Sunday, May 21

4*4*4 Elevator

A companion purchase to my Mean Cube, these are the six pieces of Jos Bergmans' 2010 4*4*4 Elevator. This one does have rotations (two, of the bottom-left piece). The two top-right pieces are shifted a couple of times in the assembly/disassembly of the cube to allow those rotations, making this a satisfyingly comprehensible construction toy. Visible in the top-left piece is one of three brass pins that help to reinforce that particular piece's joints.

Mean cube

These are the six pieces of Tom Jolly's 2004 Mean Cube that I recently acquired from Brian Menold. The small metallic circle embedded in the top of the bottom-left piece is a magnet whose complement resides on the bottom-right piece. It's a cheat meant to prevent the first piece out of the finished 4 by 4 by 4 cube from coming out too easily. There are no rotations in the assembly/disassembly but still a very challenging puzzle to own and appreciate.

Wednesday, May 17

Countdown primes

The concatenation of the integers from 1 to n have been called Smarandache numbers, whereby the concatenation of the integers from n to 1 would be reverse Smarandache numbers. No Smarandache numbers are yet known to be prime but we have two for the reverse. I prefer to call them countdown primes.

The first is 82818079787776757473727170696867666564636261605958575655545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321, first noted by Ralf Stephan in 1998. The second countdown prime was found by Eric Weisstein in 2010. We can call them countdown(82) and countdown(37765) for short.

Surprisingly, a tabulation of countdown primes in bases other than ten appears not to have been tackled by anyone so I shall remedy that herewith:

 2 — 2, 3, 4, 7, 11, 13, 25, 97, 110, 1939, ...
 3 — 2, 5, 13, 57, 109, 638, 3069, ...
 4 — 4, 106, 118, 130, 1690, ...
 5 — 2, 313, 505, ...
 6 — 2, 6, 17, 28, 33, 37, 81, ...
 7 — 373, 1825, ...
 8 — 2, 9, 47, 50, 99, ...
 9 — 2, 5, 346, ...
10 — 82, 37765, ...
11 — 2, ...
12 — 3, 4, 5, 7, 17, 58, 106, 303, ...
13 — ?

Friday, April 21

Aronson's sequence

I was made aware of Aronson's sequence by Greg Ross' Futility Closet article on it three weeks ago. A couple of things caught my eye. The first was his use of "nine billion one million second" to example the "few T-less ordinals" that "don’t arrange themselves to mop up all the incoming Ts". It would have been a little more compelling if 9001000002 was actually in the sequence — which it is not. The closest t-free ordinal that is is 9001000702.

The second thing was Greg's "We had supposed that the sentence would end with … letter in this sentence. But an infinite sentence has no end..." English number names have been well-defined only up to 10^66-1 — although I fully expect (once Mathematica debugs its IntegerName function) that that will go up to 10^306-1. There exists a longer realization but it may take some time for Mathematica to decide to incorporate it and it isn't obvious to me if the machine-generated naming scheme is potentially infinite. All strictly increasing, current English number-name sequences are necessarily finite, whether or not it is so recognized.

Sunday, April 16

Words from numbers

Last week I presented a "word" continuation puzzle. The algorithm used to create the list isn't too difficult to discover, applying English number words (one, two, three, ...) to the previous term (the zeroth assumed to be an empty string). Thus, one letter at a time, the second term from the first:

one +two

A letter gets added to the right if it doesn't already exist in the evolving string. It is deleted from the string if it does already exist. Thus the string will never contain more than one copy of any particular letter. If you noticed the double-comma near the end of my puzzle, that wasn't a typo: ourihten +onehundrednineteen results in an empty string, which +onehundredtwenty yields ohurweny, which +onehundredtwentyone yields one. Using Mathematica's built-in dictionary and ignoring already encountered words (such as one at index 121), here is a list of English found in a deep continuation:

         1 one
     21240 visaed
     45660 fads
     57242 ado
    155868 woad
    171524 aide
    271966 ad
    337664 waned
    347660 audit
    413700 and
    423066 roads
    507504 wained
    537056 goads
    557924 aid
    615808 wad
    619808 wade
    635830 wand
   1152766 mad
   1250766 moaned
   1272524 maid
   1298168 made
   2710904 maned
   3526644 mashed
  10984236 mawed
  16170624 maiden
  21730304 mated
  67092006 mead
 509056060 remands
 540798800 moated
1000080796 boards
1000146526 bards
1000152766 bad
1000298168 bade
1000530740 baud
1000558076 broads
1000562062 brands
1000748080 bandit
1000750040 band
1000816952 bandy
2000710904 baned

Why would all of our subsequent English dictionary words appear at even indices?

Sunday, April 9

What's next?

one, netw, nwhre, nwhefou, nwhouiv, nwhouvsx, whoux, wouxeigt, wouxgt, wouxgen, wouxglv, ouxgt, ouxghirtn, xghif, xghftn, ghfsi, ghfivt, fven, fvit, fviwenty, fvitone, fviy, fviwnthre, vihtyou, houwntf, houfetysix, houfixwtv, oufxvnyg, oufxvgwi, oufxvgwhry, ufxvgwine, ufxvgnehryto, ufxvgnoihre, xvgneyr, xgnhf, gnfrys, gfhiv, fvryeiht, fvti, viory, viftone, vinerytwo, vinwfthre, vinwheyfur, nwhuotf, nwhurysix, whuixfotv, wuxvryegt, wuxvgfoi, wuxvgofty, wuxvgifne, uxvgnefyo, uxvgnoifhre, xvgnhety, xgnhf, gnhftys, ghifv, vfyei, vfti, vfsxy, vfitone, vfnesxytw, vfnwithre, vnwhesxyou, nwhoutf, nwhoufy, whoufixtv, woufvsyeigt, woufvgxi, woufgxisnty, wufgxivne, ufgxiseyo, ufgxiovnhre, gxihsety, gxhnf, ghfvtyi, ghfiv, fsnye, fvti, fveghy, fvitone, fvnghytw, fvnwithre, vnwgyou, nwouhtf, nwoufegysx, woufxihtv, woufxvyiht, woufxvgi, woufxvgety, wufxvgine, ufxvgnyo, ufxvgoinhre, xvghnty, xghnf, ghfnetys, ghfiv, fvyi, fvti, fvtiohur, fvtione, fvihunrewo, fviwnthre, viwtner, wtohunf, wtfnrsix, wtfixohuv, wfxvregh, wfxvgoui, wfxvgihrten, wfxgitoul, fxgihrv, fxgvouhrn, xgvourt, xgvhin, gvourst, gh, ourihten, , ohurweny, ?

Friday, April 7

Counting t-free ordinals

Yesterday I introduced written-out English ordinals that lacked the letter "t" and I asked how many there are less than "one vigintillionth". I have come to conclude that the total number of t-free ordinals may be expressed by (c+1)^x*o, where 'c' is the number of t-free cardinals less than 1000, 'x' is the number of t-free -illions through which we traverse, and 'o' is the number of t-free ordinals less than 1000.

c = 55   (1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, 407, 409, 411, 500, 501, 504, 505, 506, 507, 509, 511, 600, 601, 604, 605, 606, 607, 609, 611, 700, 701, 704, 705, 706, 707, 709, 711, 900, 901, 904, 905, 906, 907, 909, 911)

x = 10   (10^6, 10^9, 10^15, 10^30, 10^33, 10^36, 10^39, 10^48, 10^51, 10^60)

o =  7   (2, 102, 402, 502, 602, 702, 902)

So we have 56^10*7 = 2123138423672799232.

The latest version of Mathematica has a built-in IntegerName function that does both cardinals and ordinals:

count1[ncard_] := 
 Length[Select[Range[10^(ncard - 1), 10^ncard - 1], 
   StringFreeQ[IntegerName[#, "Words"], "t"] &]]; 
m = {count1[1], count1[2], count1[3]}

{6, 1, 48}

... The number of t-free cardinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

count2[nord_] := 
 Length[Select[Range[10^(nord - 1), 10^nord - 1], 
   StringFreeQ[IntegerName[#, "Ordinal"], "t"] &]]; 
s = {count2[1], count2[2], count2[3]}

{1, 0, 6}

... The number of t-free ordinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

Do[If[StringFreeQ[IntegerName[10^(3*i), "Words"], "t"], 
  s = Join[s, m*Total[s]], s = Join[s, {0, 0, 0}]], {i, 21}]; s

{1, 0, 6, 0, 0, 0, 42, 7, 336, 2352, 392, 18816, 0, 0, 0, 131712, 21952, 1053696, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7375872, 1229312, 59006976, 413048832, 68841472, 3304390656, 23130734592, 3855122432, 185045876736, 1295321137152, 215886856192, 10362569097216, 0, 0, 0, 0, 0, 0, 72537983680512, 12089663946752, 580303869444096, 4062127086108672, 677021181018112, 32497016688869376, 0, 0, 0, 0, 0, 0, 227479116822085632, 37913186137014272, 1819832934576685056, 0, 0, 0}

... The number of t-free ordinals of 1-digit to 66-digit base-ten numbers. Finally:



IntegerName[%, "Words"]

two quintillion, one hundred twenty-three quadrillion, one hundred thirty-eight trillion, four hundred twenty-three billion, six hundred seventy-two million, seven hundred ninety-nine thousand, two hundred thirty-two

Interestingly, Mathematica has attempted to bridge the gap between the dictionary large-number names up to 10^63 (one vigintillion) and the next dictionary entry at 10^303 (one centillion):

Table[{i, IntegerName[10^i, "Words"]}, {i, 63, 306, 3}] // TableForm

 63 one vigintillion
 66 one unvigintillion
 69 one duovigintillion
 72 one trevigintillion
 75 one quattuorvigintillion
 78 one quinvigintillion
 81 one sexvigintillion
 84 one septenvigintillion
 87 one octovigintillion
 90 one novemvigintillion
 93 one trigintillion
 96 one untrigintillion
 99 one duotrigintillion
102 one trestrigintillions
105 one quattuortrigintillions
108 one quintrigintillions
111 one sextrigintillions
114 one septrigintillions
117 one octotrigintillions
120 one novemtrigintillions
123 one quadragintillions
126 one unquadragintillions
129 one duoquadragintillions
132 one tresquadragintillions
135 one quattuorquadragintillions
138 one quinquadragintillions
141 one sexquadragintillions
144 one septenquadragintillions
147 one octoquadragintillions
150 one novemquadragintillions
153 one quinquagintillions
156 one unquinquagintillions
159 one duoquinquagintillions
162 one tresquinquagintillions
165 one quattuorquinquagintillions
168 one quinquinquagintillions
171 one sexquinquagintillions
174 one septenquinquagintillions
177 one octoquinquagintillions
180 one novemquinquagintillions
183 one sexagintillions
186 one unsexagintillions
189 one duosexagintillions
192 one tresexagintillions
195 one quattuorsexagintillions
198 one quinsexagintillions
201 one sesexagintillions
204 one septensexagintillions
207 one octosexagintillions
210 one novemsexagintillions
213 one septuagintillions
216 one unseptuagintillions
219 one duoseptuagintillions
222 one treseptuagintillions
225 one quattuorseptuagintillions
228 one quinseptuagintillions
231 one seseptuagintillions
234 one septenseptuagintillions
237 one octoseptuagintillions
240 one novemseptuagintillions
243 one octogintillions
246 one unoctogintillions
249 one duooctogintillions
252 one tresoctogintillions
255 one quattuoroctogintillions
258 one quintoctogintillions
261 one sexoctogintillions
264 one septenoctogintillions
267 one octoctogintillions
270 one novoctogintillions
273 one nonagintillions
276 one unonagintillions
279 one duononagintillions
282 one trenonagintillions
285 one quattuornonagintillions
288 one quinonagintillions
291 one senonagintillions
294 one septenonagintillions
297 one octononagintillions
300 one novenonagintillions
303 one centillions
306 one billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion

Notice the terminal "s" for 10^102 to 10^303. I've alerted Wolfram to the bug. Then, starting at 10^306, all hell breaks loose!

Thursday, April 6

English t-free ordinals

one hundred second
four hundred second
five hundred second
six hundred second
seven hundred second
nine hundred second
one million second
one million one hundred second
one million four hundred second
one million five hundred second
one million six hundred second
one million seven hundred second
one million nine hundred second
four million second
four million one hundred second
four million four hundred second
four million five hundred second
four million six hundred second
four million seven hundred second
four million nine hundred second
five million second
five million one hundred second
five million four hundred second
five million five hundred second
five million six hundred second
five million seven hundred second
five million nine hundred second
six million second
six million one hundred second
six million four hundred second
six million five hundred second
six million six hundred second
six million seven hundred second
six million nine hundred second
seven million second
seven million one hundred second
seven million four hundred second
seven million five hundred second
seven million six hundred second
seven million seven hundred second
seven million nine hundred second
nine million second
nine million one hundred second
nine million four hundred second
nine million five hundred second
nine million six hundred second
nine million seven hundred second
nine million nine hundred second
eleven million second
eleven million one hundred second
eleven million four hundred second
eleven million five hundred second
eleven million six hundred second
eleven million seven hundred second
eleven million nine hundred second
one hundred million second
one hundred million one hundred second
one hundred million four hundred second
one hundred million five hundred second
one hundred million six hundred second
one hundred million seven hundred second
one hundred million nine hundred second
one hundred one million second
one hundred one million one hundred second
one hundred one million four hundred second
one hundred one million five hundred second
one hundred one million six hundred second
one hundred one million seven hundred second
one hundred one million nine hundred second
one hundred four million second
one hundred four million one hundred second
one hundred four million four hundred second
one hundred four million five hundred second
one hundred four million six hundred second
one hundred four million seven hundred second
one hundred four million nine hundred second
one hundred five million second
one hundred five million one hundred second
one hundred five million four hundred second
one hundred five million five hundred second
one hundred five million six hundred second
one hundred five million seven hundred second
one hundred five million nine hundred second
one hundred six million second
one hundred six million one hundred second
one hundred six million four hundred second
one hundred six million five hundred second
one hundred six million six hundred second
one hundred six million seven hundred second
one hundred six million nine hundred second
one hundred seven million second
one hundred seven million one hundred second
one hundred seven million four hundred second
one hundred seven million five hundred second
one hundred seven million six hundred second
one hundred seven million seven hundred second
one hundred seven million nine hundred second
one hundred nine million second
one hundred nine million one hundred second

There are 7 t-free ordinals less than 10^6, 392 less than 10^9, 21952 less than 10^15. How many are less than 10^63 (one vigintillion)?

Thursday, March 30

The sign of the four

In 1908, Matthew Burke was the head of one of twenty-two families living on the Conne River Mi'kmaq reservation in Newfoundland. Matthew's granddaughter, Margaret Burke Stewart, became the mother of sixteen children — the oldest (Catherine) ended up marrying my wife's now-deceased oldest brother (Larry). Catherine brought into that union two boys from her first marriage, Shawn and Jamie Beaupre.

Shawn (using his middle name) has been promoting himself as an aboriginal medium — Shawn Leonard. On Tuesday, Shawn teamed up with psychic/medium John Holland for a show in Moncton, New Brunswick, and they will do another tonight in Halifax, Nova Scotia. Yesterday, Holland did a Facebook interview with Shawn (click the "not now" in the pop-up sign-up if, like me, you don't do Facebook). It shows just how comfortable they are with each other in their overlapping, supportive roles.

Engineering coincidences into something that may be perceived to be meaningful is not of course everyone's cup of tea, least of all mine. This morning, Johnny Wills' Google+ photo-of-the-day theme was "four" and I quickly came up with an entry that I knew would be significantly different from the contributions of most other participants. Our brains exhibit a more-than-willing bent on assigning structure to the random bits and pieces in our lives!

Less than three hours after I posted the photo it was time for Bodie's morning walk. I have a habit of picking up any garbage that I encounter on the street so as to deposit it in a trash bin further along my route. A few houses away from my home I spotted (in light blue) just such a distraction lying in the middle of the road. Imagine my surprise as I approached to pick it up:

Tuesday, March 28

Rudolph Havermann genannt Draht

My great-great-grandfather Rudolph was born in 1800, likely in K├Ârbecke, and died in nearby Neheim in 1869. The use of genannt in my father's ancestors' names has always been bothersome to me and there are (German) explanations of its usage but I like to think that the Draht here is just an acknowledgement of Rudolph's mother's maiden name. I come to write about this man because of the mortality of his eight known-to-me children.

Rudolph's first marriage to Maria Christina Zentini (1811-1848) produced five offspring: Joseph (1837-1840), Heinrich (1839-1876), Friederich (1842-1842), Ferdinand (1843-1854), and Joseph (again, 1846-1864). Rudolph's second marriage to Maria Theresia Biermann (1813-1867) produced three more offspring: Anton (1850-1854), Heinrich Franz (1854-1854), and Maria Antonia (1858-1869). Wow, only one of these eight made it to age 18: my great-grandfather Heinrich — who died at age 36. Lucky man. Lucky me!

Monday, March 20

Fracture revisited

I noticed that the 2007 movie Fracture was on Netflix and so I thought I'd watch it again. I hadn't seen it since August 2009, a fact I was able to recover because I wrote a blog piece about a chess position in the film. I ended the item with a question about the six captured black pieces — which weren't visible in my movie-still grab. With the better-resolution Netflix version, I took it upon myself to retake a screen snippet of the board:

So the six captured black pieces were there all along, hidden in the poor contrast of my 2009 screen grab!

Tuesday, March 7

T cube

Designed by Yavuz Demirhan, realized by Brian Menold, this recently acquired puzzle more than tests my patience. The three identical pieces (two one-by-three bars attached to a one-by-five bar) fit — without any protuberances — within the (five-by-five-by-five) cubic cage. It arrived assembled and I was careful in taking it apart, but after I had reassembled it a second time it no longer came apart as I expected! And by the time I did get that disassembled I had forgotten how it was supposed to come together. Complicating things ever so slightly is that one of the frame's twenty-four inside edges is a touch less than three units long, presumably a production flaw and not part of the design. I'm getting too old for this type of toy.

Thursday, February 23

Movin' on up

When I bragged about my 100th Leyland prime find last August, I noted that I was moving up a leaderboard of probable-prime (PRP) discoverers. I currently have 131 Leyland primes under my belt and the last few days saw me take possession of position #44 on the PRP production score list, of which the above is a snippet.

Roughly, in the range where I am searching, every new find adds .01 to my production score. So to reach position #40 I have to come up with another 27 PRPs. My current rate of production is about five or six per month. So another five months.

Friday, February 17

A definite answer

Earlier today's Futility Closet titled Ahead of Schedule (since deleted) was to me more than a little disappointing. It highlights author Jack Finney's 1973 suggestion that a flying machine invented by W. J. Lewis circa 1876 might actually have flown across the skies of New York that year. Futility Closet's Greg Ross noted that when Finney included the speculation in his 1983 book, Forgotten News, he still "hadn’t received a definite answer".

So here's a definite answer: No. A flying machine invented by W. J. Lewis never flew any American skies. Finney's New York Times piece was of course mischievously disingenuous. He failed to point out that the 30 December 1876 Frank Leslie's Illustrated Newspaper (wherein Lewis's new flying machine is featured) specifically stated that what flew during "a formal test" was a small model. As the 22 September 1877 Scientific American Supplement (also featuring the machine) pointed out: "It is an easy matter to make a small flying machine shoot up a short distance into the air by means of a spring. But to make a large machine, with a steam engine, lift itself and sustain itself in the air is a problem not yet solved."

The picture of the flying contraption was reprinted yet again in a T. C. Hepworth story called Voyages in Cloudland (Frank Leslie's Popular Monthly, September 1883) wherein the Lewis machine's "travel through the air" was still nothing more than a proposal. The 27 May 1892 Velasco (Texas) Daily Times: "Dr. W. J. Lewis of San Antonio is nearing the completion of his aerial bird. He will place his bird on exhibition when finished and proved to be a success. Its [sic] only a model and says he is sure of revolutionizing navigation."

A "San Antonio history" site has a reference for 26 September 1894: "Dr. W. J. Lewis returned to San Antonio from St. Louis where he had purchased aluminum for his new invention — a flying machine with stationary wings to be powered by a machine-drive propeller." And that led me to this 29 September 1894 (preceded by "14 years ago today") text-file entry: "Dr. W. J. Lewis, of San Antonio, who is working on a flying machine, is in St. Louis. He has been studying aeronautics for 25 years. A machine was built 12 or 15 years ago, but the friction and consequent amount of power needed were too great. Dr. Lewis has been experimenting ever since trying to reduce the friction to a minimum. He believes success is not far off. Dr. Lewis' machine is constructed of bamboo covered with silk."

The 1 June 1898 Times-Picayune (from New Orleans, Louisiana) has an article that asks: "Has Dr. W. J. Lewis, of California, who is now visiting this city, discovered the principle of the flying machine?" A 30 November 1900 Macon (Missouri) Democrat newspaper article starts: "Dr. W. J. Lewis of St. Louis has invented a flying machine that flies. It is built on the order of a bird. It gives good satisfaction and by 1903 Dr. Lewis says he will have one completed — large enough to carry ..." I'll note that the 1876-1883 references all had the not-yet-doctored Mr. Lewis "of New York".

Back to Finney's 1973 speculative fiction: "So what happened after W. J. sent his machine twisting and torquing across Manhattan's skies in 1876? People just weren't a bit surprised; they'd been expecting it. Probably wondering why it had taken so long. And then, what with one thing and another, they got busy, and it just slipped the nineteenth century's mind." Give me a break!

Good mourning

Formerly, mourning was worn in England both for a longer period and of a much deeper character than is used at the present time. Two years were not considered too long a time for a father or a mother. Now custom prescribes only one year. It is also considered better form now to wear plainer and less ostentatiously heavy and expensive habiliments. Widows wear deep mourning for one year; then ordinary mourning as long a time as they may wish. Deep mourning is considered to be woollen "stuff" and crape. Second mourning is black silk trimmed with crape. Half-mourning is black and white. Complimentary mourning is black silk without crape. The different stages are less observed everywhere, outside of courts, than formerly. The French divide mourning garb into three classes, — deep, ordinary, and half mourning. In deep mourning, black woollen cloths only are worn; in ordinary mourning, silk and woollen both; and in half-mourning, black and white, gray and violet. In France, etiquette prescribes for a husband one year and six weeks; six months of deep mourning, six of ordinary, and six weeks half-mourning. For a wife, a father, a mother, six months; three deep and three half-mourning. For a grandparent, two months and a half, slight mourning. For a brother or sister, two months, one of which is deep mourning. For an uncle or aunt, three weeks of ordinary mourning, and two weeks for a cousin. While wearing deep mourning, one does not go into society, neither are visitors received. In the United States we have no fixed rules, but of late years the retirement from the world, after the loss of a near relative, has been much shortened. For one year, no formal visiting is undertaken, and no entertaining nor receiving, save in exceptional cases. Mourning (or black) is worn for a husband or a wife two years; one year deep, one year light. For parents, from one to two years; and for brothers and sisters that have reached maturity, one year. Those who are invited to a funeral, though not related, must go entirely in black, wearing black gloves and black beaver hat. To appear in hats of felt or straw, is wanting in due respect to customs.

[The Art of Dress, page 391, in The Manners That Win, Minnesota 1880.]

Thursday, February 16

Southern Ontario's great rainstorm of 1878

I got this from the Toronto Globe's Saturday, 14 September 1878 newspaper (page 8), where it is noted that it was raining steadily from late Tuesday well into Friday. Some 12 cm was reported to have fallen at Port Dover on Lake Erie. The emphasis was of course on Toronto where four lives were said to have been lost in the Don River. From my community (then village) of Weston it was reported that the iron bridge on the Grand Trunk Railway at Black Creek gave way and fell at 9 AM September 13, after fourteen hours of very heavy rain.

In many ways this was a forewarning of 1954's Hurricane Hazel and, indeed, this appears to have been the tail end of Hurricane #5 of 1878. This morning I adjusted the Wikipedia entry for that storm by replacing its September 13 extratropical placement in Virginia with one in Ontario. A remarkably modern storm path could be figured out even back then from the news of the day:

The storm originated in the Gulf of Mexico, where the barometer was low on Sept. 6th. During the latter part of that day there were high north-easterly winds and heavy rains in Florida. The disturbance hovered over Cuba and Southern Florida until the night of the 10th. It then began to travel in a northerly direction, and by the morning of the 12th it was over South Carolina, accompanied by heavy rain. During the 12th it moved at the rate of over 30 miles an hour, and by Friday morning was over the western end of Lake Ontario.

Wednesday, February 15

An unfortunate typo

A gentleman, named Mr. John Boyle, came into town yesterday, by the Northern Railroad, in search of Mrs. Margaret Thompson, a relative of his, and a resident of the village of Weston. Mrs. T., accompanied by one of her children, came to this city on Monday last, to superintend the sale of some farm produce, and intended to return the same evening, as circumstances of an urgent nature required her presence at home. Yesterday morning Mr. Boyle observed that her cows had not been attended, and remarking other things about the house which indicated the absence of the mistress, he resolved to make every possible search for her, but without success up to yesterday evening. Mrs. Thompson is a widow, and possessed of considerable property. She is described as wearing deep mourning, about 27 years of age, and of very prepossessing appearance. Mr. Boyle requested the Police to aid him in making enquiries regarding her whereabouts.

[The Globe: Toronto, Friday, 6 October 1854. The marks around the article title were already on the newspaper page prior to its being photocopied, so I wasn't the first to notice.]

Thursday, February 2

With multiplicity

It may be noted in Tuesday's "what's so special about 10928094208 in base 77" A281335 curio that the factorization contained an exponent. Arithmeticians have another way of expressing factors that have greater-than-one exponents and that way is called "with multiplicity". For example, in base 43 the smallest solution to A281335 is 10969263:

(3,8,41,23,6) = 3^6 * 41 * (8,23)

But if we insist that the factors be expressed with multiplicity:

(3,8,41,23,6) = 3 * 3 * 3 * 3 * 3 * 3 * 41 * (8,23)

Which is no longer a solution. So I created another OEIS sequence, A281336, to deal with that situation. In base 43 the smallest solution to A281336 is 12505821873:

(1,42,2,41,3,29,35) = 3 * 29 * (42,1,41,2,35)

Yes, there is no multiplicity in this particular example but that is not a necessity. As with my other sequence, I have an even bigger number as a solution for a smaller base (37) but have yet to discover it.

Wednesday, February 1

14 houses gone

The effects of 1954's Hurricane Hazel reached southern Ontario on Friday, October 15 and proceeded overnight into Saturday, October 16. One of the photographs published by the Toronto Daily Star on Monday, October 18 (page 5) was this:

Printed atop the picture was "14 houses gone, 35 people dead". (On page 1 it is 36 dead.) The montage was credited to Eric Cole and Ed Parker, though I'm not sure either one took the aerial shot. Here are three more photos scrounged from the Net that set the floodwater scene; two aerials and a remarkable ground view (credited to the Weston Historical Society, here) from the river's other side:

In the ground-view image, the submerged part of Raymore is on the far left and near the top. The two somewhat submerged structures in the right-half of the picture would be on the east (Weston) side of the Humber river. To their left, three large trees obscure Gilhaven Avenue, clearly visible (from the air) in the middle image, where you can see those trees on the other side of the river. To the left of the trees is a toppled bit of bridge abutment that is still there.

The 35 dead is still generally touted as the Raymore Drive "drowned" contribution to the total Canadian fatality count. A memorial plaque at the site instead suggested 32. Of course the Star newspaper's locations of the lost houses was conjectural. Another recreation (showing additional houses lost) suggests a different layout.

There are aerial maps of Toronto (1947-1992) so one can sort-of see what was lost and where. [I'll share my versions of 1947, 1950 and 1956.] There's a much-too-fast simulation of the disaster from the Toronto and Region Conservation Authority which (I expect) has positioned correctly all of the houses (seen in this screen grab from the Weston side of the river):

The TRCA has of course very good elevation data and this likely played an important role in their rising-water simulation. Alas, it's useless in helping us decide which houses were "lost". The story told is that a 1950-built footbridge across the river (seen in the above near the bottom right) lost its abutment on the Raymore side and the still connected wire ropes — now in the water — proceeded to collect debris in an arc emanating from the Weston side. Much water was thus redirected towards the unlucky buildings. The irony is that this bridge replaced a much-less sturdy earlier version (called in the article a swing bridge) that was deemed unsafe. Betty Kennedy (in her 1979 "Hurricane Hazel") called the replacement bridge a swing bridge also. This had me more than a little confused.  But a CBC radio interview (the site has an incorrect broadcast date) has an unidentified man calling it a swing bridge as well, so there should be no doubt that that is what the footbridge was called.

I'll do a listing of Raymore house numbers (even — river side — first) and the fatalities that I believe are associated with those dwellings:

#134: Girodat   Paul, Mary Beedham [found Oct 29]
#136: Newing   * Caroline Annie Beavan, * Gerald Norman [found Oct 22], ... 
          Salt   * Vera Frances [found 30 Jul 1955          \ ... Gerald John [found ~31 Jul 1955]
#138: Topliss   Annie May Martin, Albert
#140: Boyd   James
          Hall   Kenneth 
          LeBlanc   Alice
          Peasley   Lambert, Doris, Sylvia, (Shirley)
#142: Smith   John Clive, John William, Grace Anne Dunn [found Oct 30]
#144: Gillan   George H, Helen Stimson [found Oct 24, (child)
#148: Edwards   Joan I Jesson, Carolyn JKenneth Charles [found Oct 31], Frank K, John C
          Neil   Jean R Edwards, Susan L, Adele B, Darlene S [found Oct 24 
#152: McGarvey   Philomina Johnson, Jacqueline, Donald
#154: Brough   * Wilhelmina Helen Campbell

#143: Babbage   * Claude
          Jeffries   Edward Albert, Elizabeth Mrs. Thomas Sr.

  unbracketed number of people mentioned and assumed dead: 36
  italics: 4 missing Dec 19543 missing Aug 1955: a misreport, or who else was found?
  magenta: 3 declared dead by two newspapers on Oct 18 but no further mention found

The Toronto Star, Oct 18, page 2, misreported Caroline Newing as Katherine. Caroline's obituary gives her maiden name as Bevan. It was Beavan.

A Shirley Peasley is reported missing in the Globe and Mail, Oct 18, page 2. I found no further mention of her and can only assume that it was meant to be Sylvia.

"One Gillen child" is listed after the George and Helen 'missing' entries in the Globe and Mail, Oct 18, page 2. By the time the Toronto Star came out that day there was no such mention. I have no obituary for George, but Helen's mentions no child — nor does their combined memorial cemetery headstone.

"... and one adult, name unknown" accompanies a missing Kenneth Edwards in the Globe and Mail, Oct 18, page 2. In all likelihood this is an uninformed, duplicate reporting of the then-missing John Neil (who was in fact very much alive). Perhaps the connection hadn't at that point been made because John's dead wife Jean was reported at 148 Raymore while Kenneth's dead wife Joan was misreported at 248 Raymore.

Mrs. Thomas Jeffries Sr. is declared dead in the Toronto Daily Star on Oct 18, page 2, while the earlier Globe and Mail, Oct 18, page 2, had "Mrs. Tom Jefferies" missing — but there is a dead "unidentified woman, about 60" on page 1 that may well have been her. No obituary — but I have this mention. I'm still looking for her. I've got an 1885 Toronto-born Albert Edward Jeffries but his father's name isn't Thomas.

Afterthoughts: The 15 Aug 1959 Globe and Mail had a story on Thomas McGarvey having been charged with the holdup of a drug store, noting that in 1954 he "saw the water snatch his mother and sister". The article ends with his "father and a brother" having "escaped". [I have the obituary for his brother Donald.] The 13 Mar 1963 Toronto Daily Star reported on a gas explosion at a home on Raymore, ending the short article with: "During Hurricane Hazel, 23 persons on Raymore Dr. drowned." [Presumably a transposition typo.]

Tuesday, January 31

What's so special about 10928094208 in base 77?

One of my current recreational arithmetic side projects is creating a b-file (a longer list of the sequence) for OEIS A281335. 10928094208 is the smallest solution in base 77:

(4,2,67,10,58,31) = 2^10 * 31 * (58,4,67)

The digits of the number (on the left) are matched by the digits of its factorization (on the right). The solution for base 73 is an even bigger number — but I haven't yet figured it out!

Wednesday, January 25

Dem dry bones

When I wrote my Weston bones article eleven days ago, I was under the illusion — based on my interpretation of the 1911 newspaper articles — that the find was underneath what was to become Westminster Church School. In a cover story at WestonWeb, a Chris (from the Toronto and Region Conservation Authority) commented that the bones site (AkGv-6) was here; in other words, a little further south. This had me look again at the upper half of the photograph:

There are two buildings in the background and I thought I discerned telephone poles in the gap between them. This had to be Main St. — now Weston Rd. Below is a depiction of this part of Weston — a 1910 map (on the left) and a 1913 map (on the right). Main St. runs from upper left to lower right, cutting each map roughly in half.

Yellow structures are made of wood; red, of brick. In the three years between the maps some new buildings have appeared. Some of the wood structures have been replaced with brick. There are also some subtle displacements. In particular, the only two structures below Main St. in 1910 appear on adjacent lots in 1913. I don't think they moved the buildings to make room for the Sunday School (which appears as the big red square). Instead, it seems likely that the 1910 placements were made in error and corrected in the 1913 map. Good thing. You see, I thought those two structures were the buildings in the photograph. But there wasn't enough space between them to account for the gap in the photo. With the 1913 correction, there is now a reasonable window of Main St. (blue arrows) from some location south of the school. And this jives with Chris' placement of the bones site.

Addendum: I've just replaced the maps picture with a new version that has pasted into place (in the 1913 part) two buildings at the end of Hillcrest Rd. that appeared in an adjacent map plate and should therefore have appeared in this one. The newly added brick building now at the bottom of the map appears to be #28 Hillcrest and directories tell me that a Frank Munshaw lived there. Mrs. Munshaw was mentioned in the 1911 Globe article:

Two splendid skulls, however, were rescued from the depredations of the innocent pillagers by Mrs. Frank Munshaw, whose husband is the tenant of the property in which the skeletons were found.

I am now doubting that the two buildings in the photograph are the ones I posited (above). My current best guess is that the L-shaped structure at the bottom of lot P is the building on the left and one of the structures on lot N is the building on the right:

Sunday, January 22

Lost and found

On December 12 we had our first significant snowfall of the winter. I wrote about it here. Also noted in that post is that I lost my camera's lens cap. I'm happy to report that this morning I found it. There has been an ongoing January thaw (most of the snow is gone) but — as evidenced by the muddy tracks in my photo — I hadn't found the cap sooner because a small excavator (helping lay a new Bell Fiber cable) had been parked atop it.

Thursday, January 19

Easy does it

One thing leads to another. Once I got my library card so that I might access (online) the original newspaper articles for my Weston bones article, I used the tool to go over some family genealogy. I had previously relied on a research associate's access to archived Toronto newspapers for such information but now I was in the driver's seat!

This photo was stolen from here. My wife's grandfather, Frank Powers, had in his obituary that he was a "salesman for the Easy Washing Co. Ltd. for some years". He appears to have had a variety of jobs in his life, including (in the early 1910s) working for a printer and a jeweler. At any rate, I quickly learned that the clothes wringer at the top of the machine was called a mangle. And it would be natural to suppose that this is where the verb to mangle originated. Actually, it was the other way around.

Saturday, January 14

Dem bones

Less than a week ago I started transcribing the 1917 directory of the then-town of Weston, the part of Toronto in which I have resided now for more than fifty years. Proofreading the information (to the extent possible) has led me to look for some streets that are no longer there, which has led me to a remarkable photograph (of which the above is a crop). I'm prepared to believe that putting the boy in there with a shovel was done as a joke. One has to wonder what explanation was given at the time for the presence of these bones.

Glenn Turner (The Toronto Carrying Place, 2015) writes: "So, for some of its distance at least, Weston Road actually may be the Carrying Place. Archaeologist Dr. Shaun Austin believes that the stretch from Wilby Crescent ... to Rectory Road does in fact follow the route of the trail. He bases this on newspaper reports from 1911 of skulls and skeletons being found 'on an old Indian trail'. The burials were part of an ossuary (literally a boneyard, typical of Iroquoian nations like the Wendat and Onondowahgah) located on the site of today's Weston Park Baptist Church, ..." Alas, Weston Park Baptist Church (back then, Alexander Memorial Baptist Church, a small building at the back of today's church) is on the other side of Weston Road from the then-to-be-built (Presbyterian) Westminster Church School (later Westminster United Church, photo 1953; a parking lot, 1957-1970; by 1971, this high-rise), near where these bones were actually found.

An explanation for the (supposedly "men's") bones provided by Barb Shiells, a director of the Weston Historical Society: "Archaeologists advised that the bones were part of a native burial ground and likely dated back to 1425-1450. It was the aboriginal custom, when they were moving on to establish a new village, to hold a sacred ceremony to show respect for their dead. They would gather the previously buried remains and re-inter them in one large pit."

I'm largely unconvinced. For a little perspective, here's what happened on a more recent Toronto "ossuary" find. I've also dug up two newspaper accounts of the 1911 find and one follow-up. I've highlighted the skeletal remains' layout assessments (in blue) contradicted by a supposed expert's assessment (in red) just a few days later. Unbelievable!

The Toronto Daily Star: Friday, 28 April 1911, page 1

Workmen commencing excavations for the erection of the new Presbyterian Sunday school at the corner of Main and Mill streets, Weston, to-day, came upon over thirty skulls and skeletons at a depth of only two feet from the surface.

The property upon which the interesting relics were found is situated a short distance from the bank of the Humber River, on the old Indian trail running from Lake Simcoe to the place where Toronto now stands, and it is thought by those citizens of Weston who have seen the bones, that they probably indicate the results of an Indian battle fought many years ago.

At any rate, the oldest residents of the town are authority for the statement that the land upon which the shallowly interred remains were discovered, was never used as a white man's burying ground. More bones were turned up hourly as the work went on, and a large crowd of interested spectators lined the street. No decision has as yet been reached as to the disposition of the relics, but it is probable that they will be donated to various museums.

"The general impression in the town seems to be that these are the bones of Indians, thrown in a haphazard way into shallowly dug trenches over half a century ago, and probably long before that," said Rev. Dr. McGillivray, pastor of the Presbyterian Church, for which the Sunday school is being built. "People who have lived here for as long as 50 years can recollect no occurrence which would account for the burial of so many people in such a way."

The bones and skulls are all in a fair state of preservation and were first encountered about fifty feet from the bank of the river. No weapons or other articles that might have explained the matter have been found as yet.

The house which stands on the adjoining lot is 65 years old. The strange thing is that the bones, buried so near the surface, were not discovered long ago. Latterly, the land covering them had been cultivated as a household garden. The property was purchased by the church from a Mr. Hunter of Toronto.

A further examination of the bones and the ground in which they were found may serve to reveal some valuable historic information.

The Globe, Saturday, 29 April 1911, page 9

Bones and skulls comprising fourteen skeletons, possibly of Indians, were dug up by a road-scraper at Weston about 9 o'clock yesterday morning, shortly after excavation had been commenced for the foundation of a new Presbyterian Sunday school hall on the property at the corner of Mill and Main streets. A remarkable fact in connection with the accidental discovery of the skeletons is that they were only eighteen inches below the surface.

The reasons for the conclusion that the bones were those of Indians are many. The Weston road, which runs along beside the scene of the find, was, according to history, formerly an Indian trail. It is also close to the east bank of the Humber River, which in earlier days was navigable for flat-bottomed boats and canoes. An examination of the skulls themselves reveals unmistakable aboriginal anatomical features. Two years ago a skeleton with Indian beads was unearthed just off the main street of Weston, and four or five years ago the skeleton of an Indian chief was found wrapped in a blanket.

Mr. Hector Hart, contractor for the excavation work, told the Globe that while a scraper was at work on a drain being run through the property just acquired by the Presbyterian Church, he noticed some bones in the earth disturbed. The next turn of the scraper unearthed more bones and skulls, and before the ditch was finished fourteen skulls were found. That there is a big trench running parallel with the bank of the river and that the excavation cut at right angles was only a short section of it, and that there are probably scores of skeletons still uncovered, were the views of Mr. Hart and of Rev. A. H. MacGillivray, pastor of the Presbyterian Church.

The spot at which the skeletons were discovered is a few feet back of the church lot and lies in the property of Mr. A. T. Hunter of Hunter & Hunter, barristers, so that Mr. Hunter is the real owner of the aboriginal specimens, although a few curio seekers in town have carried off some of the more complete skulls.

Two splendid skulls, however, were rescued from the depredations of the innocent pillagers by Mrs. Frank Munshaw, whose husband is the tenant of the property in which the skeletons were found.

A large number of persons, mostly curious children, were gathered about the drain throughout the day, and the discovery was the topic of conversation over the entire town.

One resident told a Globe reporter the skeletons was not those of Indians, but the bones of gallant Canadians who fought in the rebellion of 1837 or the war of 1812. He said many lost their lives at Weston during the former trouble and that there were so many fatalities that, instead of burying the bodies in separate graves, the people dug a trench and threw them all in together.

"But," The Globe reporter remonstrated, "Canadians were too highly civilized then to be guilty of such barbarous practices as communal burials."

"Not by any means," retorted the Westonian. "They were not so highly civilized then as you might imagine. And there's a big pile of Canadians to-day no higher up in the ladder of civilization."

The bones and skulls are pretty well preserved and look as if their age should be measured, not by centuries, but by decades or years. An examination of the teeth in some of the jaw-bones shows them to be in a state of perfect preservation. They look sounder than a lot of the teeth being used in 1911 for purposes of mastication.

The skeletons, as an examination of their relative position before being taken from the ground showed, indicated that the bodies at the time of interment were thrown into the trench together without any orderly arrangement, and that the interment probably took place during or just subsequent to hostilities between Indian tribes.

Mr. Hart, who has studied Parkman carefully, stated the Huron Indians originally occupied the vicinity of Weston, but that later the tribes of the eastern States, who were more powerful, drove the Hurons back to Lake Simcoe and Georgian Bay, and that the skeltons found were probably those of the Iroquois or Mohawk Indians.

The Globe, Tuesday, 2 May 1911, page 8

Dr. Rowland B. Orr, Superintendent of the Provincial Museum, yesterday afternoon paid a visit to the scene of the unearthing of fourteen skeletons, supposed to be those of Indians, in Weston.

He told The Globe last night after his return that there was no question about their being the skeletons of Indians. The very arrangement of the bones, he says, when they were unearthed was sufficient ground for such a conclusion. He stated that the skulls were close together and that the bones forming the rest of the skeletons lay in such a manner as to form a sort of cart-wheel, with the skulls at the hub. He added that this was the customary mode of burial among the Indians centuries ago and that not only were these Indian skeletons, but they had been there for probably three hundred or more years.

Dr. Orr secured two skulls to take measurements of their various dimensions and also to preserve as curios.

Dem dry bones!