Golomb Rulers

This conference room is 11 units long and has folding partitions at positions 2, 7, and 8. This gives it a curious property: For a meeting of any given size, the room can be configured in exactly one way. A meeting of size 6 must use partitions 2 and 8 — no other setup will work exactly.

This is an example of a Golomb ruler, named for USC mathematician Solomon Golomb. It’s called a ruler because the simplest example is a measuring stick: If we’re given a 6-centimeter ruler, we find that we can add 4 marks (at integer positions) so that no two of them are the same distance apart: 0 1 4 6. No shorter ruler can accommodate 4 marks without duplication, so the 0-1-4-6 ruler is said to be “optimal.” It’s also “perfect” because it can measure any distance from 1 to 6.

The conference room is optimal because no shorter room can accommodate 5 walls without equal-sized partitions becoming available, but it’s not perfect, because it can’t accommodate an assembly of size 10. (It turns out that no perfect ruler with five marks is possible.)

Finding optimal Golomb rulers is hard — simply extending an existing ruler tends to produce a new ruler that’s either not Golomb or not optimal. The only way forward, it seems, is to compare every possible ruler with n marks and note the shortest one, an immensely laborious process. Distributed computing projects have found the longest optimal rulers to date — the most recent, with 27 marks, was found in February, five years after the previous record.

First Impressions


Shortly before its orchestral premiere in 1885, Johannes Brahms performed his fourth symphony for a small private audience in an arrangement for two pianos, played by himself and Ignaz Brüll.

After the first movement Brahms paused to assess its effect, and critic Eduard Hanslick, who was turning the pages, said, “For the whole movement I had the feeling that I was being given a beating by two incredibly intelligent people.”

Reading List

FC book covers

Just a reminder — the new Futility Closet book, Futility Closet 2: A Second Trove of Intriguing Tidbits, is available now at Amazon. It contains hundreds of hand-picked favorites from our 10-year archive of curiosities in history, literature, philosophy, mathematics, and art. Some sample items from the index:

communists, driving habits of, 73
trombones, and hair loss, 139
prayers, Canadian, private, whether, 179
James, Henry, not a refulgent fireball of novelistic rigor, 174
plunges, earthward, righted dangerously, 111
whales, exasperated, 193
Carroll, Lewis, beset by asparagus, 174
Story magazine, hard up for Ws, 19

Pair it with our first book, Futility Closet: An Idler’s Miscellany of Compendious Amusements, for twice the oddity!

Far From Home


This is a detail from the allegorical painting Taste, Hearing and Touch, completed in 1620 by the Flemish artist Jan Brueghel the Elder. If the bird on the right looks out of place, that’s because it’s a sulphur-crested cockatoo, which is native to Australia. The same bird appears in Hearing, painted three years earlier by Brueghel and Peter Paul Rubens.

How did an Australian bird find its way into a Flemish painting in 1617? Apparently it was captured during one of the first Dutch visits to pre-European Australia, perhaps by Willem Janszoon in 1606, who would have carried it to the Dutch East Indies (Indonesia) and then to Holland in 1611. That’s significant — previously it had been thought that the first European images of Australian fauna had been made during the voyages of William Dampier and William de Vlamingh, which occurred decades after Brueghel’s death in 1625.

Warwick Hirst, a former manuscript curator at the State Library of New South Wales, writes, “While we don’t know exactly how Brueghel’s cockatoo arrived in the Netherlands, it appears that Taste, Hearing and Touch, and its precursor Hearing, may well contain the earliest existing European images of a bird or animal native to Australia, predating the images from Dampier’s and de Vlamingh’s voyages by some 80 years.”

(Warwick Hirst, “Brueghel’s Cockatoo,” SL Magazine, Summer 2013.) (Thanks, Ross.)

Podcast Episode 33: Death and Robert Todd Lincoln


Abraham Lincoln’s eldest son, Robert, is the subject of a grim coincidence in American history: He’s the only person known to have been present or nearby at the assassinations of three American presidents. In the latest Futility Closet podcast we describe the circumstances of each misfortune and explore some further coincidences regarding Robert’s brushes with fatality.

We also consider whether a chimpanzee deserves a day in court and puzzle over why Australia would demolish a perfectly good building.

Sources for our segment on Robert Todd Lincoln:

Jason Emerson, Giant in the Shadows: The Life of Robert T. Lincoln, 2012.

Charles Lachman, The Last Lincolns: The Rise and Fall of a Great American Family, 2008.

Merrill D. Peterson, Lincoln in American Memory, 1994.

Ralph Gary, Following in Lincoln’s Footsteps, 2002.

Sources for the listener mail segment:

“Lyman Dillon and the Military Road,” Tri-County Historical Society (accessed 11/06/2014).

Charles Siebert, “Should a Chimp Be Able to Sue Its Owner?”, New York Times Magazine, April 23, 2014.

This week’s lateral thinking puzzle is from Paul Sloane and Des MacHale’s 1994 book Great Lateral Thinking Puzzles. Some corroboration is here (warning: this spoils the puzzle).

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!

Warnsdorff’s Rule

The knight’s tour is a familiar task in chess: On a bare board, find a path by which a knight visits each of the 64 squares exactly once. There are many solutions, but finding them by hand can be tricky — the knight tends to get stuck in a backwater, surrounding by squares that it’s already visited. In 1823 H.C. von Warnsdorff suggested a simple rule: Always move the knight to a square from which it will have the fewest available subsequent moves.

This turns out to be remarkably effective: It produces a successful tour more than 85% of the time on boards smaller than 50×50, and more than 50% of the time on boards smaller than 100×100. (Strangely, on a 7×7 board its success rate drops to 75%; see this paper.) The video above shows a successful tour on a standard chessboard; here’s another on a 14×14 board:

While we’re at it: British puzzle expert Henry Dudeney once set himself the task of devising a complete knight’s tour of a cube each of whose sides is a chessboard. He came up with this:


If you cut out the figure, fold it into a cube and fasten it using the tabs provided, you’ll have a map of the knight’s path. It can start anywhere and make its way around the whole cube, visiting each of the 364 squares once and returning to its starting point.

Dudeney also came up with this puzzle. The square below contains 36 letters. Exchange each letter once with a letter that’s connected with it by a knight’s move so that you produce a word square — a square whose first row and first column comprise the same six-letter word, as do the second row and second column, and so on.

dudeney knight's tour word square puzzle

So, for example, starting with the top row you might exchange T with E, O with R, A with M, and so on. “A little thought will greatly simplify the task,” Dudeney writes. “Thus, as there is only one O, one L, and one N, these must clearly be transferred to the diagonal from the top left-hand corner to the bottom right-hand corner. Then, as the letters in the first row must be the same as in the first file, in the second row as in the second file, and so on, you are generally limited in your choice of making a pair. The puzzle can therefore be solved in a very few minutes.”

Click for Answer

In a Word

n. one who is ignorant

n. a stupid person

n. a mistake due to ignorance

adj. that does not think

adj. lacking wit or sense

n. gross ignorance or stupidity

adj. knowing little; ignorant

n. an opponent of sloth or stupidity

Different Strokes


G.H. Hardy had a famous distaste for applied mathematics, but he made an exception in 1945 with an observation about golf. Conventional wisdom holds that consistency produces better results in stroke play (where strokes are counted for a full round of 18 holes) than in match play (where each hole is a separate contest). So if two players complete a full round with the same total number of strokes, then the more erratic player should do better if they compete hole by hole.

Hardy argues that the opposite is true. Imagine a course on which every hole is par 4. Player A is so deadly reliable that he shoots par on every hole. Player B has some chance x of hitting a “supershot,” which saves a stroke, and the same chance of hitting a “subshot,” losing a stroke. Otherwise he shoots par. Both players will average par and will be equal over a series of full rounds of golf, but the conventional wisdom says that B’s erratic play should give him an advantage if they play each hole as a separate contest.

Hardy’s insight is that the presence of the hole limits a run of good luck, while there’s no such limit on a run of bad luck. “To do a three, B must produce a supershot at one of his first three strokes, while he will take a five if he makes a subshot at one of his first four. He will thus have a net expectation 4x – 3x of loss on the hole, and should lose the match, contrary to common expectation.”

In general he finds that B’s chance of winning a hole is 3x – 9x2 + 10x3, and his chance of losing is 4x – 18x2 + 40x3 – 35x4, so that there’s a balance f(x) = x – 9x2 + 30x3 – 35x4 against him. If x < 0.37 -- that is, in all realistic cases -- the erratic player should lose.

"If experience points the other way -- and I cannot deny it, since I am no golfer -- what is the explanation? I asked Mr. Bernard Darwin, who should be as good a judge as one could find, and he put his finger at once on a likely flaw in the model. To play a 'subshot' is to give yourself an opportunity of a 'supershot' which a more mechanical player would miss: if you get into a bunker you have an opportunity of recovering without loss, and one which you are naturally keyed up to take. Thus the less mechanical player's chance of a supershot is to some extent automatically increased. How far this may resolve the paradox, if it is one, I cannot say, and changes in the model make it unpleasantly complex."

(G.H. Hardy, "A Mathematical Theorem About Golf," Mathematical Gazette, December 1945.)


“The monuments of wit survive the monuments of power.” — Francis Bacon

Bug Hunt

terletzky patent

Frustrated in catching insects in 1904, Max Terletzky hit on this rather alarming solution. A basket with an open mouth is attached to the business end of a feathered arrow; the prospective bug hunter props open the basket’s mouth, stalks his prey, and fires at it using a bow. The arrow is attached to a cord in the archer’s hand, which closes the basket doors when the arrow has intercepted the bug and reached the limit of its flight. At that point the arrow drops to the ground and the archer can draw in the cord and claim his prize.

Terletzky writes, “This particular construction of the automatic device for closing the doors of the basket is extremely strong, simple, and durable in construction, as well as thoroughly efficient in operation.” For all I know he’s right.

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