Projective Chess

In projective geometry, every family of parallel straight lines intersects at an infinitely distant point. Chess problem composers in the former Yugoslavia have adapted this idea for the chessboard, adding four special squares “at infinity.”

projective chess 1

Now a queen on a bare board, for example, can zoom off to the west (or east) and reach a square “at infinity” from which she attacks every rank on the board simultaneously from both directions. She might also zoom to the north (or south) to reach a different square at infinity; from this one she attacks every file simultaneously, again from both directions. Finally she can zoom to the northwest or southeast and attack all the diagonals parallel to a8-h1, or zoom to the northeast or southwest and attack all the diagonals parallel to a1-h8. These four “infinity squares,” plus the regular board, make up the field of play.

N. Petrovic created the problem below, published in Matematika Na Shahmatnoi Doske. White is to play and mate in at least two moves. Can you find the solution?

projective chess 2

Click for Answer
| Puzzles

Great Good Fortune

http://books.google.com/books?id=RTAVAAAAYAAJ

In Nature (January 23, p. 271) you give a letter from Mr. Scouller describing an interesting case of a rainbow, due to the image of the sun in water, which, with the ordinary primary and secondary bows, make up (there being no secondary to that formed by the reflected sun) the three which he saw. Here is a short account of what I saw long ago, almost in prehistoric times, in Scotland, where such sights ought, according to your correspondent, to be very commonly seen. I may mention that I saw at the same time, lasting some five minutes, eight well-defined rainbows of one sort or another.

In 1841, during the time of a long vacation party, spent at Oban, I walked out with my brother to Dunstaffnage, and we were on the top of the Castle, somewhere between 3 and 4 p.m., on a day in the middle of August. Not a breath of wind, bright sun over, I think, Lismore Lighthouse, dusky clouds all over Ben Cruachan and Conoll Ferry; the sea in the bay (bounded by Dunstaffnage in the west) as smooth as a pond. Gradually there appeared before us the astonishing sight of the aforesaid eight distinct rainbows, viz. primary and secondary ordinary bows; primary and secondary bows by reflected sun; primary and secondary bows formed by light from the real sun reflected from the water after leaving certain drops; primary and secondary formed by light from the sun reflected at the water, and, after leaving certain other drops, again reflected at the water. I have called the latter four distinct bows, because, although they looked like reflections of a solid set of four arcs, they were really formed by means of drops distinct from those which helped to make the first four bows. I append a sketch of what I saw.

— Percival Frost, letter to Nature, Feb. 6, 1890

| Oddities

Screwy

https://www.google.com/patents/US1087186

Patented in 1914, Socrates Scholfield’s “illustrative educational device” uses two spiral springs to demonstrate the existence of God. Or to demonstrate the tension between good and evil. Or to demonstrate the consciousness of an animal organism. Actually I’m not sure what it demonstrates, and I’ve read the five-page abstract twice.

This schematic device … provides an educational emblem of the conscious relation that must exist between the co-extensive dispensing mediums for beneficence and maleficence, in the terrestrial factory; and it clearly indicates that the attribute of maleficence, which is ascribed to the realm of the adverse medium, may, under certain changed conditions, be made subject to decrease, and to a change in its relative action; while the attribute of beneficence, which pertains to the realm of the controlling supreme governor, is unconditioned, unchangeable and everlasting.

Here’s the whole thing if you want to try it out. Be careful, I guess.

| Technology

The Steam Bicycle

http://commons.wikimedia.org/wiki/File:1884_Copeland_Steam_Cycle_%28replica%29_The_Art_of_the_Motorcycle_-_Memphis.jpg
Image: Wikimedia Commons

Visitors to Arizona’s Maricopa County Fair saw a surprising demonstration in 1884 — local inventor Lucius Copeland had added a steam engine to a bicycle to create a new vehicle that could travel 15 miles in an hour.

He sought funding for his idea but couldn’t summon enough public interest. It’s now recognized as one of the first motorcycles.

| Technology

Reduced Housing

http://commons.wikimedia.org/wiki/File:Grand_Piano_Room.jpg
Images: Wikimedia Commons

Inspired by Tennyson’s Idylls of the King, Colorado miniaturist Elaine Diehl spent 13 years and 6,000 hours creating Astolat Castle, a 9-foot dollhouse weighing 600 pounds. The weathered copper roof covers 29 rooms and 10 adjoining areas, staircases, and hallways outfitted with parquet floors, framed mirrors, tapestries, gold chandeliers, oil paintings, and fireplaces. The seven levels range from a wine cellar and an armory in the basement to a “wizard’s tower” outfitted with telescopes and zodiacal signs. The 1″ scale furnishings include seven periods and styles, including Spanish, Oriental, Tudor, 18th-century English, and Victorian.

“It’s been a wonderful hobby,” she told the Prescott, Ariz., Courier in 1990. “I used to not be able to wait to get home from work so I could play. The hours slipped by so easily, I would look at my watch and it would be three in the morning. … With this hobby, you can be in control of your own little world. In real life, you don’t have all those choices available to you.”

http://commons.wikimedia.org/wiki/File:Courtesy_of_Dr._Michael_and_Lois_Freeman.JPG
Image: Wikimedia Commons
| Oddities

A Tall Tail

http://commons.wikimedia.org/wiki/File:Halley%27s_Comet_-_May_29_1910.jpg

As skywatchers prepared for the return of Halley’s comet in 1910, their excitement turned to trepidation when astronomer Camille Flammarion warned that cyanogen gas in the comet’s tail could poison the atmosphere. The New York Times reported growing alarm among astronomers and warned, “Prof. Flammarion is of the opinion that the cyanogen gas would impregnate the atmosphere and possibly snuff out all life on the planet.” The Washington Post quoted astronomer Henri Deslandres that the comet might cause torrential rains; his colleague D.J. McAdam warned that “Disease and death have frequently been ascribed to the admixture of cometary gases with the air.”

As the fateful date approached, an ad appeared in a South African newspaper: “Gentleman having secured several cylinders of oxygen and having bricked up a capacious room wishes to meet others who would share the expense for Wednesday night. Numbers strictly limited.” In Texas, salesmen went door to door selling “comet pills” and leather inhalers. In Germany, anxious residents began wearing comet hats and carrying comet umbrellas.

On the evening of May 18, as Earth passed into the comet’s tail, hundreds marched in a candlelight parade in San Juan, and prayer vigils were held in St. Petersburg churches and on the hilltops around Mexico City. In Lexington, Ky., excited citizens held all-night services, “praying and singing to prepare … [to] meet their doom.”

Nothing happened. Well, nearly nothing: In Towaco, N.J., two men had offered to pay $10 for the best description of the event as viewed from Walkman Mountain. When the town’s weary residents returned from their vigil, they found their chicken coops empty.

| Society

Sweet Home

http://commons.wikimedia.org/wiki/File:John_Glover_-_Ullswater,_early_morning_-_Google_Art_Project.jpg

I am determined & feel sure, that the scenery of England is ten times more beautiful than any we have seen.– What reasonable person can wish for great ill proportioned mountains, two & three miles high? No, no; give me the Brythen or some such compact little hill.– And then as for your boundless plains & impenetrable forests, who would compare them with the green fields & oak woods of England?– People are pleased to talk of the ever smiling sky of the Tropics: must not this be precious nonsense? Who admires a lady’s face who is always smiling? England is not one of your insipid beauties; she can cry, & frown, & smile, all by turns.– In short I am convinced it is a most ridiculous thing to go round the world, when by staying quietly, the world will go round with you.

— Charles Darwin, letter to his sister, July 18, 1836. He was on board the Beagle, bound for Ascencion. He had written the previous December, “How glad I shall be, when I can say, like that good old Quarter Master, who entering the Channel, on a gloomy November morning, exclaimed, ‘Ah here there are none of those d—-d blue skys’.”

| Science & Math

Reunion

http://commons.wikimedia.org/wiki/File:MUTCD_W8-15P.svg

A Russian problem from the 1999 Mathematical Olympiad:

A father wants to take his two sons to visit their grandmother, who lives 33 kilometers away. His motorcycle will cover 25 kilometers per hour if he rides alone, but the speed drops to 20 kph if he carries one passenger, and he cannot carry two. Each brother walks at 5 kph. Can the three of them reach grandmother’s house in 3 hours?

Click for Answer
| Puzzles

Budget Trouble

http://commons.wikimedia.org/wiki/File:Open_Clip_Art_Library_Piggy_Bank.svg

An energetic boy got a piggy bank for his birthday. He decided that from then on he will number every bill he gets from his grandparents (1, 2, …) and put it all in his bank. During the first half year he got 2 bills, but at the end of this period he pulled out 1 bill (chosen at random). In the next 1/4 year he got 2 more bills, but at the end of this period he pulled 1 bill chosen at random from the 3 bills in his bank. In the next 1/8 year he repeated the same routine etc. (each period is half the length of the previous period). What is the probability that any of the bills he got during this year will remain in his bank after a full year of the above activity? Paradoxically the probability is 0, even though it is clear that he only spent half of his money. Can we offer the boy good financial advice without making him cut his expenses?

— Talma Leviatan, “On the Use of Paradoxes in the Teaching of Probability,” Proceedings of ICOTS 6, 2002

| Science & Math