Monday, June 20

Cubic Lock


These are the four pieces of Cubic Lock by Goh Pit Khiam as realized in exotic woods by Brian Menold. A union of the two pieces on the left (top, the key made up of 10 cubies; bottom, 9 cubies) is inserted into a union of the two pieces on the right (top, 23 cubies; bottom, 19 cubies) and by means of a strategic to-and-fro of the key, the pieces are shifted into place. It's not obvious that the finished 4*4*4 cube has an internal void until one counts and adds those cubies. A good show-off puzzle as one is unlikely to forget the assembly once one has put it together a few times.

Wednesday, June 8

The wall


I went to Raymore Park (on the other side of the Humber river) last Friday to see what progress had been made on the erosion-control retaining-wall they were putting in place (on my side of the river). Contrary to the intelligence in my previous entry on this, I can now see that the wall will not be so much "on top of the now-in-the-river foundation" as I had supposed but (rather) much higher and more-closely hugging the slope bedrock — which is actually being exposed for a more stable conglomeration. And that storm drain interruption will end up being a barrier to my ever walking along the full length of the wall. At any rate (depending on the wall-top width), it may be too high to be safe. The bits of white floating through the air (more evident in the second photo) is tree fluff.

Monday, May 23

577, 5569, ...

This incipient-sequence comment in T.D. Noe's A138290 recently caught my eye. So, prime p such that 2^(p+1)-2^q-1 is composite for all positive q < p. Carlos Rivera had it published as Puzzle 437 in 2008. Therein, Giovanni Resta notes that he had checked up to 15373 without finding a third term. All of my processors were busy of course but I hadn't been using Catherine's computer in a while (too slow for my needs), so...

This morning (after several weeks of number crunching) I noticed that her machine had come up with the third term: 29251. To give you an idea of the computing involved: on my latest iMac, p = 577 (174-digit numbers) verified compositeness for all terms immediately; p = 5569 (1677-digit numbers) took a minute; and p = 29251 (8806-digit numbers) needed more than four hours! Alas, I'm still one term shy of a new OEIS sequence contribution.

Monday, May 9

Mount Dennis


The part of Toronto in which I live used to be a town called Weston. I live in the south part of Weston, just northwest of a community called Mount Dennis. This photograph was taken in the very northern part thereof, looking down Weston Road towards Toronto's CN Tower. My camera is at maximum zoom, giving the intersection at Jane Street (200 meters away) and the streetscape beyond a seriously foreshortened look. The Tower is 10.8 km distant. I was on my way to Jane's Walk.

Thursday, April 21

Humber river retaining wall

When I speculated one month ago that the current armourstone toe of the valley slope on the east side of the Humber river (not far from my home) was going to be replaced with a higher version, I wasn't entirely correct. It may end up being higher but (more importantly) it will also be further into the river.


Roy Murray continues his Raymore Park updates on the construction (March 30, April 12, April 18, and a follow-up). In the last of these he lets on that the retaining wall will end up on top of the now-in-the-river foundation! This is actually good news for my side of the river because it means that I might be able to access this new stretch of real estate from the south (behind Denison Park) and walk it up to the footbridge (across the river) that lies beyond. Of course Roy worries that the now-narrower river channel will negatively impact his side of the river during flood events. Here's a bird's-eye Apple Maps view of the area (I've put a red square around my house):

Saturday, April 9

8613, ...

In numbers of the form 2^n-2^m-1, m<n-1, for what values of n is the sum of the values of m for which that number is prime equal to n?

Tuesday, April 5

She said, he said

On page 20 of the 23 November 1939 Cleveland Plain Dealer is a Good Morning from Claire MacMurray column called "Thanksgiving Nightmare" which tells the tale of a "Mrs. Amos Pinchot" writing a bit of verse whilst in a dream state:

Hogamus Higamus
Men are Polygamous
Higamus Hogamus
Women Monogamous

The men/women being polygamous/monogamous was certainly nothing new. A Google Books search finds references going back into the 1800s. What was new are the two words hogamus and higamus. Who coined them?

The Reader's Digest reprinted the poem and its origin in its May 1940 issue, giving the nonsense words wide circulation. Any attribution to folk after this date is therefore moot. No one appears to have come out of the woodwork to say: Hey, that's my misappropriated poetry! But we do have something else — someone suggesting that Claire MacMurray lied about the poem's authorship.

In 1942 James Grier Miller published a book called "Unconsciousness" in which he retells the story of the writing of the verse — but from the perspective of a man! The switch was accomplished by means of a footnote stating: "Mrs. Amos Pinchot has repeatedly been incorrectly said to have been the author of this quatrain. She denies any responsibility for it, however, and the true author appears to be shrouded in anonymity." Mr. Miller was certainly no stranger to referencing assertions, so it's somewhat disconcerting to see him not do so here.

There were two women who could have claimed to having been Mrs. Amos Pinchot: Gertrude Minturn (who married Amos in 1900) and Ruth Pickering (who married him in 1919). Gertrude died in May 1939 — six months before Claire MacMurray's story appeared. If Ruth Pickering denied authorship of the poem, when and where?

It strikes me as disingenuous to hold Claire MacMurray to a standard that we do not demand of James Grier Miller. Either of them could have promulgated an untruth. [A third possibility is that Ruth Pickering — in denying authorship — did so. A fourth possibility is that Claire MacMurray was referring to Gertrude Minturn.] Yet James Miller's unsupported claim somehow demeans that of Claire MacMurray to an extent that Garson O'Toole calls her story "fanciful". No matter. In the realm of word/phrase origins, early citation is everything. If one is at all uneasy about crediting Mrs. Amos Pinchot for the creation of those fanciful words, attribute them instead to Claire MacMurray*!

* "Claire MacMurray" was her pen name. She was born Bessie Claire McMurray on 12 Feb 1899 in Huntington IN. In 1916 she swam from Liverpool to Havana (Illinois) in under four hours. Marrying Edward Howard II in 1923, they had three sons (born 1925, 1927, and 1929). Claire saw publication of her starting-in-1936 newspaper columns as two books (1941, 1944, subsequently amalgamated into a third). The first of those books inspired the 1941 NBC radio series "We're Five in the Family". Claire died 31 Jul 2003 in Cleveland OH.

Sunday, April 3

L'Anse aux Meadows to Stormy Point

Here's a dozen distance/direction quotes culled from the online stories:

National Geographic (Mar 31): "hundreds of miles south of"
New York Times (Mar 31): "about 300 miles south of"
Huffington Post (Apr 1): "about 300 miles further south than"
Daily Mail (Apr 1): "400 miles (643km) south west of"
Canada Journal (Apr 1): "300 miles (480km) south of"
Gossip Monthly (Apr 1): "about 600 km south of"
CTV News (Apr 1): "approximately 500 kilometres from"
The Telegram (Apr 1): "almost 500 km south of"
The Register (Apr 1): "300km away from"
Digital Journal (Apr 1): "about 500 kilometres (300 miles) south of"
Canadian Press (Apr 1): "about 600 kilometres from"
The Japan Times (Apr 2): "about 700 kilometers (430 miles) away"

To give credit to Digital Journal's Karen Graham, she added "on the south-west coast". My measurement in Google Earth is 501.13 km at a heading of 214.96°, which corresponds (roughly) to southwest by south (SW by S = 213.75°).

Wikipedia now sports an entry on Point Rosee: "approximately 600 kilometres (370 mi) south of L'Anse aux Meadows". I'm curious how long it will take before it mentions Stormy Point.

Saturday, April 2

Stormy Point, Millville NL


When the (possible) new Vikings-settlement story broke on Thursday it was given a dateline location of Point Rosee. This dealt amateur geographers a good measure of grief because Point Rosie (by Rosée Harbour, in Fortune Bay) is (by way of English corruption) actually Point Enragée, on the Burin peninsula — nowhere near the archaeological site.

It took a little time to determine the site's exact location. And having that, I learned that it was actually called Stormy Point. So from whence comes the "Point Rosee"? It could be a local name. However, it's possible — likely even — that the archaeologists decided to disguise the location so as to keep the public at bay.*

I have to say, Stormy Point strikes me as a poor location for a settlement. It's totally open to the elements!

* Addendum: I just finished watching the Nova episode on this. Indeed, early in the program they called the location "secret".

Thursday, March 31

Cyclopean primes

After watching the Numberphile Glitch Primes and Cyclops Numbers video yesterday, I thought it a bit of a waste (because of an overly narrow definition) that there should only be one binary Cyclops number that is prime. What if we allow the solitary zero to be anywhere (except of course leftmost) in a binary all-ones number? Well, we would have the terms in Antti Karttunen's A095078, which I'm now calling Cyclopean primes. It would be easy to generate ten or twenty thousand of these and call it a day, but I'm interested in — given that these numbers are of the form 2^n -2^m -1 with m< n-1 — how many primes (values of m) there are for any given n. So, my table starts like this (indices n, bracketed m's):

 1
 2   {0}
 3   {1}
 4   {2,1}
 5   {3,1}
 6   {4,2,1}
 7
 8   {6,5,4,2}
 9   {7,5,3,1}
10   {5,2,1}
11   {3}
etc.

There are 38218 solutions up to (and including) n=10000, so an average of 3.8 solutions for a given n. The so-far largest number of solutions is 20, occurring at n=2850 and n=9510.