What's so special about 6987952709035199279905674?

Éric Angelini came up with this yesterday:

abcdefghij... (letters are distinct digits):
a is divisible by 1 with remainder 0
ab is divisible by 2 with remainder 1
abc is divisible by 3 with remainder 2
abcd is divisible by 4 with remainder 3
etc.

The largest such integer would be 95674103:

9/1 leaves a remainder of 0

95/2 leaves a remainder of 1

956/3 leaves a remainder of 2

9567/4 leaves a remainder of 3

95674/5 leaves a remainder of 4

956741/6 leaves a remainder of 5

9567410/7 leaves a remainder of 6

95674103/8 leaves a remainder of 7

Much more interesting is if we dispense with the "distinct digits" limitation. Let us allow any digit. The largest integer is now 6987952709035199279905674:

6/1 leaves a remainder of 0

69/2 leaves a remainder of 1

698/3 leaves a remainder of 2

6987/4 leaves a remainder of 3

69879/5 leaves a remainder of 4

698795/6 leaves a remainder of 5

6987952/7 leaves a remainder of 6

69879527/8 leaves a remainder of 7

698795270/9 leaves a remainder of 8

6987952709/10 leaves a remainder of 9

69879527090/11 leaves a remainder of 10

698795270903/12 leaves a remainder of 11

6987952709035/13 leaves a remainder of 12

69879527090351/14 leaves a remainder of 13

698795270903519/15 leaves a remainder of 14

6987952709035199/16 leaves a remainder of 15

69879527090351992/17 leaves a remainder of 16

698795270903519927/18 leaves a remainder of 17

6987952709035199279/19 leaves a remainder of 18

69879527090351992799/20 leaves a remainder of 19

698795270903519927990/21 leaves a remainder of 20

6987952709035199279905/22 leaves a remainder of 21

69879527090351992799056/23 leaves a remainder of 22

698795270903519927990567/24 leaves a remainder of 23

6987952709035199279905674/25 leaves a remainder of 24

Oaklands

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This is my first-year high school class of 1964/1965. Last Saturday, one of these teenagers hosted a high school reunion celebration in his residence at 50 Yorkville Avenue. If one were to look roughly northwest from the top of that building, they might see something like this:

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I have marked the De La Salle college location in yellow. This Apple maps representation is ~2018 because construction on the townhouses (just west of Del's fussball pitch) on Avenue Road is in its starting phase. Additionally, there are now townhouses to the south of the field on Oaklands Avenue. The city of Toronto did a detailed heritage property background report in 2015. Here's a closer bird's-eye view:

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There is a worthwhile article on De La Salle, written in 1980 by William Stephenson, called All Things Right & Dutiful. Somewhat surprisingly for such fare, it contains a paragraph on the once-upon-a-time corporal punishment meted out to some of the students.

I was unfortunately unable to attend this 52nd anniversary (counting from our graduating year) get-together. Many of my reasons noted here were of course tongue-in-cheek.

Top ten reasons why I am not attending my high-school reunion

#10. I don't want to belong to any club that will accept me as a member. [Groucho Marx]

#9. I'm still recovering from the last time I was talked into attending a get-together.

#8. I can't think of a bigger social evaluative threat. (Not that I care.)

#7. I am under house arrest!

#6. I flipped a coin and it came up tails.

#5. The delta-driven fourth wave (and other respiratory delights).

#4. Elbow-bump performance anxiety.

#3. My dog wasn't invited.

#2. My shoes are on their last legs.

#1. Mama told me not to come. (That ain't the way to have fun.)

3816547290

I've been looking very hard for the earliest occurrence in print of the number 3816547290 as the solution of a famous divisibility challenge. For an explanation, see OEIS A181736. David Gauld, the author of that sequence, wrote about this number way back in December of 1984 (problem 15 on page 17, here; the following month it appeared in Games magazine, page 2). I recently asked David wherefrom he got that number. He replied that it was in the 1983 book "Les nombres remarquables" by François Le Lionnais. Indeed:

There's more! A Google book search for 3816547290 comes up with a Russian hit, Nauka i zhizn' (Science and life) - page 148, supposedly 1980:

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Once I figured out the trailing numbers, this did look like a bona fide mention. The snippet view, alas, showed nothing. And Google's date assignments can't generally be trusted, so I longed for an inside-the-journal look. Playing with it (try searching for 381654), I eventually conjured up this snippet:

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Showing a '1980' at the top-right made me hopeful that this is an actual earlier posing of the divisibility curiosity than that of Le Lionnais. If that were so, it would not be unreasonable to suppose that this dissemination was responsible for Le Lionnais (being the well-connected individual that he was) coming to be acquainted with 3816547290.

A variant of the divisibility puzzle that excludes the digit 0 appeared just prior to Le Lionnais in an "Acorn User" competition in November 1982:

Acorn User: November 1982, page 71

Acorn User: February 1983, detail from page 55

The February 1983 answer to the puzzle is significant for noting that the problem appeared in the 'Brainteaser' column of the Sunday Times (in 1982), which is how David Wells credited 381654729 in "The Penguin Dictionary of Curious and Interesting Numbers" (1986). More relevant (to me) is the assertion that it was first posed in Scientific American "five years ago". Of course! That would make it Dr. Matrix (Martin Gardner) in Chautauqua (Mathematical Games, Scientific American, December 1978, page 23):

"Jaime Poniachik of Buenos Aires, who edits an excellent Spanish puzzle magazine titled The Snark, happened to be visiting the Chautauqua Institution at the time of the ASMOF demonstration. When it was his turn to ask a question, he said he had a friend in the U.S. with a curious social security number. Its nine digits include every digit from 1 through 9, and they form a number in which the first two digits (reading from left to right) make a number divisible by 2, the first three digits make a number divisible by 3, the first four digits make a number divisible by 4 and so on until the entire number is divisible by 9. What is the number?"

Gardner's solution (Mathematical Games, Scientific American, January 1979, page 24):

"The social security number is 381-65-4729. Adding 0 at the end gives the unique solution to the same problem with the ten digits from 0 through 9."

So 3816547290 was implicitly noted as of this date and the explicit Russian mention a year or so later is no longer a surprise. In "The Magic Numbers of Dr. Matrix" (1985), Gardner cleans up the ambiguity of his "ten digits" solution by adding "..., and the ten-digit number divisible by 10." There is considerably more in the book but I will quote only a couple of paragraphs:

"Michael R. Leuze worked with a computer to examine solutions to this problem in number systems other than base 10. He found that there are no solutions in any odd base or in base 12. In base 2 there is the trivial solution 1. In base 4 there are two solutions, 123 and 321; in base 6, 14325 and 54321; in base 8, 3254167, 5234761, and 5674321. In base 14, as in base 10, there is a unique solution: 9 12 3 10 5 4 7 6 11 8 1 2 13. Leuze conjectures that there are no solutions in higher bases."

"Leuze sent a thorough analysis of the problem in negative bases. For other generalizations, see 'Progressively Divisible Numbers', by Stewart Metchette, in Journal of Recreational Mathematics, Vol. 15, No. 2 (1982-83), pages 119-22. The problem surfaced again in 'What's In a Number?', Mathematical Gazette, Vol. 67 (December 1983), pages 281-282. The editor added in a footnote that when the problem appeared in London's Sunday Times in 1982, irate readers made the same careless mistake some of my readers did — they complained that the answer was not unique because they had worked on the problem with an eight-digit calculator!"

Finally, Gardner reveals the true author of the puzzle:

"Jaime Poniachik (the person who posed the problem in my column) told me the problem had been invented by his wife, Lea Gorodisky, who worked with him on the Snark."

Spirit talker

Canada's Aboriginal Peoples Television Network (APTN) premieres tonight season 2 of Spirit Talker, "a documentary series that will follow Mi’kmaq medium Shawn Leonard as he travels from coast to coast using his psychic abilities to connect the living with the dead and bring hope, healing, and closure to indigenous communities." I've watched all of season 1 so I will certainly watch at least this first episode, not so much for what Shawn says and does (which is all fairly predictable, tired old shtick) but more for enlightening glimpses of behind-the-scenes. CTV News Atlantic (Halifax NS) did this promo piece (Keeping Up with Katie Kelly) introducing the series:

Katie: When you watch the show, you you [unintelligible] ... You know so much. People are shocked by what you say. Is it real? Is it all real?
Shawn: It's absolutely 100% real, yea. This isn't like a scripted show in any way. This is like... They just put me in front of people and let me do my thing.

In spite of being unscripted, Shawn's on-site production team includes some seven other people whose bread and butter depends on how good they can make Shawn look. Shawn himself is not of course an unwitting pawn of the deceased. His art, such as it is, likely goes back to his 1995 marriage to Melissa McClinchy who brought a dabbling in pseudoscience to their relationship (which lasted until 2014). Their paranormal pastime had become a public persona focus for Shawn as he strove* to incorporate his Indigenous ancestry into the act, dropping (for example) his Beaupre surname from his online identity. In 2015, Shawn celebrated an Indigenous wedding to a Bonny Martell but this ended bitterly a mere eighteen months later. Around this time he may have made "Shawn Leonard" his legal identity. The current status of his subsequent relationship to a Michelle Belanger (not the occult author), touted at one point as his fiancée, is unknown to me.

Perhaps a good indicator of Shawn's reach is this photograph ...

... taken from his Facebook header, impressive for the sheer number in his pre-pandemic Spirit Talker "Tribe" online course group and (of course) their gender bias. I have mentioned Shawn once before in my blog, here. Apparently season 3 of Spirit Talker (with an Ontario focus) is already half-way done.

* Shawn is still striving. In the Katie Kelly interview he states: "I'm a very proud Mi'kmaq person." Then, as if to justify the assertion: "I never grew up on reserve. I grew up near to the reserve in Shubenacadie, Nova Scotia." This must be his stock response when folk on the reserves that he visits ask him which reserve he is from.

Hurricane Larry

Hurricane Larry landfall @ 1:20 AM, local time

This image shows the "Larry makes landfall on Newfoundland" advisory location as we moved this morning into the 20th anniversary date of the 9/11 attacks. It is not lost on me that on the attack's 9th anniversary, 11 years ago (also a Saturday), my brother-in-law Larry passed away. The hurricane's landfall is 37 km east-southeast of St. Bernard's.

A CBC news story on the hurricane mentions a "statement" that said "Larry made landfall at 1:30 a.m. NT just west of Long Harbour, N.L., on the Avalon Peninsula." I have no reason to suppose that my National Hurricane Center landfall location is incorrect, so the "just west of Long Harbour" translates to "50 km west of Long Harbour". That's on the other side of Placentia Bay at that latitude, on the Burin Peninsula. So not really a very good description. I'm aware that the official location coordinates might themselves be subject to error bars (especially this far north when much of the eye wall will have dissipated) but that is an entirely different discussion.

CTV's (Canadian Press) version of the event has Larry "arriving near South East Bight around midnight local time, according to the National Hurricane Centre in Miami." South East Bight is very close to the landfall coordinates and when one is looking for a location place-name, it will do. But what about the "midnight local time"? The NHC's summary has "1150 PM AST" (Friday) ... "0350 UTC" (Saturday). Why did the NHC use AST? Likely because that (currently) is Miami time. Technically Miami is in the EDT time zone, but Eastern Daylight Time = Atlantic Standard Time. Coordinated Universal Time is 4 hours ahead of Atlantic Standard Time. But right now, Atlantic Canada is observing Atlantic Daylight Time so landfall would have been at 12:50 AM ADT. Furthermore, Newfoundland Time is an additional half-hour ahead, so 1:20 AM local time. What made the Canadian Press think that AST was local time? It's an easy mistake to make if you have no patience for diligence.

My 1500th Leyland prime find

This morning I found my 1500th Leyland prime. My 1000th was only eight months ago! I have 579 red values found using Mathematica (from 3 October 2015 to 3 July 2020), followed by 921 blue values found using xyyxsieve & pfgw64 (from 6 July 2020 on). Compared to my graph for the 1000th, everything is compressed: horizontally by the addition of eight months, vertically by the addition of eighteen integers ranging from 300000 to 301441 decimal digits and two more composed of 386548 and 386561 decimal digits.

Ad war

I had been fighting a battle with Google ads for I don't know how long. If it was an animated gif and had an x in the top-right corner, I would smash that x, click "send feedback" and choose "not interested in this ad" from the four options.

When I initially started this, I chose "ad was inappropriate" to more properly indicate my feeling that anything animated was an annoyance — distracting me from reading the page — but the nuance was always that there was something more politically incorrect about the ad and I gave up on the sentiment. Sometimes, after I had flagged a handful of identical such ads on the same page with "not interested", I would finally do a "seen this ad multiple times". And in spite of Google assuring me that it would try to stop showing me these ads, I would in fact see the same stupid ads from the same advertisers again and again.

Early this morning I finally gave up on the experiment. The only noticeable change over all these many months had been that I was seeing more ads for women's apparel. So much for artificial so-called intelligence. I'm now using the Magic Lasso ad-blocker extension in Safari and all I can say is: Why didn't I do this a long time ago?

Opening day

The FreshCo that (two months ago) I anticipated opening, opened today. I was there just after 9 AM checking it out. I had washed my hair and changed my hole-riddled T-shirt so that folk wouldn't mistake me as a harbinger of a zombie apocalypse. It was the first time in over seventeen months that I have been inside a grocery outlet! I picked up a couple of 2L milks, a couple of 750G yogurts, one head of lettuce, a chunk of beef, and a few smaller items. These were distributed in two bags that I had brought to the store. The ten-minute walk home was somewhat tiring, my back apparently no longer suited to such a task. One sour note (noticed after I got home) was that the box of Milk-Bones I purchased only because it was shelf-tagged at $1.25 (a very good price) showed up as costing $2.49. The $5-off opening special (on orders over $35) made up for that annoyance.

The Huron and Ontario ship canal

After the Frederick Moffat house aside in my last blog article, I wondered about other early settlers in my neighbourhood, in particular those on my street: Sykes Avenue. Additionally, I wondered about the street name's origin. A possibility soon emerged: William Sykes (1815-1872) who in his later years became resident engineer of the Huron and Ontario Ship Canal Company in Toronto. An 1869 prospectus of that venture is made available by Kingston's Queen's University, here. A more-than-interesting aspect of the proposed canal was its debouchure near Toronto:

"The Southern division will commence in the Humber Bay, at a point about 25 chains west of the mouth of the Humber River, where it is proposed to make three lift locks in the solid ground, elevating the canal 45 feet and carrying it, by means of a stone aqueduct, over the Lake Shore Road, Great Western Railway and side road. From the Northern extremity of this aqueduct, the canal will be continued, through the solid ground, to the village of Lambton, a distance of three miles, where it will join the river and receive its water supply. Following the valley of the Humber River for about three miles farther, to Weston, it will there intersect the Grand Trunk Railway, which is carried over the valley of the Humber by a viaduct about 60 feet above the water level. There the canal will be brought, by the necessary excavation in the bed of the stream, underneath the Railway, at a level to afford 100 feet clear headway for the passage of masts of vessels; its elevation to the natural level being effected, after passing beneath the viaduct, by three lift locks."

It is difficult to imagine the plan's impact on Weston had it ever come to pass. How might history have been changed, for example, in light of the flooding caused by the hurricanes of 1878 and 1954? This section of the canal's meander ends up in Lake Simcoe, called "Lac Taronto" by early French map-makers who appropriated an Aboriginal word referring to the area between the eastern part of Orillia and Atherley. The proposed canal's waterway route from Toronto to Lake Simcoe (shown in red) mirrors in part a supposed Toronto Carrying Place trail.

There exists a perceptual mechanism known as "figure—ground": We focus on the former to the exclusion of the latter. I believe now that the Carrying Place trail is such a figure, the ground (ironically) being its nearby waterways. This figure has been honed by far too many occasions of uncritical thinking, relying largely on the portion of the trail (from Bloor Street to Weston) with which we are most familiar. I gave a hint of the problem that this entails in my 2017 article on Weston's origin "across the river":

"There's another plaque at the northwest corner of Little Avenue and Weston Road ... detailing — in a map — the 'Toronto' Carrying Place, a portage trail running from the Humber River at Bloor Street to the Holland River, a considerable journey. You may be forgiven if you think that the dashed line segments represents an actual path, but in fact we know very little about the journey. The trail is just a supposed connection between the start and end points, ignoring the many places along the way where a canoe might sensibly be set in water."

In fact, the trail isn't the figure; it's the ground. The waterway route is the real figure! The supposed "trail" is really just a bunch of portage elements, of which the stretch from Bloor Street to Weston misleads one into thinking that this is all about walking, not canoeing; that this is all about a land route starting in Toronto, not a waterway route ending in Taronto.

Update: I have discovered that the Huron and Ontario ship canal is a decade-later rehashing of the Georgian Bay canal proposed by R.B. Mason and Kivas Tully in 1858. The chart/map "shewing the several proposed routes of the Toronto & Georgian Bay ship canal between the city of Toronto and Lake Huron" is from their survey. Embarrassingly, what I had taken to be the figure of Indigenous roads in the Huron & Ontario ship canal folded-map segment (which lacked the legend) turned out (by reference to the Toronto & Georgian Bay ship canal map) to be the ground (ridges) of watersheds. It is actually an interesting correspondence: The ridge of a watershed is high terrain that encounters no flowing water. This is ideal for a no-maintenance transportation corridor!