Magic Space

knecht most-perfect square

Craig Knecht, whose “terraformed” magic squares we explored in 2013, has begun to experiment with applying “magic” properties to David Hilbert’s space-filling curve.

The Hilbert curve finds its way to every cell in the square above by following the pattern shown at the lower left. Knecht divided that path into eight-cell segments, as shown in (a), and then sought solutions in which each colored eight-cell panel produced the magic sum of 260 while each of the eight ordinal positions across the eight panels did so as well. For example, in (b), a large red digit 1 marks the “first” position in each panel; the hope was to find values for these eight cells that would sum to 260, and likewise for all the “second” cells, the “third” ones, and so on.

The result, shown in (c), is a “most-perfect” magic square: Each colored panel sums to 260, and every set of cells that are 8 spaces apart on the Hilbert curve also sum to 260.

The next step was to apply this idea in three dimensions, and recently Knecht made the breakthrough shown below — a 4×4×4 “most perfect” number cube. The 64 numbers in the cube can be broken into 36 2×2 subsquares in each dimension, as shown. In all 108 of these subsquares, the four constituent numbers total 130. And as with the two-dimensional square above, a Hilbert curve can be drawn through the cube that visits each cell once, and cells that are eight cells apart on this curve sum to 260.

One of Knecht’s correspondents pointed out that the cube is even magicker than he had supposed: The “wraparound” subsquares (for example, 5, 28, 44, and 53 on the top of the cube) also sum to 130, as does each set of four corners, making a total of 192 2×2 subsquares that sum to 130.

“So in summary … making the Hilbert space-filling curve path have this magic property of values 8 spaces summing to the magic constant + this 2×2 planar criteria produces a very interesting cube!”

knecht most-perfect cube

Black and White

spencer chess problem

G.B. Spencer devised this ingenious diagram in 1906: It contains 16 separate chess problems, one on each rank and file. In each case, ignore the pieces not on that rank or file and find a way for White to mate in two moves.

For example, the solution to the problem on the first rank is 1. Bd4 Ke2 2. Ng3#. What are the other 15 solutions?

Click for Answer

Viewpoint

https://commons.wikimedia.org/wiki/File:Karl_D%C3%B6nitz.jpg

Captured by the British near the end of World War I, submarine commander Karl Dönitz was a prisoner at Gibraltar when news of the armistice came through. As he and a fellow German officer watched the celebrations “with infinitely bitter hearts,” a British captain joined them on deck.

Dönitz waved his arm in a gesture to encompass all the ships in the roads, British, American, French, Japanese, and asked if he could take any joy from a victory which could only be attained with the whole world for allies.

‘Yes,’ the Captain replied after a pause, ‘it’s very curious.’

In his memoirs, Dönitz wrote, “I will hold the memory of this fair and noble English sea officer in high regard all my life.”

(From Peter Padfield, Donitz: The Last Führer, 2013.)

Parrot Talk

https://commons.wikimedia.org/wiki/File:Commanville_-_Souvenirs_sur_Gustave_Flaubert_-_Illustration_p._10.png

At his death in 1880, Gustave Flaubert left behind notes for a Dictionary of Received Ideas, a list of the trite utterances that people repeat inevitably in conversation:

  • accident: Always “regrettable” or “unlucky” — as if a mishap might sometimes be a cause for rejoicing.
  • Archimedes: On hearing his name, shout “Eureka!” Or else: “Give me a fulcrum and I will move the world.” There is also Archimedes’ screw, but you are not expected to know what that is.
  • bachelors: All self-centered, all rakes. Should be taxed. Headed for a lonely old age.
  • book: Always too long, regardless of subject.
  • classics: You are supposed to know all about them.
  • forehead: Wide and bald, a sign of genius, or of self-confidence.
  • funny: Should be used on all occasions: “How funny!”
  • gibberish: Foreigners’ way of talking. Always make fun of the foreigner who murders French.
  • handwriting: A neat hand leads to the top. Undecipherable: a sign of deep science, e.g. doctors’ prescriptions.
  • jury: Do everything you can to get off it.
  • metaphors: Always too many in poems. Always too many in anybody’s writing.
  • original: Make fun of everything that is original, hate it, beat it down, annihilate it if you can.
  • pillow: Never use a pillow: it will make you into a hunchback.
  • restaurant: You should order the dishes not usually served at home. When uncertain, look at what others around you are eating.
  • taste: “What is simple is always in good taste.” Always say this to a woman who apologizes for the inadequacy of her dress.
  • war: Thunder against.
  • young gentleman: Always sowing wild oats; he is expected to do so. Astonishment when he doesn’t.

“All our trouble,” he wrote to George Sand, “comes from our gigantic ignorance. … When shall we get over empty speculation and accepted ideas? What should be studied is believed without discussion. Instead of examining, people pontificate.”

(Thanks, Macari.)

Cover Story

https://www.google.com/patents/US2930053

Making a bed is a tiresome chore, observed inventor Richard Nowels in 1959, because so much time is spent in walking and bending: “It is necessary to pull the covers from both sides of the bed, from the bottom end of the bed, and possibly also from the top end of the bed.”

A better solution, he decided, is to hide an electric motor under the bed and connect it by cords to the edges of the bedclothes. Now when you get out of bed you can turn on the motor and draw the sheets and blankets immediately into their proper positions.

It’s handy in the middle of the night, too: “Kicked-out covers may be automatically re-tucked into place by the bed occupant merely by flipping a switch.”

A Reflexive Rainbow

sallows color table

From Lee Sallows: The international color code is used to mark the values of electronic components such as resistors. It assigns a distinct color to each of the 10 decimal digits, as seen in the center column of the table at right: 0 = BLACK, 1 = BROWN, …, 9 = WHITE.

Lee’s table has an ingenious reflexive property. The letters in the left-hand column are associated with the values -1 to -9, and those in the right-hand column with the values 1 to 9.

Now spelling the name of each color produces a sum that matches the number represented by that color:

sallows color sums

“This is more remarkable that it may seem,” Lee writes, “because the numbers assigned to the letters are now restricted to single-digit values only.”

Podcast Episode 98: The St. Albans Raid

https://commons.wikimedia.org/wiki/File:St_Albans_Raiders.gif

Seemingly safe in northern New England, the residents of St. Albans, Vermont, were astonished in October 1864 when a group of Confederate soldiers appeared in their midst, terrorizing residents, robbing banks, and stealing horses. In this week’s episode of the Futility Closet podcast we’ll tell the story of the St. Albans raid, the northernmost land action of the Civil War.

We’ll also learn about Charles Darwin’s misadventures at the equator and puzzle over a groundskeeper’s strange method of tending grass.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on the St. Albans raid:

Dennis K. Wilson, Justice Under Pressure: The Saint Albans Raid and Its Aftermath, 1992.

Robin W. Winks, The Civil War Years: Canada and the United States, 1998.

Stuart Lutz, “Terror in St. Albans,” Civil War Times Illustrated 40:3 (June 2001).

Rick Beard, “When the Rebels Invaded Vermont,” New York Times, Oct. 17, 2014.

“A Reminiscence of the St. Albans Raid,” Montreal Daily Witness, April 5, 1878.

“Confederate Raid on St. Albans, Vt.,” Pittsburgh Gazette Times, Oct 21, 1914.

“Leader of Raid on St. Albans, Vermont, Centre of Controversy at Champlain Celebration,” Boston Evening Transcript, May 9, 1912.

Edgar Andrew Collard, “Of Many Things …,” Montreal Gazette, March 28, 1969.

“English View of the St. Albans Raid Case,” Halifax Morning Chronicle, Jan. 24, 1865.

Listener mail:

Wikipedia, “Line-Crossing Ceremony” (accessed March 18, 2016).

R.D. Keynes, ed., Charles Darwin’s Beagle Diary, 2001.

Jacqueline Klimas, “Navy Leaders Try to Stamp Out Hazing, But Many Sailors Question the Rules,” Military Times, July 2, 2013.

Wikipedia, “Plimsoll Shoe” (accessed March 18, 2016).

This week’s lateral thinking puzzle is from Paul Sloane and Des MacHale’s 1998 book Ingenious Lateral Thinking Puzzles.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!