no. After some time looking for online instructions on how to enable the menu bar it finally dawned on me that this might be a display issue. Indeed, shoving the cursor to the top of the screen and clicking I was able to engage a menu that had merely been cut off. I played with the Displays in System Preferences without success. Finally I managed to get a proper desktop by engaging the Samsung TV's display preferences — specifically the Screen-Fit option.

I spent some time updating the software. The pre-installed Keynote, Pages, and Numbers refused to update and, after a half-hour-or-so online chat with a couple of Apple employees, I was advised to apply for the Apple Up-to-Date program, which I did. Within an hour I received redeem-codes for the programs, installed them on my iMac, after which the updates on the Mac Pro proceeded without further ado.

Saturday, January 25

Divide by the sum of the base-7 digits

Consider a number divided by the sum of its base-7 digits. If it goes evenly, repeat the procedure with the quotient. Continue thus until either the quotient arrived at is not evenly divisible by the sum of its base-7 digits, or the number 1 is reached.

There are only a finite number of positive integers that reach 1, the largest being

written in base ten, or

written in base seven.

This number takes 2046 steps to reach 1. I have plotted the count of positive integers (out of a total 693633 available) that take n steps (from n = 0 to n = 2046) to get to 1 in a graph. The smaller, darker-blue plot at the bottom-left is the count of positive integers (out of a total 15095 available) that take n steps (from n = 0 to n = 440) to get to 1 in the base-10 version of the procedure.

Thursday, January 2

When adders multiply

I had occasion, yesterday and today, to do some Google Books sourcing of quotations involving snakes doing arithmetic. It's a bit of a challenge getting full quotations from snippet views but it can often be done.

This list is not meant to be comprehensive. I just wanted to put in one place a number of items more-or-less bound by the word-play under scrutiny. Here and there, I've taken the liberty to make small editorial alterations that involve punctuation. [My thoughts are in square brackets.]

Oliver Corwin Sabin, Washington News Letter, 1900.

Another writer tells us that the first man was called Adam, and a curse laid upon him by his Creator because he falsified the word of God by adding to it, and his name therefore was probably Add-am.
The writer further argues in this puerile vein that the serpent in the garden of Eden was cursed by God for the same reason, and hence we have the word adder applied to a very deadly serpent. The writer was not aware of the fact that may be learned from any good dictionary of the English language, that adder is a corruption of the Anglo-Saxon word atter, which meant deaf, and was applied to a small, venomous snake that was reputed to lie with one ear flat to the ground, and its tail in the other ear, that its sleep might not be disturbed by noises.
Any concordance to the Scriptures will inform the writer of this curious Adamic theory, that the Hebrew word Adam, has three meanings, the same being respectively, earthly, ruddy, beautiful, and it never had any reference to addition, although he being the father of all living things might well have given a name that referred to multiplication.

Ambrose Bierce, Kings of Beasts (The Snake), 1902; taken from Collected Works, 1912. 

I can tell you, though, about the snakes in the Garden of Eden, all exceptin the one which was tempted by Eve. When they had all been made, Adam he called them together and give them their names, and then he waved his arms and said: 'Now go 4th into all the waste places of the earth and multiply.'
"They all slided away only but jest one, which lay still and shook its head, real sad. Then Adam he said: 'Why dont you do as I said? Off with you to once!'"
"But the snake, it spoke up and sed, the snake did: 'If you please, sir, Ime willing to go 4th, but I cant multiply. Ime a adder. You told me so your self.'"

Brewers Journal (attributed to Baltimore American), 1907.

"I wonder why the snakes a man sees when he's been drinking multiply so fast?"
"I suppose because the kind of snakes he sees are adders."

Irvin S. Cobb, Who's Who at the Zoo, Hampton-Columbian Magazine, October 1911.

The secretary bird moons around seriously, doing sums in its head and, as if to add to the mathematical aspect of Zoo life, there are the rabbits multiplying rapidly and the great Egyptian asp which is considerable of an adder itself.
If you resent that last allusion as bordering dangerously upon a pun, do not blame me. I stole it from an English comic weekly ...

[Cobb's piece was much reprinted.]

Webster Method, 1918.

Rabbits multiply and some snakes are adders, but G.F. Butt, of John S. Metcalf Co., says, it takes a shark to figure elevator contracts.

[I don't get it.]

Journal of the American Institute of Homeopathy, 1922.

There is a story that you all probably recollect, that when Noah finally realized that his Ark was at last on the firm ground of Mount Ararat he opened the door and said to his cargo, "Now all you animals get out and go multiply." They all did, all but two snakes which did not move from their corner. Then Noah again opened the door and said, "Did you fellows hear what I said, go out and multiply." Whereupon one of the snakes, raised its head and replied: "We can't, Noah, we are adders."

[Currently the earliest incorporation of Noah.]

International Congress on Accounting, 1933.

You remember that Noah was shooing the animals out of the ark two by two, and saying to them in a helpful spirit, "Now, get out, be fruitful and multiply" and when he thought the Ark was empty he looked round and saw that it was not cleared, because in one corner there were two little snakes. He scolded them and said, "Get out! Be fruitful and multiply"; and they said "We cannot; we're adders."

The Michigan Technic (Stresses and Strains), October 1934.

After the flood was over, Noah went back to see if all the animals were out of the Ark. He found a pair of snakes in a corner weeping copiously. When he asked them what the matter was, they answered: "You told us to go out and multiply upon the earth and we cannot, for we are adders."

[This was reprinted in a number of trade journals in the 1930s and 1940s (mostly verbatim, except 'matter' is 'trouble', which makes me doubt this is the original, but at least it has a firm date; a Google-assigned 1930 date for one of the copies can't be assumed belonging to its provided snippet).]

Proceedings of the Grand Division, 1941.

I would like to tell you a little story, in that I have referred to mathematicians. The gentleman in that corner over there a while ago touched light on Noah, the man who built the ark to save a pair of each who were threatened by the great flood that lasted 40 days.
When the waters started to rise in this flood, Noah laid his gangplank and he said, "Come ye all here and take shelter."
After the flood had subsided, Noah said to these passengers on this boat, "I want you to go into all the corners of the earth and multiply."
So after the boat was thoroughly cleaned, so he thought, all the passengers had dispersed, there were two little snakes over in the corner. Noah said, "I thought I told you all to go into the corners of the earth and multiply."
One of the little snakes said, "Well, you did, Mr. Noah, but we are adders."

Jane Hedges (to William D. Hedges), WWII Letters of Three Brothers and Their Sister, October 1942.

I just heard a silly joke. "The Lord told two little snakes, to go forth and multiply. Instead, they went off to a corner and cried. Why? Because they were adders." That's the kind of stuff that circulates around this office.

The Michigan Technic (Ambrose McHigan), April 1946.

It is widely known that rabbits and guinea pigs are some of the most rapid multipliers. The snake out-does them though. Some snakes are adders at birth. The honors, however, go to the tiny organism, amoeba. It divides to multiply.

New Zealand Law Journal, 1947.

... He told all the animals to go out and multiply. Afterwards, he met two snakes, and they were just as he had seen them before, so he told them that they must multiply. They said, 'Well, we are sorry, Mr. Noah, but we are adders.' 'That is no good,' said Noah, 'you must multiply.' He was very pleased next time he saw them that there were a lot of little snakes about them. He said, 'Oh, I see you have learned to multiply.' They said, 'No, Mr. Noah, we did it by logs.'

[If the date is correct, this is currently the earliest mention of using 'logs' to solve the adders' dilemma.]

John Raymond Shute, The Seer (So Spake the Snakes), 1950.

"On that great day when the ark gently grounded on the highlands of the mountains, our mighty ancestor threw wide its doors and bade every pair of animals to go forth into the earth and multiply, so that there might be abundant increase. Out onto the land poured the foxes, the jackals, the lions and every manner of animals that had for many days and nights remained aboard the ark. Finally every pair was upon the good earth save a pair of snakes who lingered in the shade of the deckhouse. 'What now,' roared Noah, 'go ye forth also and multiply in the earth.' But the reptiles hung their heads and did make answer, saying: 'Kind sir, this we cannot do, for you see we know naught of multiplying, for we are adders.'"

Iron Age, April 1953.

Another new joke which we printed here some time back (the one about the two snakes in Noah's Ark who couldn't go forth and multiply because they were adders) has brought forth a torrent of two letters with an engineering flavor.
Mr. D.J. Foskett wrote from London that "the story was inaccurate because the snakes were found in the middle of the forest which was then cut down so that, although adders, they could multiply by logs." And Mr. W.E. Brainard of Hughes Aircraft, California, wrote that Noah should have chopped down a tree and built a table from its logs. Adders can easily multiply if they have a log table." One more case of great minds and hands across the sea.

Bryan Morgan, Total to Date, 1953.

... in fact, were like those snakes which were loath to repopulate the world after the Flood: they said that they were adders and found it hard ...

[I can't nudge snippet-view to show me anything of this passage.]

Journal and Proceedings of the Institution of Agricultural Engineers, 1954.

You might remember the story of Noah when the Ark came to rest on dry land. He was evacuating the animals and to each one he said: "Go forth. Go forth and multiply."
He said that to the waiting Snake who replied "We can't multiply because we are adders."
Then Noah noticed Mrs. Snake with a family of little snakes.
"What is this?" he said, "I thought you couldn't multiply."
"Oh, we did that by logarithms," replied Mr. Snake.

Anatol Rapoport, The Language of Science (Wistar Institute symposium address), April 1959.

On releasing the animals from the ark, Noah bid them to go forth and multiply. Suddenly two little snakes spoke up, "But we can't multiply — we are adders." Thereupon Noah constructed a table from rough-hewn lumber and said, "Here is a log table. Now you adders can multiply."
The puns on "adders" and "multiply" are obvious. However there are two more puns, barbs aimed at the viscera of the mathematician: "log table", i.e., a logarithmic table is a device which reduces multiplication to addition.

[Rapoport's address was reprinted in Readings for Technical Writers (Blickle & Passe, 1963) and The World of Words (Kottler & Light, 1967).]

George Gamow, Matter, Earth, and Sky, 1965.

At this point the author cannot resist telling a story of what happened when Noah's Ark, loaded with animal couples of all species, finally landed at Mt. Ararat. As a lion and a lioness, a deer and a doe, a rooster and a hen, etc., were coming onto solid land, Noah was giving them instructions to spread out and to multiply. Next spring, walking around the land which was again flourishing after the Great Deluge waters receded, Noah was enjoying the sight of little lion cubs, yearlings, baby chickens, etc., running around. But then he found a couple of snakes that had no offspring. Didn't I tell you to multiply?" asked Noah in anger. "Why didn't you?" "Sorry, Sir," answered the snakes, "we can't multiply because we're adders." A year passed, and Noah, walking around his property, found under a big table (built for the family's outdoor meals) the same two snakes — surrounded by a dozen or so newly-born offspring. "You told me last year," said Noah in surprise, "that you couldn't multiply because you're adders. How is it that you managed to multiply?" "Oh, sir," said the papa adder proudly, "you see, we've found a log table!" Let us hope that this story will persuade those students who do not know how to use the log table to learn this art.

[A very verbose retelling of the pun. I include it because author Gamow had a following.]

Inland Members' Service Bulletin, December 1968.

An Adder That Multiplies. With that we've come full circle. We realize that a computer adds by adding, multiples [sic] by adding, subtracts by adding, and divides by subtracting by adding. It's a cross between a snake and a rabbit — an adder that multiples [sic]. Get it! And man can these things add!

Sarah Fullton, via Scot Morris, Omni (Games), July 1979.

To bear offspring, Noah's snakes were unable.
Their fertility was somewhat unstable.
He constructed a bed
Out of tree trunks and said,
"Even adders can multiply on a log table."

[cf. Michael Stueben, Twenty Years Before the Blackboard, 1998:

To reproduce, Noah's snakes were unable.
Seems their fertility was somewhat unstable.
  So he made them a bed
  Of tree trunks and said,
"Even adders can multiply on a log table!"

... attributed, alas, to Anonymous. There are forces afoot attempting to obscure one's 15 minutes of fame.]

Sunday, November 10

Define, divide, and conquer

I had done a table of smallest positive integer whose name has a given number (3-758) of letters and was anxious to tackle the largest-number analogue. The initial terms had a modicum of fame by being part of the Google Labs Aptitude Test (second-last item). I checked the sequence in the OEIS where it sported nine terms (3-11 letters) and the opinion that "beyond this point the terms are too ill-defined to include". So the first thing I did was provide a more tractable outcome for the sequence, accomplished simply enough by adding "less than 10^66" to the sequence definition.

I realize that a largest-number sequence limited in scope by a hard cut-off like this is somehow less natural (if words in any language could ever be considered natural) but it does no harm to the initial terms and provides a firm framework for many more. I derived the terms for 12-42 letters manually, more or less by inspection. I contributed the terms thus far as a b-file and noted the final (758 letters) number which I had previously encountered as a contribution from Eric Brahinsky on Jeff Miller's Word Oddities site, though since removed. Finally, I added the keyword "hard" because I did not see my way to solving for a lot more terms. Well, I hadn't at that point tried.

Once I did set about expanding the b-file, things just fell into place: terms for 631-701 letters were derived by brute force on my computer. Manually, I derived the terms for 757-702 letters (backwards). Then I had my computer calculate sections of 28 terms each surrounding (10^3n-1)*10^(66-3n) for an appropriate range of n. This gave me terms for 50-77, 84-111, 120-147, 153-180, 187-214, 225-252, 258-285, 291-318, 323-350, 353-380, 383-410, 413-440, 444-471, 475-502, 507-534, 539-566, and 568-630 letters.

These sections provided the anchors for what remained. Considering endings of 10*10^9 provided the terms for 43, 78, 112, 148, 181, 215, 253, 286, 319, 351, 381, 411, 441, 472, 503, 535, and 567 letters; endings of 10*10^12: 44, 79, 113, 149, 182, 216, 254, 287, 320, 352, 382, 412, 442, 473, 504, and 536 letters; endings of 10*10^33: 45, 80, 114, 150, 183, 217, 255, 288, and 321 letters; endings of 10*10^24: 46, 81, 115, 151, 184, 218, 256, 289, and 322 letters; endings of 10*10^36: 47, 82, 116, 152, 185, 219, 257, and 290 letters.

The remainder were done piecemeal, in this order: 48, 83, 117, 186, and 220 letters; 49, 118, and 221 letters; 119 and 222 letters; 223 letters; 224 letters; 443, 474, 505, and 537 letters; and finally, 506 and 538 letters.

After all that, I removed the keyword "hard" from this OEIS sequence. My table is here.

Saturday, August 31

The 1%

This propaganda blurb is courtesy the Toronto Hydro-Electric System, accompanying my most recent every-other-month bill. The 1% struck me as unusually low. Looking over the source document, the relevant fact should be gleanable from the last column of this table, but (alas) electricity is grouped with water and fuel, which (in total) constitute 3.1% of the average 2009 household expenditure, the latest available. Why use the Canada-wide table instead of the Ontario one? Because Ontario's is slightly higher at 3.3%. How does Toronto Hydro arrive at 1%? Presumably by suggesting that electricity is one third of the water-fuel-electricity expenditure. In our household (which uses natural gas for cooking and heating), electricity actually represents half of that water-fuel-electricity total.

Our entire 2009 income was just 46% of that average household expenditure suggested by Statistics Canada. We didn't, but suppose we spent every penny of our earnings that year. What fraction would have been for electricity? The answer is 3.7%.

Sunday, August 25

Square Split Subtract

There are three (base-ten) 38-digit squares that can be split (somewhere) in such a way that (the difference between the two parts)2 is the original number. One of them is the square of 3636363636363636365:

Square: 36363636363636363652 = 13223140495867768604958677685950413225
Split: (here, into two 19-digit parts) 1322314049586776860 ' 4958677685950413225
Subtract: 4958677685950413225 - 1322314049586776860 = 3636363636363636365

What are the other two solutions?

Saturday, August 3

Cooking the books

Velikovsky, von Däniken, Sitchin,
Baking pseudohistorical fiction.
Lack of critical thinking
Has their soufflés a-sinking.
There be too many cooks in the kitchen!