Sunday, January 19, 2020


I don't take a lot of photographs of people I don't know, but when I do, I prefer to do it on the sly. So it was somewhat unfortunate that Weston's "pigeon man" decided to break the fourth wall and acknowledge me across the intersection. The upraised hand, possibly a gesture of "peace"; the piercing stare, not so much.

Saturday, January 18, 2020

Anagrammatic sums

Éric Angelini asked about anagrammatic sums on MathFun on January 12: "Let a + b = c and a < b < c and a, b, c = anagrams of each other." Halfway down his sausage article, he lists the Gilles Esposito-Farèse calculation for 3- to 5-digit results: 1 @ 3-digit, 25 @ 4-digit, and 648 @ 5-digit. In that spirit, here are 17338 @ 6-digit results. I count 495014 @ 7-digit and 17565942 @ 8-digit.

The idea for these has been around a few years. Claudio Meller's A160851 appears to be a (currently) somewhat misguided attempt at enumeration, while Rajesh Bhowmick's A203024, fleshed out by Charles Greathouse, provides a seemingly complete listing of sums, including 9449 6-digit terms. My 17338 6-digit results yield only 9443 distinct sums. Why six fewer?

Apparently 6-digit sums are the first that allow the sums to be twice one of the addends (i.e., a = b). In A023086 we see that there are twelve such. It turns out that six of these are the six that are not in my 17338 sums (because I did not allow a = b):

251748 = 2 * 125874
257148 = 2 * 128574
285174 = 2 * 142587
285714 = 2 * 142857
517482 = 2 * 258741
825174 = 2 * 412587

The other six are included because they each had an alternate solution:

517428 = 2 * 258714 = 241587 + 275841
571428 = 2 * 285714 = 142857 + 428571
571482 = 2 * 285741 = 158724 + 412758
825714 = 2 * 412857 = 241587 + 584127
851742 = 2 * 425871 = 127584 + 724158
857142 = 2 * 428571 = 142857 + 714285 = 275418 + 581724 = 285714 + 571428

Addendum: Éric and Gilles had created A331468 for their triples.

Tuesday, January 14, 2020

Mysterious lights in St. John's Cemetery on the Humber

Photo taken from the middle of Denison Park, looking south, the evening darkness fast approaching, two bright lights appear in St. John's Cemetery on the Humber and, after a short while, go out. I suppose one hundred years ago when the park and surrounding area were still all fields, one might have thought such an occurrence — especially at night — to be mysterious, especially if the lights reappeared in the same location at irregular intervals — but never on for very long.

Of course I've seen this phenomenon innumerable times over the last forty years, as have many of the local residents. I've never given it much thought because the explanation was always a tad obvious, all the more so if one was in the cemetery when the lights came on. I was reminded of it only because I came across an article by Clark Kim in the 30 October 2013 edition of the York Guardian:

"There was a story that Denison Cemetery was haunted," said [Cherri] Hurst [of the Weston Historical Society]. It was also known as St. John’s Cemetery on the Humber where members of the Denison family are buried. 

Lights would mysteriously be seen in the cemetery at night. It turned out those lights were turned on and off by whiskey runners trying to hide their stashes of alcohol.

"Weston was a dry town for a long time. It was a small town. You couldn’t get away with stuff," she said.

Note the emphasis on the lights being turned on and off. Behind the south end of the cemetery is an almost 200-metre stretch of much lower ground before it rises again to where West Park Healthcare Centre has its presence. I don't know what the road layout was one hundred years ago, but today there is a curved West Park Receiving road off Emmett Ave. that presents a gated "employee entrance" a little along. Beyond the gate, one arrives at a 50-metre stretch of straight road that then abruptly turns right. That straight stretch is what allows the lights to "turn on" and the sharp right, to "turn off". In the following Google map, a yellow arrow extends the straight stretch of road over the low ground, through the cemetery, through Denison Park, stopping at the park's north end (at Lippincott St. W.):

Monday, January 13, 2020


A lot of rain here on Saturday had the "Raymore" island on the east side of the Humber (just above the weir; photo taken on Friday) ...

... somewhat inundated with river water on Sunday:

By this morning, the torrent had subsided:

Tuesday, January 07, 2020

The Fu Yao fruit bag mystery

Yesterday morning, walking Bodie up the north side of Clouston Ave., I saw a bag by a tree, across the street at the southwest corner of Clouston Ave. and Centre Rd. Thinking that it was trash, I crossed the road, determined to pick it up so as to deposit it in the trash bin at Denison Park on my way back. It wasn't trash but, rather, a plastic bag full of fruit. Nestled against the tree were three Canadian quarters. I left the bag of fruit but decided to take the money home. This morning, I had another look:

The bag was from Fu Yao Supermarket which has two locations in Toronto, neither of them anywhere near the northwest part of the city where I reside. Inside the bag were nine apples, a few grapes, and some sort of flattish bread-like item:

I'm at a loss to imagine a scenario that could account for all this.