Friday, January 09, 2015


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 08, 2015


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 01, 2015

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.