Wednesday, December 23

Double Noeud

The photo shows the seven pieces and cage (crafted by Brian Menold) of Gregory Benedetti's Double Noeud. First the two lower-left pieces are together (with some shifting) fitted into the cage. Then the two lower-right pieces are together (again, with some shifting) inserted and rotated into place. The three remaining pieces will now complete the cube. I like this much better than Triagonal Agony.

Thursday, December 10

The underpass

The photo (taken in 2012) shows Catherine aiming her camera at a billboard showing an artist's conception of what the local Denison Road underpass would look like once completed. Note that the completion date was predicted to be spring 2013. It was finally opened last Saturday! Some other misrepresentations on the billboard are the sloping grassy areas which have been replaced with high retaining walls and the dedicated bicycle lanes which are nonexistent. The treed landscape backdrop was always pure fiction. Here's my photo — taken yesterday — of what the thing actually looks like:

Friday, November 27

Triagonal Agony

This is another puzzle just purchased from Brian Menold. Designed by Laszlo Molnar, the six pieces form a cube. But that's not all. You have to construct that cube in a box that has only two triangular openings on adjacent faces. Just creating the cube is not easy. After some time I managed to luck into a solution but got confused trying to place a couple of its pieces into the box. Removing them from the box I found that I could no longer recreate my cube. Argh! Once I did finally construct the cube again I smartly drew myself a placement picture. I held the cube a number of ways and decided which might be the final two pieces into the box. Setting those aside I tried to position the other four pieces inside the box. I managed three only and then had a difficult time getting them out again. Extremely frustrating!


The six pieces of my just-purchased Pirouette. This is another Jos Bergmans creation crafted by Brian Menold. The two pieces in the bottom corners of the photo are together pirouetted into the structure at top-left before the other three pieces can be inserted to complete the cube. Very nice.

Friday, October 16

A senior moment

My first waking hours as an official senior citizen did not go well. Having taken Bodie for his morning walk, I was crossing the street intent on depositing his poop bag into the Denison Park garbage container across from the cemetery. I kept a wary eye on the cemetery because a dog that does not like Bodie was in there. That was a mistake. Somehow I failed to negotiate the curb and ended up face-down on the sidewalk. My camera took a hit but there was no significant damage: a small crack on the lens housing is all. My right knee seems to have borne the brunt of it.

My third Leyland prime find

13350^9739 + 9739^13350 is prime! This one falls into the same smallish gap (between two previously-known primes) as my second find.

Saturday, October 3

My first Leyland prime find

Back in May when I wrote my Indexing the Leyland primes I saw an opportunity to discover some of my own Leyland primes simply by looking where no one else was. My plan was to first check the gaps between the smallest 954 known Leyland primes in order to verify that these gaps contained no additional primes. That process is ongoing. Somewhat bored with the tedium, about a month ago I also started checking some small gaps beyond Leyland prime #954. That search has now borne fruit: 13739^4600 + 4600^13739 is prime!

Friday, September 11


This is a picture of some elderberries on a tree in our front yard. We also have two more trees in the back, although one of those is still on the mend after a raccoon tried climbing its slender trunk last year — giving it a serious bend. Larry used to make elderberry wine.

Monday, September 7

Still missing

In (I think) 1960, back in my home town in Germany, my mother purchased for me my first copy of Micky Maus. It might have been #4 of that year's series. Subsequent issues were also bought for me, until I had a small stack, and these I brought with me to Canada when we emigrated in May of that year. Thus my hoarding craze began. For the few years that I managed to hold onto those comic books I was somewhat annoyed at not having those missing (#1-3) issues.

In 1967 I started collecting Scientific American magazines. Soon aware that its then-current format started in May 1948 I began seeking out back issues. Thirty-some years later I had garnered not only those but additional monthlies going back to 1920 and assorted weeklies going all the way back to 1845.

In November 2011 I scored 3246 pdf issues of Scientific American Weekly (out of the 3349 that were published prior to 1910, not including the Supplement). Over the past three days I have added (cropped from Google-available bound volumes) 88 of the missing 103 issues to my collection.

That leaves 15 still missing (date / volume / number):

1866-04-28   14  18  $
1867-06-22   16  25
1872-12-21   27  25  $
1895-11-23   73  21
1902-07-05   87   1  $
1902-07-12        2  $
1902-07-19        3  $
1902-07-26        4  $
1903-05-16   88  20  $
1904-02-13   90   7  $
1904-06-11       24  $
1904-06-18       25  $
1904-06-25       26  $
1904-09-24   91  13  $
1907-10-26   97  17

The $ signs mark the 12 issues that are available from Scientific American for $8 US each.

Sunday, August 23

American English

Today's Futility Closet quotes an Americanized Hamlet by writer Albert Edward Wilson, guessing that "it might be from the 1930s". I decided to have a look...

In the 1 Sep 1945 Adelaide (Australia) The Mail is a piece — apparently from a reader — titled Uncle Samlet:

   So Hollywood is to film "Hamlet" up to date with Cary Grant playing the Gloomy Dane as a modern man, and speaking to a 1945 script (writes "Jacques").
   Some years ago Sir Barry Jackson produced a Hamlet, who wore plus-fours, and smoked cigars, but Shakespeare's script wasn't altered. Sir Herbert Tree gave the Prince of Denmark a beard, a score of women (including Bernhardt) played the part, but this screen version will strike a new note. Can you imagine the "To be, or not to be" soliloquy in modern American? Something like this, maybe:—

"To quit or not to quit: that's what I'm up against.
Ought I to stick the darn thing out,
Let old man Fortune make a monkey out of me,
Or take a crack against this gang of twerps
And swipe the lot of them..."
   And so on.

Very similar to the start of the A.E. Wilson version so one might suspect that it was taken from him. By the way, the Cary Grant (Hitchcock) version of Hamlet was not to be. I wonder how the soliloquy was handled in Irving Fiske's (1941) Hamlet in Modern English? At any rate, the very idea of an Americanized Hamlet occurs much earlier. Here's a 1917 version in the Richmond Guardian:

   An authority estimates that in about 20 years the American language will be unintelligible to English ears. Then we shall, doubtless, hear the latest American representative of Hamlet delivering the famous speech something like this:—

To quit or not to quit — that's what I'm up against
Is it the cheese to sit still in the game,
And take whatever's coming to you yet,
Or to put up a rough-house 'gainst a bunch of troubles
Till they are down and out?
Who'd stand for all the hardest kind of luck,
The frozen face, the main guy's jollying,
The fly cop's club, the handling of a lemon
When any old time he can chase himself
Into the boneyard?

Say what?

Friday, August 21

Trust, but verify

The phrase is generally attributed to (president) Ronald Reagan by way of a Russian proverb. Reagan may well have popularized the expression (in English) but it appears in print somewhat earlier; for example, translated from a "Khrushchevian motto" in the Jan/Feb 1966 issue of Problems of Communism. I'm going to suggest prior usage much further back in time, albeit significantly out of context. The following quotation negates the 'trust' part, but doesn't that make more sense?

"The secret of success in the practical study of facts is to observe carefully and patiently; to take nothing on trust, but verify the truth of every appearance; to test the accuracy of every impression, and bring each inference to the trial of fact without daring to make any generalisation."

[The Lancet: To the medical student. London, 13 Sep 1879]

Thursday, August 6

The refund

The 3 TB hard drive on my late-2012 iMac failed in September 2014, nine months past the one-year Apple warranty included with the computer's purchase. It cost me over $400 (Canadian) to have it replaced. I was a little pissed at the time because hard drives ordered directly from manufacturers are generally guaranteed for five years, so this seemed like a drive manufacturing defect for which I was made to pay because I purchased it through a third party.

Fast-forward another nine months: Apple officially recognizes a problem with those drives and initiates a replacement program. Since I already had my drive replaced, I was offered a full refund on the aforementioned expense instead. Great! I phoned Apple and they mailed me a check. Or at least they said they did. After five weeks, I informed them of not having received it. One of my options was to have that check cancelled and have the funds transferred electronically instead. I decided to try this. I phoned Apple with my banking details and, after an initial mixup over the so-called transit number, found the money in my bank account on July 30. Great!

Then, yesterday, I received a check in the mail for those same funds, initiated on the same day as my wire transfer — July 30 — so this was not the check mailed a month earlier.

Of course I had no intention of cashing the check but I thought it would be a courtesy to let them know of it, since the check's creation at the same time that the funds were being wired electronically pointed to an obscure failing of Apple's refund procedure. I provided the check date, number, and the reference numbers that were on the printout that accompanied the check. The slightly snarky email response was: "I can confirm that we successfully canceled the check, so we appreciate your assistance in discarding it, and not attempting to cash it." There seemed to be no recognition that the check they had cancelled was (presumably) the older check and not this newly generated one! Wow.

Monday, July 13

It's bigger than we thought

Phil Plait's Bad Astronomy blog (for Slate) this morning had me in a wee bit of a tizzy. One might have expected better. Plait's contention that the very preliminary New Horizons Pluto diameter of 2370 ±20 km somehow betters the previous estimate — which he thought was 2368 ±20 km — is just silly and (likely because a few of his readers said so) he has since done a couple of updates to soften the headline.

It gets worse. It is true that there was — and continues to be — a lot of confusion about Pluto's diameter. I don't know who first thought to add the ±20 to the 2368 km but a number of current online planetary sources had it. However, the 2368 almost certainly comes from an article about the variation of methane abundance in Pluto's atmosphere. The authors suggest:

For the sake (and amusement) of making predictions, we "guess" here a radius of 1184 km, based on our considerations on its likely lower (1180 km) and upper (1188 km) limits.

So their estimate is actually 2368 ±8 km! In his second update Plait says: "Emily Lakdawalla in the article linked in the first update says that the new measurements reduce the uncertainty quite a bit, and I'm inclined to agree." Really? Why would Lakdawalla suggest this? Last year Emily Lakdawalla authored an article titled When will we know which is bigger, Pluto or Eris? In it, Lakdawalla quotes Alan Stern (principal investigator of the New Horizons mission) saying that "we already know the radius to ± 20 kilometers", to which she explains (in brackets) that "they know the diameter to ± 40 kilometers". Wow!

Wednesday, July 8

Strength in numbers

For many years now I have been an admirer of the mechanics of swallowwort growth. If two or more of the spindly stalks are sufficiently close together, they will (on contact) braid themselves into a more sturdy (rope-like) superstalk. The additional strength of this structure allows the plants to reach higher, before weight (eventually) arches them over. If under a tree, that extra push may be all that is needed to reach a lower branch. The vine (Cynanchum rossicum) is invasive in North America and there is plenty of it in the local (private) cemetery. A relative of American milkweed, the thickets created by the plant have given it the moniker dog-strangling vine.

Friday, June 26


Spin is my first acquaintance with a Jos Bergmans creation, realized here in exotic woods by Brian Menold. Bergmans' polycube constructions are for me a natural progression from my old standby, Stewart Coffin's Convolution, and its subsequent improvement, Involute. Spin is a pleasure to take apart and put back together.

Monday, June 15

Japanese tree lilac

At this time of the year, my walks to and from the park are distracted by these two mature Japanese tree lilacs — Syringa reticulata — in bloom. The pervasive fragrance of the showy flowers is to me quite pleasant, an opinion not apparently shared by everyone. One person described it as vaguely fishy but — as another commenter there pointed out — it is musky, not muskie.

Sunday, June 14

Free ride

Today was Metrolinx's appreciation day for the Bloor and Weston communities putting up with the rail service construction's inconveniences. In spite of a steady rain, the event seemed well attended. I picked up my free round-trip ticket and went up to the downtown platform where I watched a full up train come and leave without me. I then proceeded to the airport platform for what turned out to be more of the same. One might have expected a small ramp up of services (considering the freebie) but I guess appreciation only goes so far. Standing in a crowded passenger vehicle for an hour is not my idea of a worthwhile venture. I left the platform, availed myself of a free hot dog, and sauntered home.

Sunday, June 7

Weston Station

A couple more pictures of the new Union Pearson Express train. Above, a train in Weston Station — headed downtown (the top of the CN tower is visible above the front of the train). The station is far from ready. I had to dodge workers to get to the platform — and the new footbridge across Lawrence Avenue West was still closed! Below, a train has left the station going to the airport — about to enter the tunnel that allows Weston's King and Church streets not to be interrupted by the UP service. John Street was a third major Weston thoroughfare (roughly where the train is in the photo). It being too close to the Lawrence rail overpass, the railroad section of John had to be sacrificed to the tunnel approach's downslope. In lieu of the road, residents have been promised a footbridge.

Saturday, June 6

UP and away

The Union Pearson Express trains started their passengers-accepted runs this morning. My first photo shows a three-parter shortly after it had left Weston station (one of only two stops: a 7-minute walk from my home) on its 25-minute ride from the airport to downtown Toronto. The outrageously high fares are mitigated somewhat if one buys a transit debit card for one's payments. My second photo, a few minutes later, shows a two-parter from Toronto about to arrive into Weston station:

Tuesday, May 26

Indexing the Leyland primes

A Leyland number is x^y+y^x, technically x >= y > 1 except that (x,y) = (2,1) has lately been admitted. Ordered by size, every Leyland number has associated with it an index number. Leyland primes are Leyland numbers that are prime, proven or probable. Most of the known Leyland primes can also be indexed — but not all of them! The difficulty lies in the manner in which Leyland primes have historically been discovered: restricting y to small values while allowing x to be very large. This method divorces finds from their Leyland number indices (i.e., their size), with the possibility of unchecked primes existing between two known examples. Of the currently 1092 known Leyland primes, I believe that the smallest 954 (or slighly more) are indexable.

There's a Numberphile video on Leyland numbers and Leyland primes. In it is mentioned the currently largest known Leyland prime: Serge Batalov's (x,y) = (328574,15). What might be this 386434-digit prime's index? Step one is to figure out the number's Leyland index. Using a Mathematica program to count, I believe it to be Leyland #3808683611. Step two is to fit the Leyland number indices of the 954 indexable primes to a curve:

The suggestion here is 17*index^2.23 as a decent fit. This equation is not meant to be exact: a database of further-along primes might necessitate adjusting the multiplier and exponent somewhat, but for our purposes it is good enough. What Leyland prime index will generate a Leyland number index of ~3808000000? The number 5553 comes close. So, I expect the currently largest known Leyland prime to be roughly #5550 of all Leyland primes. That leaves thousands of smaller Leyland primes still to be discovered!

Saturday, May 16


Year after year, I see here a lot of the same bird species (and, I rather suspect, the same individuals where those species are poorly represented). I first saw this trumpeter swan in the river on May 7, and again the following day. I thought it had moved on, but there it was yesterday — flying a couple of circles before heading upriver. I know there are trumpeters on the lake (Ontario) but this is the first one I've seen in this locale. Ontario trumpeters sport yellow wing-tags for easy identification.

Thursday, April 16

O Canada

The image is a section of Canada's northwest, part of a new political map of Canada, a copy of which I have placed here. It's a big picture, so be patient. For me, it loads and handles ok in Chrome and Firefox — not so much in Safari. For those who didn't know, I'll point out that all of the islands in Hudson Bay — including its southerly James Bay extension — are part of Nunavut.

Wednesday, April 1

My hood

The picture is an Apple 3D representation of where in Toronto I live and take my walks. I'll try to describe the map in words without recourse to symbols planted onto the image. The bird's eye view looks west toward Raymore Park on the farther side of the Humber river, scene of significant devastation when hurricane Hazel hit in 1954. A built-in-1995 footbridge across the river is visible at the top. The main street is Weston Road, seen curving at the right toward the highrises. Just beyond is downtown Weston — once a village (the orange lot near the bottom right of this 1878 map is my reckoned property), then town, outside of Toronto. The railroad corridor shown under reconstruction at bottom right will hopefully be finished this year.

If you look for an h — let's call it a chair — with legs abutting Weston Road, my street — Sykes Avenue — is the seat and front leg of that chair. My residence is on the south side of the leg part, sixth house in. Sykes runs into Denison Road West, the back of the chair, which then curves and continues until it is stopped by St. John's Cemetery on the Humber, a private cemetery that — strictly speaking — is in Mount Dennis, the community south of Weston. To the right (north) of the cemetery is Denison Park. It is behind this park — looking down — that I get my photos of Raymore island (as I call it) sitting in the deeper water held back by the curved whitewater of Raymore weir, downstream to its left (south).

The cemetery and most of the streetscape is on high ground, contrasted with the low ground of Raymore Park, adjacent bits on the nearer side of the river including the autumn-hued treescape above (west of) the cemetery, and the large school building and mega-housing structures at bottom left, below (east of) the cemetery.

A photo I took this morning of a beaver in the river returning to its den under the island:

Wednesday, March 11

Visitors from infinity, reprise

When I posted my Visitors from infinity on New Year's day, I had been working with a database of two-and-a-half billion (10^9) numerical correspondences. On February 6, I completed a computation taking the Yellowstone permutation to five billion — but only as another two-and-a-half-billion-term file, because I cannot (with only 64 GB RAM) store all five billion terms into Mathematica at the same time.

The limitation of that shortcoming is that when I map n into A098550(n) or A098550(n) into n, any time the mapping crosses over into the other-file regime I have to reboot Mathematica with that other file. Just the reading of it takes about nine hours and working the 302 currently unknown-outcome trajectories (with minima < 10^4) backwards (towards the left in the graph) might take another day or two on each reading. But I finally completed the task yesterday! The picture shows all 302 orbits with their minima synced (although many of the more-or-less random-hued paths are obscured by others).

One can see in the graph that my backward reach is limited to five billion, while moving forward (to the right) always ends in a point beyond (sometimes well beyond) that. If you are interested in the raw data, it is still here — and I have placed individual graphs for all 302 chains here.

Friday, February 6

(101*10^120072-87)/7 is prime

A good part of my dabbling in recreational arithmetic is searching for large primes. My previous best was 2*(10^39960+10^63+10^36+10^12)+2223 which has 39961 digits and dates to October 2011. Today I bettered that with (101*10^120072-87)/7 which is made up of 120074 digits — large enough to be on page three of Henri & Renaud Lifchitz's top probable prime records. The find arises out of an observation I made for A254005.

Friday, January 9


It's no secret that there are coyotes in Toronto. Over the years, neighbourhood acquaintances have told me they had seen one — generally down by the river, but just a few days ago in the local cemetery wherein a lot of folk walk/run their dogs. I don't always take my camera with me anymore — the way I used to when going for a walk — because having Bodie with me constrains my picture-framing ability. But this morning I did have my camera and certainly glad of it.

Initially, I spotted one coyote walking up the east side of Raymore Island:

A minute or so later, I noticed another one on the west side:

They both looked to be in good health, in spite of the bitter cold of the last few days. I like this more-distant capture of that second coyote rounding the upstream tip of the island:

Thursday, January 8


Twenty-five years ago I was still buying books, magazines, journals, and newsletters, as though I didn't know that the internet was just around the corner. Of course I didn't know. One of those newsletters was Cubism For Fun and there must have been a contest therein for me to have submitted to them an offbeat musical puzzle, complete with a mixtape. I never heard any more about it — until two days ago, when a Christian Halberstadt emailed me and asked if I was the author of it. I wasn't sure; it had been so long!

Unbeknownst to me, my "mystery" puzzle was published in 1997. Christian supplied me with a copy. He and his friend, Rolf Braun, had subsequently made an effort to solve it, or at least deconstruct some of the verbiage. Here is a reprint (with a few minor editorial accruements) of my 1990 submission to CFF:

Chet Ritter was a Mohawk from Kanesatake. He was young, handsome – but had a somewhat brutish attitude. He was well-educated; certainly well-read. He had trouble holding onto money, friends, or a place to live. However, unlike most people with that misfortune, he never succumbed to the despair and depravities of society’s outcasts. Instead, you could find him at the library – reading some mathematical journal and, headphones on, listening to some recent compact disc. (Remember when all you could find at a library was books?)

It was thus that I first met him some two years ago, listening to Bruce Cockburn’s Inner City Front and studying Raymond Smullyan’s Chess Mysteries of Sherlock Holmes. I placed in front of him a note on which I had written: CAN YOU CHAT? After turning around to look at me, but before removing his headphones, he crossed out my CAN, wrote underneath it the word KEN, placed quotation marks around the word CHAT, and printed his answer on the handed-back note: OUI. C’EST MOI! I must have looked a little puzzled, for he explained, “All my friends call me Cat.” I smiled as the joke sank in.

I think it was his witticism, this perversity, that attracted me to him… A child-like honesty in the face of come-what-may.

We had many delightful discussions on our haphazard encounters, detailing everything from the failings of democracy to the meaning of life. Much of what we dealt with were problems and puzzles in the realm of recreational mathematics. He was most interested in perusing a few of the less-well-known periodicals and newsletters to which I subscribed. I think he even wrote a couple of letters-to-the-editor to some of these.

Several months ago, he disappeared. He had lost his most recent part-time job, availing himself to the charities of one of his women-friends and spending the nights (near as I could tell) under a bridge by the river. Some weeks prior, he had handed me a slip of paper titled “DUODENE” on which were listed the following twelve items:

Who Cares?
School Days
Black Diamond Bay
10:15 Saturday Night
Me And The Boys
The Walk
Norwegian Wood
Get Off Of My Cloud
Planet Claire
The Magnificent Seven

I recognized at least a few of these as songs. In addition, Cat gave me a good-quality, blank 60-minute tape, saying simply, “You’ll need this.” I reacted to his generosity by asking, “Did you pay for this?” Cat shot back: “You gonna pay anybody for recording this music?” Embarrassed, I said no more.

It took a little doing, but I finally got that tape together. I had to have help however from a few of my more musically-inclined acquaintances; at least half of the songs (and three of the artists) were unknown to me. The first eight selections fit neatly on ‘side one’ of the tape. By this time I had a pretty good idea what the organizing principle was. ‘Side two’ confirmed my hunch although (I have to admit) I listened to Bob Dylan’s Hurricane three times before I heard it. When next I saw Cat, I gloated, “Some of the song-placings are context-sensitive. In fact Joey is downright ambiguous… You could have put it somewhere else.” Cat, grinning from ear to ear, jabbed a finger gently into my ribs and said, “But then, the songs wouldn’t have fit on the tape.”

That was a week before Cat’s disappearance and the second-last time I saw him. He never said good-bye but did mention a certain “home-sickness”. Life in the big city had (I think) gotten to him. “You know,” he once said, “this world is run by jerks for jerks.”

Thursday, January 1

Visitors from infinity

Sometimes a sequence in the OEIS doesn't do justice to the bigger picture. The blue trajectory, forward from the zero point to where it ends on the right is A251412. This integer sequence also has a backward history. Combined with an infinite number of like-minded sequences (four of which are shown; the identifying numbers at the forward end are the trajectories' minima at point zero) coming in from infinity at the left and going back out to it at the right, these trajectories meander for position in number space. Any outgoing trajectory running into an integer-point of an incoming one would of course merge with it. (Well, it was a single trajectory all along. We just didn't know it.) What if an outgoing trajectory were to run into its own tail? In that case, the trajectory is seen to be — not infinite — but finite. There are currently 34 known finite orbits in this mapping (which includes 7 fixed points, orbits of length 1). The currently longest orbit is one of length 91.