A Bimagic Queen’s Tour

walkington semi-bimagic queen's tour
Image: William Walkington (CC BY-NC-SA 4.0)

A queen’s tour is the record of a chess queen’s journey around an empty board in which she visits each of the squares once. If the squares are numbered by the order in which she visits them, then the resulting square is magic if the numbers in each rank and file sum to the same total. It’s bimagic if the squares of these numbers also produce a consistent total.

William Walkington has just found the first bimagic queen’s tour, which also appears to be the first bimagic tour of any chess piece. (William Roxby Beverley published the first magic knight’s tour in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science in 1848.)

Note that here the long diagonals don’t produce the magic sum, as they would in a magic square. This constraint is normally dropped in a magic tour — in fact, of the 140 magic knight tours possible on an 8×8 board, none have two long magic diagonals, and no bimagic queen’s tour with qualifying diagonals is possible on such a board either.

More details, and an interesting description of the search, are on William’s blog. He has been told that a complete list of such bimagic queen’s tours is within reach of a computer search, though the field is daunting — there are more than 1.7 billion essentially different semi-bimagic squares possible on an 8×8 board, and each allows more than 400 million permutations.

| Science & Math

The Isolator

https://manifold.umn.edu/read/the-perversity-of-things-hugo-gernsback-on-media-tinkering-and-scientifiction/section/69697807-3c5f-4de5-aa3b-d070728205f9

Irritated with distractions in his editorial work, Hugo Gernsback designed a helmet “to do away with all possible interferences that prey on the mind”:

The first helmet constructed as per illustration was made of wood, lined with cork inside and out, and finally covered with felt. There were three pieces of glass inserted for the eyes. In front of the mouth there is a baffle, which allows breathing but keeps out the sound. The first construction was fairly successful, and while it did not shut out all the noises, it reached an efficiency of about 75 per cent. The reason was that solid wood was used.

In a later version he omitted the wood and added an air space between layers of cotton and felt, achieving an efficiency of 90 to 95 percent. Even the eyepieces are black except for a single slit, to prevent the eyes from wandering. “With this arrangement it is found that an important task can be completed in short order and the construction of the Isolator will be found to be a great investment.”

(He even designed an ideal office, with a soundproof door, triple-paned windows, and felt-filled walls, in which to wear this — see the illustration at the link below.)

(Hugo Gernsback, “The Isolator,” Science and Invention 13:3 [July 1925], 214ff.)

| Technology

Memorial

https://commons.wikimedia.org/wiki/File:Anthem_Veterans_Memorial_(15585427880)_crop.jpg
Image: Wikimedia Commons

The Anthem Veterans Memorial, in Anthem, Arizona, consists of five white pillars representing the Army, Marine Corps, Navy, Air Force, and Coast Guard. Each pillar contains a slanted elliptical opening, and the five are arranged so that at 11:11 a.m. on Veterans Day, November 11, the sun’s light passes through all five and illuminates the Great Seal of the United States, which is inlaid among 750 red paving stones engraved with the names of veterans.

https://www.youtube.com/watch?v=euiFY-RxeRE

| Art · History · Society · Technology

An Odd Word

https://commons.wikimedia.org/wiki/File:Ha_ha_wall_diagram.jpg
Image: Wikimedia Commons

In E.M. Forster’s 1907 novel The Longest Journey, the description of the country estate Cadover contains a surprising term:

The lawn ended in a Ha-ha (‘Ha! ha! who shall regard it?’), and thence the bare land sloped down into the village.

A ha-ha is indeed the term for a sort of buried wall adjoined by a sloping ditch — it will keep deer out of your garden without blocking the view. But how it came by that name seems uncertain. Possibly it’s a shortened form of “half and half” (half wall, half ditch), and possibly it’s named for the cries of its observers — the earliest usage in the Oxford English Dictionary is John James’ 1712 translation of Antoine-Joseph Dézallier d’Argenville’s Theory and Practice of Gardening — he refers to “a large and deep Ditch at the Foot.., which surprizes..and makes one cry, Ah! Ah! from whence it takes its Name.”

In Terry Pratchett’s novel Men at Arms, a ha-ha is accidentally specified to be 50 feet deep. The result is called a hoho, and it claims the lives of three gardeners. In Snuff, two characters go for a walk in the countryside and “navigate their way around the ha-ha, keep their distance from the ho-ho and completely ignore the he-he.”

| Language

Moving Words

In October 1996, Parade magazine published the results of a vanity license plate contest that received more than 7,000 entries. Here are the 10 winning plates:

  1. IRIGHTI
  2. RUD14ME?
  3. HAHAHAHA
  4. XQQSME
  5. IM12XL
  6. ULIV1S
  7. AXN28D+
  8. VAN GO
  9. H2OUUP-2
  10. TI-3VOM

What are their meanings?

Click for Answer
| Language

Bending the Rules

New York zoning rules limit the height of skyscrapers, so Oiio Studio has proposed an innovative solution: Bend the building into a horseshoe. Designer Ioannis Oikonomou’s “Big Bend” building would be the “longest” building in the world, at 4,000 feet, but it would stand only 200 feet taller than One World Trade Center, currently the city’s tallest building.

“If we manage to bend our structure instead of bending the zoning rules of New York we would be able to create one of the most prestigious buildings in Manhattan,” the firm says in its building proposal. “The Big Bend can become a modest architectural solution to the height limitations of Manhattan.”

Whether that can be done remains to be seen. The project remains in the proposal stage.

| Oddities · Technology

For What It’s Worth

In 1882 Anton Chekhov published eight “Questions Posed by a Mad Mathematician.” Here are the first three:

  1. I was chased by 30 dogs, 7 of which were white, 8 gray, and the rest black. Which of my legs was bitten, the right or the left?
  2. Ptolemy was born in the year 223 A.D. and died after reaching the age of eighty-four. Half his life he spent traveling, and a third, having fun. What is the price of a pound of nails, and was Ptolemy married?
  3. On New Year’s Eve, 200 people were thrown out of the Bolshoi Theater’s costume ball for brawling. If the brawlers numbered 200, then what was the number of guests who were drunk, slightly drunk, swearing, and those trying but not managing to brawl?

The full list appears in The Undiscovered Chekhov, translated by Peter Constantine (1998). No answers are provided.

| Literature · Science & Math

Laconic

At the 1961 Solvay conference on physics, Abdus Salam overheard this conversation between Richard Feynman and Paul Dirac:

Feynman extended his hand towards Dirac and said: ‘I am Feynman.’ It was clear from his tone that it was the first time they were meeting. Dirac extended his hand and said: ‘I am Dirac.’ There was silence, which from Feynman was rather remarkable. Then Feynman, like a schoolboy in the presence of a Master, said to Dirac: ‘It must have felt good to have invented that equation.’ And Dirac said: ‘But that was a long time ago.’ Silence again. To break this, Dirac asked Feynman: ‘What are you yourself working on?’ Feynman said: ‘Meson theories’ and Dirac said: ‘Are you trying to invent a similar equation?’ Feynman said: ‘That would be very difficult.’ And Dirac, in an anxious voice, said: ‘But one must try.’

“At that point the conversation finished because the meeting had started.”

(Abdus Salam, “Physics and the Excellences of the Life It Brings,” in Ideals and Realities: Selected Essays of Abdus Salam, 1987.)

| Science & Math