Another Puzzling Commute

A few weeks after his first confusing journey home from the train station, Smith again finishes work ahead of schedule and takes an early train home. This time he arrives at his suburban station half an hour early. Again, rather than wait for the chauffeur, he starts walking home. And as before, he meets his chauffeur on the road, who picks him up promptly and takes him home. How many minutes early do they reach the house this time?

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Image: Wikimedia Commons

Canadian puzzle designer Jeff Widderich invented this game in 2007. The goal is to place a digit 1-9 in each white cell so that each crossword-style “word” contains a straight, that is, a set of consecutive numbers in some order. For example, the top row of five digits might contain 62534, but not 91548.

One other constraint: Each full row or column must contain no repeated digits. That means, for example, that each of the two long vertical “words” will contain all nine digits. The digits in black cells count toward this constraint — the 9 in the black cell near the center means that no 9 appears elsewhere in its row or column. Can you complete the rest of the diagram?

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An Unsolved Puzzle

Sam Loyd’s 1903 Eighth Book of Tan explores the world of tangrams, the pastime of constructing specified shapes from a given set of seven pieces:

The book includes a few “paradoxes,” two of which I’ve mentioned here before. But here’s another:

“The seventh and eighth figures represent the mysterious square, built with seven pieces; then with one corner clipped off, and still the same seven pieces employed.”

The book includes no solution. The square on the left is just the regular “block” formation above. But if anyone has discovered how Loyd produced a “clipped” square using the same seven pieces, I haven’t been able to find it.

Elegant Variation

Students are sometimes taught never to use the same word twice in a sentence. This can lead to trouble: If a writer uses a synonym merely to avoid repeating a word, the reader can be left wondering whether there’s some significance in the change. H.W. Fowler called this affliction elegant variation and added, “There are few literary faults so widely prevalent.” He gives some examples:

The Bohemian Diet will be the second Parliament to elect women deputies, for Sweden already has several lady deputies.

Mr. John Redmond has just now a path to tread even more thorny than that which Mr. Asquith has to walk.

“What has Bohemia done that its females should be mere women?” Fowler asks. “And can Mr. Asquith really have taught himself to walk without treading?”

Charles W. Morton called this the “elongated yellow fruit” school of writing, after a famous second reference to a banana in the Boston Evening Transcript. (Sub-editors at the Guardian began using the term “povs” after one writer referred to carrots as “popular orange vegetables.”) Morton cited some further examples:

billiard balls = “the numbered spheroids”
Bluebeard = “the azure-whiskered wifeslayer”
Easter egg hunt = “hen-fruit safari”
milk = “lacteal fluid”
oysters = “succulent bivalves”
peanut = “the succulent goober”
songbird = “avian songster”
truck = “rubber-tired mastodon of the highway”

In A Slight Sense of Outrage, Morton wrote that the sin “lies somewhere between the cliché and the ‘fine writing’ so dreaded by teachers of English Composition. … It does bespeak an author who wishes to seem knowledgeable, and versatile. … It can also bespeak an author who is merely pompous.”

Getting Acquainted

Sheep can be trained to recognize human faces, even from photographs. In a 2017 study at Cambridge University, researchers trained sheep to recognize photographs of four celebrities (Fiona Bruce, Jake Gyllenhaal, Barack Obama, and Emma Watson). They learned to distinguish a celebrity’s face from another face 8 out of 10 times, and their performance dropped only about 15% when they were shown photographs taken at an angle. When an unfamiliar photograph of a handler was inserted randomly in place of a celebrity, they chose that photo 7 out of 10 times.

“During this final task the researchers observed an interesting behaviour. Upon seeing a photographic image of the handler for the first time — in other words, the sheep had never seen an image of this person before — the sheep did a ‘double take’. The sheep checked first the (unfamiliar) face, then the handler’s image, and then unfamiliar face again before making a decision to choose the familiar face, of the handler.”

Sheep are long-lived and have relatively large brains, so it’s hoped that studying them will shed light on illnesses such as Huntington’s disease.

Also: Pigeons can distinguish Monet from Picasso, and rats can distinguish spoken Japanese from spoken Dutch. “A previous study by Porter and Neuringer (1984), who reported discrimination by pigeons between music and Bach and Stravinsky, and the present study suggest that pigeons have abilities that enable them to identify both musical and visual artists.”


Many people know that you can form a pentagon by tying a strip of paper in a simple overhand knot.

Stephen Bleecker Luce’s seamanship manual of 1863 tells how to fold a blade of grass into an octagon (below):

It is first doubled short over itself, then 1 under 2, — leaving a space, then 2 over 1, and down through the centre of the triangle; next 1 over 2, and down through the centre, coming out on the opposite side, and so on until an octagonal figure is formed.

That’s pretty terse, but I think I’ve almost managed to do it tonight. Keep your eye on the drawing of the finished piece, and don’t form and flatten the finished shape until you’ve done all the weaving.

(Via The Ashley Book of Knots.)

Math Notes

The sum or difference of any pair of the numbers {150568, 420968, 434657} is a square:

420968 + 150568 = 7562
420968 – 150568 = 5202
434657 + 420968 = 9252
434657 – 420968 = 1172
434657 + 150568 = 7652
434657 – 150568 = 5332

Miracles and Agents

Jones tells a mountain to hop into the sea and it does so. Has he performed a miracle?

Well, no, writes University of Birmingham philosopher George Chryssides. If Jones repeats his feat, then he’s revealed an underlying causal principle that’s amenable to study just like the rest of the natural world. If he doesn’t repeat the feat, then there’s no support for the idea of a link between his command and the mountain’s movement — we know only that the two events coincided, not that one caused the other.

“In order … to determine the answer to the question, ‘Did Jones move the mountain?’ … we must ascertain whether similar effects would follow similar putative causes,” Chryssides writes. “Either an allegedly miraculous event is a violation of scientific law, in which case it could not be performed by an agent, or else it is performed by an agent, in which case it could not be a violation of scientific law.”

(George D. Chryssides, “Miracles and Agents,” Religious Studies 11:3 [September 1975], 319-327.)