Happy New Year!

## Saturday, December 24, 2022

### Porch visitors

 furries

## Friday, December 23, 2022

### The fire is so delightful

 let it snow, let it blow, let it go

## Monday, December 19, 2022

### Binary complement sequences

On Friday, Joshua Searle posted to the Sequence Fanatics Discussion list a neat procedure: take the binary complement of an integer multiplied by 3. Iterate. For example, starting with 3 we get the binary of 9 (1001), the complement of which (0110) is 6. Continuing, from 6 we get the binary of 18 (10010), the complement of which (01101) is 13. Arriving at zero, we stop.

0 3
1 6
2 13
3 24
4 55
5 90
6 241
7 300
8 123
9 142
10 85
11 0

Eleven steps to get to zero. The largest integer reached is 300 at step 7. We can shorthand the sequence data for 3 with (11,7,300) [steps to reach zero, steps to reach a maximum, the maximum]. Here are the statistics for integer starts up to 28:

0 (0,0,0)
1 (1,0,1)
2 (2,0,2)
3 (11,7,300)
4 (12,8,300)
5 (1,0,5)
6 (10,6,300)
7 (3,1,10)
8 (4,2,10)
9 (13,9,300)
10 (2,0,10)
11 (19,15,300)
12 (80,28,328536)
13 (9,5,300)
14 (2,1,21)
15 (15,11,300)
16 (16,12,300)
17 (81,29,328536)
18 (14,10,300)
19 (11,7,300)
20 (12,8,300)
21 (1,0,21)
22 (6,2,72)
23 (83,31,328536)
24 (8,4,300)
25 (73,21,328536)
26 (22,5,661)
27 (79,27,328536)
28 (7572,2962,123130640068522377168864228132316865867184046004226894)

Note the large number of steps and maximum in the last one. Somewhat more surprising are subsequent records. Do all integer starts eventually reach zero? Tom Duff has provided a list of the progressive record number of steps to get to zero:

1: 1
2: 2
3: 11
4: 12
9: 13
11: 19
12: 80
17: 81
23: 83
28: 7572
33: 7573
74: 7574
86: 7578
180: 7580
227: 664475
350: 664882
821: 3180929
3822: 3180930
4187: 3180931
5561: 3181981
6380: 3181988
6398: 3182002
22174: 3182226
22246: 120796790
26494: 556068798
34859: 556068799
49827: 556068871
70772: 556068872
103721: 572086553
104282: 572086610
204953: 1246707529
213884: 1246707552
225095: 1246707555
407354: 1246707602
425720: 87037147316

Update: Tim Peters has been working on extending the outcome of start numbers up to ten million. Here are a few results from his effort:

1671887:   128018188027
6264400:   918217008016
6524469: >2000000000000
7404616:  3306609997544

## Saturday, December 10, 2022

 iPhone 6 on warp drive

Catherine's almost-seven-year-old iPhone 6 was dying/dead so I ordered a replacement.

Starting it up went well enough although I did spend some fifteen minutes prior looking for a paper clip. Only later did I discover the iPhone 13's accompanying SIM-tray ejector tool:

## Thursday, December 08, 2022

### Products with embedded indices

Éric Angelini did a "smallest multiplication" bit yesterday that I felt was worth extending.

0 0 = 0 * 1
1 10 = 2 * 5
2 12 = 3 * 4
3 132 = 6 * 22
4 84 = 7 * 12
5 152 = 8 * 19
6 126 = 9 * 14
7 170 = 10 * 17
8 198 = 11 * 18
9 195 = 13 * 15
10 1008 = 16 * 63
11 1100 = 20 * 55
12 1218 = 21 * 58
13 713 = 23 * 31
14 1416 = 24 * 59
15 1150 = 25 * 46
16 1612 = 26 * 62
17 1728 = 27 * 64
18 1820 = 28 * 65
19 1914 = 29 * 66
20 1020 = 30 * 34
21 1216 = 32 * 38
22 1221 = 33 * 37
23 2345 = 35 * 67
24 2448 = 36 * 68
25 2925 = 39 * 75
26 12600 = 40 * 315
27 1927 = 41 * 47
28 2898 = 42 * 69
...

The column of indices on the far left is shown embedded (in bold) in their adjacent products. The constraint on the multiplier and multiplicand is that they must be distinct nonnegative integers with the multiplier the smallest such not yet used and the multiplicand the smallest such that yields the embedded index. A chart extending the products is here.

## Friday, November 18, 2022

### Coyote sighting

Spotted early this morning walking west down our street, just as Bodie and I were leaving for our walk. Already too distant to be distinct by the time I managed the photo, I have added a helpful arrow. At the bottom of the shot are some of the two dozen bags of leaves where our driveway meets the road, ready for today's yard-waste pickup. Resourceful squirrels will tear the paper off the bottom of the bags in order to steal the mostly dry leaves. I've had to re-bag one of them because it was beyond repair.

## Wednesday, November 16, 2022

### Paper delivery

Looking east up Denison Rd. W., at the spot where it curves south, I believe I recognize the newspaper delivery driver's tire-tracks in this morning's freshly fallen snow.

## Saturday, November 05, 2022

### Tuva

I've been watching VagaBond for a while now. His videos provide welcome counterpoint to the tragic criminality of Russia's invasion of Ukraine, and this one, in Tuva, is better than the usual freighthopping fare.

## Friday, November 04, 2022

### Twin concatenations

Éric Angelini came up with this idea of listing products (one per line) where the multipliers introducing the line are the positive integers and the smallest possible multiplicands (after the initial line) are deduced from the products, given that the concatenations of the products (to the right of the equal signs) must be the same as the concatenations of the multipliers and multiplicands (in order, to the left of the equal signs), from the beginning, as far as it will go.

On the MathFun mail list, Éric shared Gilles Esposito-Farese's version that makes the multipliers the nonnegative integers (up to 50), instead of the positive-integers versions shown on Éric's blog:

0 * 0 = 0
1 * 0 = 0
2 * 5 = 10
4 * 63659 = 254636
5 * 119 = 595
...

For some reason Gilles skipped 3. Which makes his output for lines 4 to 50 incorrect. Ouch!

Here is my correction:

0 * 0 = 0
1 * 0 = 0
2 * 5 = 10
3 * 846 = 2538
4 * 1 = 4
5 * 1283 = 6415
6 * 2 = 12
7 * 1194673 = 8362711
8 * 118342264792783 = 946738118342264
9 * 88 = 792
10 * 7839881 = 78398810
11 * 7127164651933786 = 78398811171271646
12 * 432781551 = 5193378612
13 * 332908885487176222196 = 4327815511333290888548
14 * 512587299724651848 = 7176222196145125872
15 * 664831 = 9972465
16 * 11550979 = 184815664
...

Here is my listing up to 1000. Please let me know if you see an error!

## Friday, October 14, 2022

### Non sequitur

I don't watch a lot of Netflix, likely because after wading through a bunch of popular/trending items I don't find anything palatable. Yesterday I tried a different approach. I looked through the critically acclaimed films and (eventually) came across The House:

Although the movie left me slightly discombobulated, I thought it generally worthwhile. For some reason, about half-way through, Harry Mathews' The Sinking of the Odradek Stadium came to mind. I once owned a copy of it. In the early 1980s (I think), I lent it to Brad (I think), a workplace acquaintance. I lost track of Brad at some point. I never got the book back!

## Tuesday, September 13, 2022

### Delicious Dishes (one-reeler)

Delicious Dishes is a film short that was uploaded to the Prelinger Archives in 2003. If you're a watcher of TCM (Turner Classic Movies), you may have — as I recently did — viewed it there. There is precious little real information about the one-reeler on the internet and at least one person (Tony Bensley) was struggling with misinformation. So I decided to have a look.

The first thing that I found was that the film is listed in Graham Webb's 2020 Encyclopedia of American Short Films, 1926-1959: Delicious Dishes: How to Make Them (copyright 24 March 1933, M & M Distributing Co.). The Library of Congress (Copyright Office) has an entry in its 1933 Catalog of Copyright Entries - Part 1 (p. 159) which locates the incorporated company in Asbury Park NJ. Good start! I have a distributor name/place and a very specific date.

I subsequently found Tony Bensley's reply on the YouTube version of the film noting that an Arnold Morris (often incorrectly assigned as the individual in the short) appeared on Late Night with David Letterman on 4 March 1987 and that Arnold mentions (starting @1:12) his father being in the same business as himself. Time to put on my genealogy hat and do some Morris family research.

The April 1930 Asbury Park NJ census has them at 703 Comstock St.:
Arnold's father is Nathan Morris. Nathan's occupation is given as "auctioneer" (of jewelry).

The April 1940 Asbury Park NJ census has them at 1106 Monroe Ave.:
Nathan's occupation is now "kitchen gadgets" (manufacturing).

Nathan K. Morris (24 July 1901 – 29 July 1972) must be the pitchman in the film. Subsequent research found Malcolm Gladwell's 30 October 2000 article about Ron Popeil, The Pitchman, written for The New Yorker. Gladwell suggests that Nathan "partnered with his brother Al ..." That's it! The partnership must have been incorporated as M & M (i.e., Morris & Morris), the distributing company of the film.

There is a 1930 Asbury Park NJ census "Al" at 710 1st Ave.:
... but he was "Abe" in the Atlantic City NJ census in 1940:
Nathan's slightly-older brother's name was actually Abraham. So, was Gladwell wrong about Nathan's brother's name? Initially I thought so, construing a 1930 mistranscription of Abe into Al. However, searching for Nathan Morris in "The Billboard", I soon found this mention of his brother "Al":
 27 March 1943: click to enlarge
Al could be Abe's middle name! I finally clued in after seeing a Billboard short (29 May 1943) on Nat's son, Lester, being in the army, as were "Archie and Rube Morris, sons of Al Morris". Presumably, middle names for Leonard and Edward. I think they were very comfortable with using different names, formally or informally, as the social situation may have demanded. By the way, I've been unable to find Nathan's middle name, but I will point out that there was a 1900 New Jersey Legislature assembly bill (No. 315) entitled "An act to change the name of Morris Kederisky and family to 'Kidders Morris' ...". Kidders was of course Nathan's father. A Geni entry has the surname spelled "Kadaretzky". It is not clear to me how or why "Kidders" morphed into "Kidder" but it seems haphazard. A 1901 New Jersey birth index record has a "Kidder" Morris but the 1905 census has "Keddres". A 1908 accident on a railroad train and the 1910 census has him as "Kidder".
 5 July 1947: click to enlarge
 5 November 1950: click to enlarge
 N.K. Morris Mfg. Co. advert in The Billboard, 30 June 1951
 N.K. Morris Mfg. Co. advert in The Billboard, 7 April 1956

## Sunday, September 11, 2022

### A million-digit Leyland prime (spot check)

As expected, one of my slower processes has rediscovered (this afternoon) Gabor Levai's 1000027-digit Leyland prime. I thought that it might be a good time to see how far I've come. It seems that I've completed about 49% of the numbers that I wanted to test since I started 132 days ago, so, on average, I'm testing about 220 numbers per day. It means that, at best, I'll be done by February 2023 but a lot depends on my transferring processes from my three slow computers to others as they become available.

I had installed differing operating systems on those machines to see what would happen. I saw no improvement. So I'm thinking now that the slow speed arises from the fact that these three Mac minis were purchased (in order to save money) with only 8 GB RAM, which may be insufficient for the task at hand (it had been adequate for testing 300000-digit numbers). I will now terminate three of the six processes on each machine to see if there is any speed improvement on the remaining three processes. Later this month I will have completed 14 processes on two other machines, which allows me to transfer the 9 terminated processes.

## Thursday, September 01, 2022

### A million-digit Leyland prime (power interruption)

This morning, a 23-minute power interruption allowed a couple of Toronto Hydro employees to do some maintenance work on a close-to-my-home transformer. I have all 18 computers (104 processes currently working on my million-digit Leyland prime search) on battery back-up units but the amount of time here was just long enough to shut down seven of them (39 processes). Another 5 to 10 minutes might have killed them all.

By early afternoon I had saved the output of the interrupted programs and reinitialized their new search ranges. Three of the seven affected computers were the ones that were taking 18.5 hours per candidate in my May 27 reality check, so I installed new (differing) operating systems on those machines, hoping for a speed improvement. I won't know for a couple of days if that will pan out.

## Saturday, August 27, 2022

### The tooth abscess

Back on April 20, I developed a terrible toothache that came to be associated with a swollen lower-right jaw. Over the next five days I consumed more Acetaminophen and Ibuprofen pills than likely I had over several decades. On April 25, my dentist (whom I had not seen since December 2019 because of the pandemic) had a look.

He prescribed a seven-day regimen of Amoxicillin. Within 24 hours the situation improved. He also recommended that I see an endodontist, which I did on May 10. The endodontist gave me three options: do nothing, have a certain dead tooth pulled, or let him do a root canal that involved (at least, pending complications) two three-hour sessions. I don't know that I can last one three-hour session. The last time I had a root canal the doctor left me alone for ten or so minutes and I chose that time to have a panic attack! Unfortunately, that set in motion similar feelings of anxiety, subsequently, every time that I have had my mouth worked on in a dentist's chair.

Mulling over my options, I did nothing over the course of the summer. On July 7, I managed to complicate things by breaking off part of a tooth in the upper-left part of my mouth. As annoying as this was, I did not see my dentist about it. Around August 20, I noted that the area around my dead tooth was getting reinfected. Salt-water rinses were not helping. Fortunately, this time the formation of the abscess was not accompanied by pain and, by August 22, I felt that my own body was dealing with it and that, perhaps, the worst was over. Nevertheless, I was fearful that my immune system was not capable of conquering the bacterial pathogen and I thought it prudent to see if I could acquire an antibiotic.

Constrained somewhat by my unwillingness to travel any distance, I thought that making an appointment with my family doctor right here in Weston was an option. He could write me a prescription and I could get it filled nearby. But my call to his office on August 23 did not go as planned. The receptionist, upon discovering that my last appointment had been way back in November 2012 (I was as surprised as she was: Had it really been that long?) felt that she needed to ask the doctor if he wanted to engage with me. The next day she left a message with their (apparently new) policy: Due to being overburdened with Covid, my family doctor was not taking back patients he hasn't seen in over five year!

I spent the rest of Friday making my potato/carrot/spinach soup complemented with some lentils, roast-beef gravy, and leftover frozen bits from a long-ago spiral ham. I had several bowlfuls and tried to get some late-afternoon shut-eye. When I came to later that evening, I was feeling much better! The swelling in my gums had receded a little and I had the sense that my immune system was going to be able to handle this after all. Perhaps it was some remnant antibiotic in the ham!

Update: Monday came and went without hearing again from the receptionist. It appears that I have lost my dentist!

## Saturday, August 20, 2022

### Bucket list #4

My original five-piece bucket list is here.

4. Tim Hortons (I know. How can something so ubiquitous be so difficult to reach.)

 1931 Weston Road, this morning (after visiting Shoppers Drug Mart)
 Six doughnuts: \$6.49

## Saturday, August 13, 2022

### The new backyard residents

The area of our backyard shown in the above has become a focus of activity of late, first by Bodie wading into the growth sniffing out something-or-other, then by the scurrying of some smallish animals (that I took to be mice) into the crevasses of the railway tie that is there (on the right), as well as the corners of the shed (on the far left) which sported some previously undisclosed openings into its underpinnings.

It wasn't just mice. Unperturbed by my presence, this afternoon I spotted a somewhat larger intruder, lazily munching away at some item in the (left) grassy area:

I am wondering now if the opossum that I thought I saw back in June was misidentified.

## Tuesday, August 02, 2022

### Wikipedia (reprise)

It has been more than a decade since I last made a Wikipedia edit. Just now I tried again. I changed Michelle Josef's birth year from 1954 to 1953. I have a December 1998 Globe & Mail article describing Josef as "a 45-year-old". And if that isn't substantive enough, I have a 15 March 2012 tweet from Michelle Josef herself saying: "I have been alive 21549 days."

Assuming that Michelle made the tweet on her 59th birthday (i.e., birthday anniversary), this would have her being born on 15 March 1953. How did Michelle arrive at the day count? 59*365 = 21535. Add 15 days for the leap years 1956 to 2012 and we have 21550 days. Why did Michelle calculate one day less? Perhaps she knew that century years (ending in 00) were not leap years and therefore did not count the year 2000 as one. Unfortunately, century years divisible by 400 are exceptions to the century-years rule and 2000 was a leap year. There are of course other possibilities. For example, (2012-1956)/4 = 14, or Michelle tweeted the day before her birthday, or some other scenario.

Update: On 18 October 2022, a Rickyharder added "March 21" to my "1953" Wikipedia year correction. My own investigation suggested that the date was March 20, based on a family source. I don't know which date is correct. Assuming my date, Michelle tweeted five days before her birthday, so her day count was four days too many. Go figure.

## Monday, August 01, 2022

### Sum & erase (integer version)

One of the drawbacks of Éric Angelini's sum-and-erase is that we are dealing with number strings, not integers. This is because we allow leading zeros. There's an easy and obvious fix: Don't allow leading zeros! Everything else is the same but, if the erasure of all instances of a digit (1-9) that is contained in the sum-of-digits moves one or more zeros to the leading (left) edge then those zeros are erased as well. Now the only number that is allowed to start with zero is zero itself. And 0 by itself disappears because it is the left-most digit of 0 and the sum-of-digits is 0. And although total erasure (the empty string "" in Éric's version) happens here as well, we can define it to be zero so as to keep our sequencing in the integer realm. Conveniently in Mathematica, FromDigits[{}] == FromDigits[{0}] == 0.

0 => "" = 0
1 =>
"" = 0
2 =>
"" = 0
3 =>
"" = 0
4 =>
"" = 0
5 =>
"" = 0
6 =>
"" = 0
7 =>
"" = 0
8 =>
"" = 0
9 =>
"" = 0
100..00 => 0
=> "" = 0

Summarizing the procedure:

1. Total the number's digits and concatenate this sum to the right of the number.
2. If the first digit in our number is one of the digits in our sum, delete every instance of it.
3. If (after deletion) there now appear leading zeros in our number, delete these as well.
4. If all digits or even all non-zero digits have been deleted, the subsequent term is zero.
5. Otherwise, the subsequent term is the concatenation of the remaining digits.
6. To generate a sequence of terms, iterate. Most starts eventually end up at zero.

In Éric's procedure, I had found (in addition to the fixed-point c0) six cycles. In this integer version, I have [index.  name = c&start-string  (cycle length)  {smallest precursor}]:

0.  c0  (1)  {0}

1.  c86  (80858)  {16}

2.  c30323322046333587  (1634)  {5916}

3.  c48822444886224973  (29)  {23675}:

0 48822444886224973
1 4882244488622497385
2 488224448862249738598
3 488224448862249738598115
4 488224448862249738598115122
5 488224448862249738598115122127
6 488224448862249738598115122127137
7 882288622973859811512212713718
8 882288622973859811512212713718137
9 2262297359115122127137113714
10 226229735911512212713711371497
11 226229735911512212713711371497113
12 226229735911512212713711371497113118
13 6973591151171371137149711311818
14 6973591151171371137149711311818123
15 6973591151171371137149711311818123129
16 6973591151171371137149711311818123129141
17 6973591151171371137149711311818123129141147
18 6973591151171371137149711311818123129141147159
19 6973591151171371137149711311818123129141147159174
20 97359115117137113714971131181812312914114715917418
21 73511511713711371471131181812312141147151741818
22 73511511713711371471131181812312141147151741818153
23 73511511713711371471131181812312141147151741818153162
24 351151113113141131181812312141141514181815316211
25 351151113113141131181812312141141514181815316211124
26 51151111114111181812121411415141818151621112411
27 111111114111181812121411411418181162111241111
28 11111111411118181212141141141818116211124111197
29 48822444886224973

4.  c77078088077807837313303137333430853003033013  (43)  {83123}

5.  c886054637  (140)  {256311}

6.  c5869099118496111114109711  (85)  {346715}

Also six cycles (so far), in order of smallest precursor. I have a distance-to-cycle chart. For positive terms it is the number of steps needed to get to zero. Negative terms are for starts that will reach some other cycle. Terms that are 0 are for integers that are part of a cycle.

## Sunday, July 31, 2022

### Bodie's weight

 click to enlarge

The vertical scale is kilograms. Red dots are vet readings. Blue dots are home (bathroom scale) readings. I don't really trust the bathroom scale to be accurate but it will do. We just want to be cognizant of inordinate weight gain. So every two weeks Bodie looks forward to jumping into my arms. Unfortunately, the process also monitors my personal weight gain.

## Saturday, July 30, 2022

### Aerial action

Just beyond my neighbour's yard, a northern mockingbird lets a hawk know it's not welcome.

## Friday, July 29, 2022

### Don't it always seem to go ...?

It has been a year since I last received delivery of a couple of big bottles of Welch's grape juice. I should have ordered more in the following months! By the time I did, they were out of stock. I can't tell if the supply shortage is temporary or permanent.

## Thursday, July 28, 2022

### Cycles in Éric Angelini's sum-and-erase

Éric's article on sum-and-erase is here. I wanted a place where I could post known cycles, so I will do that in this listing [index.  name = c&start-string  (cycle length)  {precursors}]:

0.  c0  (1)  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 00, 01, 02, 03, 05, 06, 07, 08, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, ...}

1.  c3374  (583792)  {04, 09, 25, 31, 63, 67, 69, 77, 92, 96, 99, 023, 027, 031, 040, 042, 044, 049, 067, 074, 075, 081, 082, 090, 093, 099, 105, 109, 116, 133, 138, 148, 152, 161, 162, 163, 168, 174, 194, 197, 198, 206, 216, 253, 257, 258, 285, 307, 314, 327, 352, 370, 382, 389, 398, 422, 423, 430, 463, 477, 482, 483, 496, 503, 521, 543, 544, 568, 581, 582, 590, 592, 599, 604, 605, 610, 613, 630, 635, 639, 645, 656, 665, 667, 668, 676, 681, 682, 685, 686, 688, 708, 722, 736, 737, 742, 758, 766, 773, 782, 801, 809, 810, 814, 821, 839, 841, 848, 849, 856, 858, 867, 875, 884, 885, 894, 902, 906, 923, 932, 944, 958, 977, 982, 988, 0119, 0135, 0144, 0147, 0152, 0163, 0166, 0176, 0191, 0199, 0203, 0207, 0216, 0230, 0235, 0244, 0253, 0256, 0259, 0267, 0270, 0275, 0279, 0299, 0301, 0310, 0314, 0317, 0333, 0349, 0361, 0375, 0378, 0385, 0387, 0400, 0402, 0404, 0409, 0420, 0426, 0429, 0440, 0448, 0482, 0484, 0486, 0490, 0498, 0521, 0539, 0559, 0562, 0578, 0592, 0595, 0607, 0619, 0645, 0656, 0658, 0665, 0668, 0670, 0671, 0673, 0686, 0698, 0704, 0705, 0713, 0716, 0735, 0736, 0738, 0740, 0750, 0767, 0768, 0776, 0794, 0801, 0802, 0810, 0813, 0814, 0817, 0819, 0820, 0824, 0829, 0831, 0847, 0855, 0856, 0866, 0867, 0900, 0903, 0909, 0911, 0916, 0919, 0925, 0929, 0930, 0938, 0957, 0958, 0982, 0990, 0991, 0992, 0999, 1005, 1009, 1016, 1019, 1027, 1050, 1056, 1059, 1061, 1062, 1082, 1090, 1091, 1099, 1106, 1109, 1119, 1120, 1163, 1168, 1191, 1192, 1201, 1210, 1222, 1238, 1239, 1250, 1253, 1254, 1264, 1268, 1277, 1285, 1289, 1299, 1307, 1327, 1333, 1335, 1337, 1339, 1352, 1353, 1364, 1365, 1370, 1373, 1376, 1378, 1393, 1396, 1422, 1495, 1520, 1526, 1528, 1530, 1545, 1554, 1557, 1559, 1566, 1575, 1578, 1593, 1595, 1604, 1613, 1618, 1628, 1629, 1630, 1631, 1637, 1648, 1649, 1668, 1679, 1681, 1685, 1686, 1724, 1728, 1729, 1739, 1754, 1760, 1762, 1767, 1776, 1792, 1793, 1794, 1823, 1825, 1832, 1840, 1843, 1853, 1866, 1867, 1894, 1904, 1907, 1908, 1911, 1912, 1921, 1923, 1929, 1933, 1934, 1938, 1970, 1987, 1992, 2006, 2019, 2028, 2047, 2048, 2053, 2058, 2060, 2068, 2074, 2082, 2085, 2091, 2094, 2099, 2109, 2126, 2133, 2134, 2157, 2162, 2163, 2168, 2169, 2173, 2188, 2194, 2208, 2216, 2222, 2230, 2234, 2249, 2254, 2264, 2280, 2291, 2295, 2299, 2302, 2307, 2314, 2320, 2324, 2342, 2344, 2347, 2348, 2351, 2357, 2358, 2361, 2370, 2384, 2389, 2391, 2394, 2395, 2398, 2413, 2414, 2416, 2418, 2429, 2441, 2458, 2471, 2480, 2481, 2492, 2503, 2524, 2531, 2542, 2545, 2554, 2556, 2565, 2576, 2584, 2604, 2624, 2642, 2643, 2644, 2655, 2668, 2679, 2686, 2687, 2704, 2740, 2757, 2758, 2763, 2764, 2775, 2777, 2784, 2786, 2793, 2802, 2805, 2818, 2820, 2837, 2839, 2843, 2848, 2850, 2856, 2875, 2881, 2884, 2897, 2904, 2906, 2912, 2917, 2918, 2921, 2925, 2929, 2948, 2951, 2952, 2964, 2992, 3007, 3019, 3033, 3054, 3057, 3061, 3070, 3071, 3089, 3091, 3095, 3098, 3109, 3117, 3125, 3129, 3136, 3140, 3146, 3148, 3163, 3164, 3167, 3171, 3176, 3177, 3199, 3213, 3218, 3231, 3237, 3244, 3254, 3256, 3273, 3287, 3298, 3299, 3303, 3316, 3321, 3327, 3330, 3335, 3352, 3353, 3371, 3374, 3387, 3396, 3398, 3457, 3462, 3476, 3502, 3509, 3512, 3515, 3523, 3528, 3532, 3533, 3546, 3551, 3564, 3578, 3604, 3611, 3612, 3619, 3644, 3654, 3655, 3668, 3675, 3686, 3687, 3700, 3701, 3707, 3709, 3713, 3724, 3731, 3734, 3743, 3747, 3757, 3767, 3770, 3774, 3775, 3776, 3789, 3792, 3794, 3804, 3809, 3812, 3814, 3821, 3837, 3851, 3855, 3856, 3858, 3859, 3866, 3872, 3873, 3885, 3888, 3896, 3905, 3911, 3922, 3929, 3936, 3938, 3942, 3950, 3963, 3967, 3969, 3972, 3983, 3986, 3989, 3992, 3996, 3998, 4019, 4023, 4035, 4053, 4063, 4069, 4091, 4096, 4097, 4099, 4102, 4109, 4123, 4139, 4146, 4155, 4163, 4164, 4176, 4196, 4202, 4218, 4220, 4228, 4235, 4239, 4253, 4282, 4283, 4299, 4300, 4301, 4307, 4347, 4348, 4349, 4361, 4368, 4370, 4371, 4374, 4376, 4378, 4380, 4381, 4384, 4387, 4394, 4399, 4416, 4437, 4438, 4439, 4452, 4458, 4461, 4483, 4491, 4492, 4506, 4516, 4524, 4527, 4531, 4542, 4548, 4578, 4584, 4598, 4612, 4614, 4616, 4625, 4627, 4628, 4641, 4661, 4668, 4671, 4686, 4698, 4701, 4710, 4731, 4767, 4776, 4793, 4796, 4815, 4826, 4834, 4843, 4850, 4866, 4888, 4890, 4911, 4914, 4922, 4924, 4929, 4935, 4938, 4939, 4941, 4942, 4959, 4960, 4970, 4992, 4993, 4995, 5003, 5019, 5030, 5031, 5033, 5034, 5035, 5038, 5039, 5053, 5063, 5067, 5088, 5091, 5109, 5113, 5131, 5134, 5142, 5163, 5166, 5176, 5186, 5196, 5198, 5218, 5236, 5244, 5249, 5263, 5264, 5274, 5286, 5296, 5299, 5307, 5333, 5339, 5361, 5366, 5370, 5371, 5384, 5388, 5391, 5393, 5428, 5429, 5439, 5446, 5448, 5455, 5456, 5464, 5465, 5483, 5484, 5487, 5493, 5496, 5503, 5545, 5546, 5554, 5555, 5562, 5566, 5578, 5582, 5584, 5604, 5616, 5625, 5630, 5637, 5648, 5652, 5656, 5661, 5665, 5668, 5686, 5694, 5720, 5742, 5758, 5760, 5761, 5763, 5767, 5776, 5779, 5781, 5782, 5785, 5794, 5797, 5810, 5814, 5825, 5827, 5832, 5845, 5852, 5854, 5863, 5866, 5870, 5871, 5900, 5929, 5930, 5937, 5938, 5964, 5979, 5982, 5992, 5997, 6004, 6005, 6014, 6019, 6021, 6040, 6041, 6050, 6056, 6065, 6078, 6091, 6099, 6100, 6104, 6106, 6107, 6109, 6130, 6134, 6155, 6160, 6174, 6177, 6225, 6235, 6236, 6237, 6244, 6252, 6253, 6263, 6268, 6277, 6286, 6287, 6288, 6298, 6299, 6300, 6307, 6309, 6318, 6321, 6351, 6370, 6372, 6377, 6379, 6404, 6405, 6416, 6417, 6429, 6430, 6440, 6453, 6461, 6473, 6485, 6489, 6490, 6499, 6545, 6554, 6559, 6569, 6572, 6578, 6588, 6595, 6596, 6605, 6610, 6623, 6628, 6641, 6659, 6680, 6682, 6691, 6698, 6705, 6713, 6714, 6718, 6723, 6743, 6753, 6781, 6794, 6803, 6804, 6806, 6812, 6823, 6826, 6830, 6834, 6837, 6852, 6854, 6860, 6862, 6873, 6894, 6899, 6904, 6913, 6915, 6916, 6922, 6923, 6929, 6938, 6942, 6949, 6961, 6968, 6975, 6986, 6992, 6994, 6999, 7008, 7019, 7035, 7037, 7056, 7067, 7073, 7076, 7078, 7080, 7084, 7087, 7091, 7099, 7102, 7109, 7147, 7150, 7158, 7163, 7174, 7178, 7187, 7195, 7205, 7227, 7230, 7235, 7240, 7244, 7253, 7260, 7263, 7272, 7279, 7288, 7294, 7297, 7299, 7305, 7318, 7340, 7358, 7361, 7365, 7366, 7377, 7385, 7458, 7490, 7493, 7495, 7499, 7502, 7508, 7516, 7517, 7526, 7529, 7530, 7542, 7547, 7561, 7563, 7571, 7574, 7578, 7581, 7587, 7604, 7611, 7614, 7620, 7621, 7626, 7627, 7635, 7655, 7657, 7662, 7668, 7672, 7675, 7680, 7682, 7686, 7703, 7706, 7708, 7714, 7718, 7722, 7729, 7737, 7751, 7754, 7758, 7762, 7765, 7773, 7798, 7824, 7825, 7833, 7841, 7849, 7860, 7863, 7866, 7892, 7904, 7909, 7912, 7920, 7929, 7931, 7936, 7938, 7940, 7941, 7942, 7945, 7950, 7958, 7978, 7987, 7990, 7992, 7999, 8001, 8009, 8010, 8012, 8013, 8017, 8019, 8024, 8029, 8031, 8041, 8047, 8053, 8067, 8090, 8091, 8100, 8108, 8109, 8129, 8143, 8152, 8156, 8160, 8162, 8163, 8173, 8179, 8180, 8204, 8224, 8229, 8234, 8235, 8239, 8242, 8244, 8253, 8256, 8279, 8292, 8299, 8307, 8313, 8323, 8324, 8331, 8332, 8341, 8344, 8358, 8361, 8369, 8370, 8374, 8385, 8391, 8401, 8402, 8403, 8407, 8408, 8422, 8436, 8446, 8452, 8456, 8464, 8480, 8492, 8506, 8517, 8518, 8527, 8531, 8534, 8537, 8561, 8562, 8566, 8581, 8604, 8626, 8633, 8644, 8647, 8657, 8662, 8669, 8672, 8679, 8692, 8696, 8707, 8716, 8723, 8724, 8726, 8727, 8731, 8736, 8742, 8765, 8767, 8770, 8772, 8776, 8788, 8791, 8794, 8810, 8835, 8840, 8851, 8878, 8887, 8891, 8893, 8905, 8918, 8929, 8938, 8944, 8952, 8961, 8976, 8981, 8983, 8992, 9002, 9006, 9017, 9020, 9025, 9029, 9038, 9044, 9058, 9060, 9077, 9088, 9092, 9104, 9105, 9128, 9143, 9144, 9149, 9163, 9178, 9180, 9182, 9189, 9194, 9198, 9211, 9227, 9229, 9235, 9244, 9247, 9253, 9272, 9284, 9288, 9292, 9302, 9307, 9336, 9347, 9361, 9362, 9363, 9370, 9377, 9379, 9382, 9389, 9397, 9398, 9407, 9416, 9435, 9444, 9462, 9468, 9475, 9478, 9483, 9508, 9516, 9518, 9556, 9565, 9567, 9568, 9578, 9583, 9604, 9615, 9617, 9618, 9626, 9633, 9641, 9645, 9658, 9662, 9668, 9686, 9688, 9689, 9698, 9704, 9707, 9709, 9710, 9749, 9750, 9761, 9767, 9770, 9776, 9790, 9794, 9801, 9866, 9870, 9902, 9914, 9918, 9922, 9937, 9938, 9968, 9970, 9974}

2.  c083433  (120621)  {092, 245, 351, 395, 413, 433, 468, 571, 802, 0382, 0479, 0588, 0594, 0691, 0848, 0884, 0902, 0920, 1038, 1142, 1170, 1369, 1380, 1412, 1421, 1433, 1479, 1482, 1560, 1584, 1645, 1666, 1701, 1710, 1802, 1848, 1884, 1986, 2096, 2148, 2235, 2251, 2325, 2352, 2450, 2477, 2512, 2521, 2555, 2601, 2634, 2649, 2688, 2761, 2955, 2970, 2999, 3358, 3479, 3510, 3519, 3538, 3544, 3583, 3699, 3702, 3769, 3779, 3797, 3802, 3848, 3884, 3917, 3949, 3981, 3994, 4012, 4103, 4112, 4121, 4122, 4138, 4160, 4165, 4183, 4286, 4650, 4707, 4770, 4790, 4792, 4802, 4825, 4870, 4909, 4933, 4990, 5078, 5108, 5277, 5427, 5479, 5602, 5786, 5802, 5808, 5823, 5848, 5880, 5884, 6335, 6353, 6385, 6411, 6456, 6465, 6479, 6491, 6591, 6645, 6669, 6694, 6696, 6802, 6848, 6884, 6946, 6964, 6966, 7014, 7098, 7228, 7282, 7291, 7444, 7579, 7588, 7597, 7759, 7802, 7816, 7848, 7884, 7903, 7946, 8002, 8020, 8021, 8288, 8376, 8428, 8479, 8482, 8708, 8780, 8828, 8842, 8870, 8882, 9134, 9168, 9273, 9277, 9515, 9551, 9581, 9802, 9806, 9848, 9884, 9999}

3.  c222227222772202078  (1723)  {695, 825, 905, 0655, 0857, 0864, 1385, 1857, 2695, 3078, 3285, 3857, 4189, 4857, 5778, 5787, 6054, 6504, 6685, 6856, 6857, 6865, 7864, 8265, 8459, 8572, 9005, 9050, 9857}

4.  c332  (14072)  {323, 332, 3238, 3283, 3328, 5133, 7132}

5.  c3933  (10538)  {333, 3339, 3393, 3933}

6.  c5464644657500000011711019071641751  (49)  {243, 2439, 6782}

Roughly, in order of precursor frequency. Strictly speaking, the bold numbers are not precursors (to cycles) but, rather, cycle terms themselves. I include them for the sake of completeness. Are there any more cycles?

Update: In a 14 Sep 2023 video seminar, Neil Sloane @38:33 mentions Michael Branicky finding (in my notation) c00020203143 (20173) and c00660657510634435617071915 (46), precursors not mentioned.

## Saturday, July 23, 2022

### My tired lip, Angela

I don't recall when exactly I read Martin Gardner's article on mnemonics [Scientific American, October 1957; 'The Scientific American Book of Mathematical Puzzles & Diversions', 1959; reprinted as 'Hexaflexagons and Other Mathematical Diversions', 1988] but it was likely sixty years ago when I lived on Claremont Street and came across the 1959 book on puzzles and diversions, borrowed from what must have been the 'Boys & Girls Library' at Manning Ave. & Robinson St. I didn't get into Martin Gardner and Scientific American as an obsession until a few years later, December 1967 being my first store-purchased issue.

I was reminded of this by Futility Closet's recent 'Words and Numbers' which adjusts slightly Gardner's version (above) and gets into some π mnemonics. I had myself created one of these back then: "My tired lip, Angela, my lip." Twelve digits. Not too shabby and I recalled it after all these years, although I did attempt at first to place the 'tired' in front of the second 'lip'. I also remembered that the final p=9 wasn't actually π's twelfth digit but a rounding-up approximation. There is a much more intensive explanation/application of this particular use of mnemonics, here.

## Saturday, July 16, 2022

### The fire at #3

 1:25 AM (click to enlarge)

 aftermath, in the light of day

Police said that this was arson, by a woman walking south on (very nearby) Weston Road.

## Friday, July 01, 2022

### The Townsend fire

Allan Reed and Frances Montgomery were married in 1937 in a Goderich Presbyterian church. Their third child, Eleanor, was born 8 January 1944 in nearby Dungannon, having been preceded by her siblings James and Jewel. A good Christian upbringing was no doubt an integral part of all three children's lives. Nevertheless, Eleanor found herself pregnant at age 17. She was Mrs. Pat Moorehead a decade later when Canadian Queen of the Fiddle was released. Eleanor's daughter, Catherine, was soon to be part of Eleanor's circa-1973 marriage to Graham Townsend, musician extraordinaire, and the 1975 birth of a half-brother, Graham (Gray) Jr.

The young Gray picture (above) is taken from a YouTube video where he is wearing dark glasses. It is well known that Graham Sr. was visually impaired from birth. Not well known is the fact that Graham's father, Fred Townsend, also struggled with poor eyesight in his youth and, by the time he reached his twenties, became convinced that he was going blind. In the 1901 U.K. census, Fred's mother, Alice Townsend (1878-1962), as well as Alice's mother, Dinah, are both described as "partially blind". Dinah's husband, William Townsend (1833-1921) is described as being "blind since 1864".

 Gray at age 14

In 1984, Catherine had her first child, making Eleanor a grandmother. In 1991, Eleanor's mother, Frances, died at the age of 76. A year or so later, the Townsends moved to Barrie. In 1993, Catherine had a second child, Isabella, by a second-marriage to a Mr. Monti, who is otherwise unknown to me.

 Gray at age 19

On 25 October 1998, at the Orangeville Convention Centre, hundreds of musicians and fans turned out to pay tribute to Graham and Eleanor Townsend. None too soon. On 3 December 1998, Graham Sr. died at age 56 from the effects of colon cancer after struggling with it, stated one source, for two years. According to Donna Douglas, Graham "had sought out alternative therapies", somewhat surprisingly, "with the financial help of friends in the music industry". On 22 December 1998, Eleanor's father, Allan Reed, died at age 72. A week or so later, Catherine Monti is preparing to give birth to her third child. Her older offspring "is away at camp" (possibly a mental health treatment facility) and the younger one, Isabella, is with gran'ma Eleanor. Perhaps Mr. Monti drove Isabella to the Townsend home in Barrie. Perhaps Eleanor picked her up in Dungannon.

In the wee hours of 31 December 1998, a fierce fire gutted the Townsend home at 56 Springdale Drive. The remains of 54-year-old Eleanor and 5-year-old Isabella were found in the living room, dead (it was determined later) from smoke inhalation. 23-year-old Gray managed to escape. "Isabella Johnson, a neighbour and friend who has lived next door since the Townsends moved in about six years ago, said she was awakened by Gray's yelling. She then heard a 'popping' noise as the fire engulfed the home..." [Montreal Gazette: 2 January 1999]. Gray ended up in the hospital with undisclosed injuries. It seems plausible that he had received burns. But how exactly, I have not come across anyone mentioning.

After an investigation by Barrie police and the Fire Marshal's Office, it was "revealed the fire was deliberately set by [Eleanor Townsend] in her vehicle which was parked in the garage" [Toronto Star: 16 March 1999]. "She was distraught over the death of her husband Graham, ... She was on prescription medication at the time of the fire, Detective Bryan Shultz said in a news release." As previously explained, the death of her husband was not a surprise and there is evidence that she was actually coping rather well: "Following the death of her husband, [Eleanor] Townsend said in an interview last month that she planned to work with her son at getting her late husband's works recorded" [Montreal Gazette: 2 January 1999]. The bit about being "on prescription medication" strikes me as not being obviously relevant. They should perhaps have stated the kind of prescription and how much of it was in her system. Instead, for a "fire [that] was purposely set in the seat of a car, parked in the garage of the house" [The Globe and Mail: 17 March 1999], what might have been her motive? Would not Eleanor taking care of her granddaughter and (presumably) looking forward to another grandchild have been important factors to consider? The authorities surely must have had additional evidence that implicated her.

It appears that they did. The official opinion may have had something to do with statements made by Gray: "'Mum was really not well and I was trying to keep things together when dad died', said her son, ..." [The Globe and Mail: 17 March 1999]. In fact, a family friend, Steve Fruitman, described Gray as being "terribly despondent over his father's death" [Gerry Taylor, New Brunswick Telegraph Journal: 5 January 1999], a situation not unlike the one Gray ascribes to his mother. Since it was thought important to point out that Eleanor was on medication, what about Gray? When he was in the hospital after the fire, were blood tests performed to examine his drug and alcohol levels?

Barrie fire chief Jim Lemieux had suggested that "Eleanor Townsend at first escaped the fire, but it seems she realized Isabella was trapped inside 'and went right back in'" [Toronto Star: 1 January 1999]. This is echoed in Taylor's Telegraph Journal article, but it included Gray in the escape: "The fire, of undetermined origin, was well advanced before Eleanor and her 23-year-old son, Gray, awakened and rushed from the house. Outside, Eleanor remembered that her five-year-old granddaughter was still asleep in the house and rushed back inside where she perished with the child." This information could only have come from Gray.

Ten weeks later: "[Gray] lived with [Eleanor Townsend] and she alerted him to the fire. He escorted his mother and his niece Isabella on to the deck at the back of the house. At some point, after being separated from her son, [Eleanor] Townsend and Isabella re-entered the home ..." [Toronto Star: 16 March 1999]. It appears that the story had changed. Neither version addresses how Gray might have sustained injuries but at least the later account has him taking charge and implies that Eleanor was aware of the fire before he was. "Family members said they believe Ms. Townsend went back into the house to look for the family dog and the girl followed her. The dog died in the fire." [The Globe and Mail: 17 March 1999]. That would have been Eleanor's brother, Jim Reed, and and his then-wife, Marina, showing that they were not prepared to doubt Gray's retelling of his escape from the fire.

A comment from someone Gray was dating "in the early 90s" [taken from the Gray-at-age-19 video] suggests that he "was patient, charming, disciplined and loved to teach. He also ran 5 miles a day every morning!" Perhaps all that ended with the move to Barrie. A new home, perhaps new friends. Had Gray finished high school yet? Was he, like his grandfather, struggling with increasingly bad eyesight and was, as a result, unable to get a driver's license? Did he resent his mother, Eleanor, being able to drive a vehicle at her whim, spending time with her daughter and grandchildren?

Gray died in 2003, presumably a short time after making available "a number of reel to reel tapes of material that Graham and Eleanor had taped on various occasions that were never released before" [Randy Morrison: liner notes on the 2004 album 'The Talented Townsends']. Somewhat remarkably (for an individual who is the product of musical royalty and is himself an entry in The Canadian Encyclopedia), I am not aware of anyone detailing anything concrete about his personal life after the fire or the circumstances of his death.

Update: Marlene Frost, my genealogy research partner, managed to acquire for me Randy Morrison's email. Randy kindly informed me that Gray died on the day of "New Year’s Eve of an overdose of pills and alcohol". That would have been the fifth anniversary of the day of the fire. And while Randy said that Gray's vision was fine (I had conjectured that Gray was going blind as a reason for some emotional turmoil that might have made him set fire to Eleanor's car), Randy understood Gray to have been bipolar: "Graham had to really keep a watch on him and after Graham died Gray was really lost and out of control."

Final thought: Graham Sr. had died. Presumably the Townsend estate now belonged to Eleanor. She may have had a will, or not. What happens to the estate after her death? She had two children, so maybe it was split between them. Or maybe Gray gets it all. One thing seems certain: Eleanor being named as the person who set the fire surely facilitated the transfer of insurance money for the burned-out house.

## Tuesday, June 28, 2022

### Less is more

Last month I described my disappointment at the real-time performance I was getting in my million-digit Leyland-prime search. I have now done some testing on my wife's recently-acquired iMac, a 3.7 GHz 10-core Intel i9, slightly faster than my own two 3.6 GHz versions. Her computer was running only 4 of the million-digit search processes, because it was a replacement for a failing older 4-core iMac. Supplementing those processes by additional ones, I wanted to see how that would impact search time:

4 processes   4.2 hours   1.05 hours/process
5 processes   4.9 hours   0.98 hours/process
6 processes   5.9 hours   0.98 hours/process
7 processes   7.5 hours   1.07 hours/process
8 processes   9.7 hours   1.21 hours/process

In other words, on a 10-core iMac I can run 8 searches significantly faster simply by doing 4 at a time! Running 6 processes appears to be ideal.

## Monday, June 27, 2022

### Animal sightings

It's been an incredible couple of weeks for me spotting by my home animals that are usually seen much further afield. On June 11, it was ravens in our backyard maple tree. And on the same day, a chipmunk made it momentarily onto the edge of our deck. Both are rare in our neighbourhood and, as far as I know, never on our street. Blue jays are generally in flocks flying overhead but I was peripherally aware of (more by sound than by sight) a mating pair in the area. The photo, taken on June 18, shows one in our mulberry tree that could be an offspring of that pair. On June 20, the neighbour's side-yard hummingbird feeder, just over the fence, sported a hummingbird. And on June 23, I saw an opossum [caveat] crossing the road to the edge of our driveway. I don't think I've ever encountered a hummingbird or an opossum. As fate would have it, only the blue jay managed to make it into my camera.

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