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.
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.”
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.)
W.H. Auden won first prize for mathematics at St. Edmund’s School in Hindhead, Surrey, when he was 13. He recalled being asked to learn the following mnemonic around 1919:
Minus times Minus equals Plus;
The reason for this we need not discuss.
1 + 2 = 3
1×2 + 2×3 + 3×4 = 4×5
1×2×3 + 2×3×4 + 3×4×5 + 4×5×6 = 5×6×7
1×2×3×4 + 2×3×4×5 + 3×4×5×6 + 4×5×6×7 + 5×6×7×8 = 6×7×8×9
In general, the sum of the first (n+1) products of consecutive n-tuples of consecutive integers is equal to the product of the next n-tuple.
Suppose n students are sitting at n desks in a classroom. They’re asked to stand, mill around at random, and then sit again. What is the probability that at least one student will find herself in her original seat?
Intuition says that the probability ought to drop as the number of students increases, but in fact it remains about the same:
In fact, Pierre Rémond de Montmort showed in 1708 that it’s
… which approaches 1 – 1/e, or about 0.63212. Whether there are 10 students or 10,000, the chance that at least one student returns to her own seat is about 2/3.
Antoni Zygmund once asked if the World Series shouldn’t be called the World Sequence? And shouldn’t a combination lock be called a permutation lock? John Von Neumann once had a dog called Inverse. It would sit when told to stand and go when it was told to come. Von Neumann pronounced the term infinite series as infinite serious.
— Michael Stueben, Twenty Years Before the Blackboard, 1998
Starting in the 1970s, neurobiologist Otto-Joachim Grüsser spent 10 years collating the light sources in 2,124 paintings selected at random from Western art originating between the 14th and 20th centuries. He found that in most paintings considered Western works of art, especially those painted around the time of the Scientific Revolution, the light falls from the left.
“At the beginning of modern Western art during the early Gothic period, a preference for diffuse illumination or light sources distributed around the painted scene was found,” Grüsser noted. “In a minority of paintings from the fourteenth century that show a clear light direction, a bias to the left side is present. This left-sided preference increased at the expense of diffuse or middle light sources up to the sixteenth and seventeenth centuries and declined thereafter. In the twentieth century, the diffuse or middle type of light distribution again became dominant.”
It’s not clear what to make of this. It seems reasonable that a right-handed artist might favor light falling from the left, but why should this vary with time? Grüsser found that the left-handed Leonardo da Vinci applied light sources from varying angles, and Hans Holbein the Younger, also a dominant left-hander, favored light falling from the right.
“From such observations in the works of these two left-handed painters who painted, drew, and wrote with the left hand, one gains the impression that the distribution of left, middle, and right light direction in left-handed painters deviates significantly from the average distribution of light found in the paintings of other contemporary painters. It would be interesting to study the drawings and paintings of other confirmed left-handed artists, who worked exclusively with the left hand.”
(Otto-Joachim Grüsser, Thomas Selke, and Barbara Zynda, “Cerebral Lateralization and Some Implications for Art, Aesthetic Perception, and Artistic Creativity,” in Ingo Rentschler, Barbara Herzberger, and David Epstein, Beauty and the Brain, 1988.)
Suppose you have three identical Dewar flasks labeled A, B, and C. (A Thermos is a Dewar flask.) You also have an empty container labeled D, which has thermally perfect conducting walls and which fits inside a Dewar flask.
Pour 1 liter of 80°C water into flask A and 1 liter of 20°C water into flask B. Now, using all four containers, is it possible to use the hot water to heat the cold water so that the final temperature of the cold water is higher than the final temperature of the hot water? How? (You can’t actually mix the hot water with the cold.)
Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.
The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they’re different. Their chances of winning are equal.
The first die has 5 red faces and 1 blue face. What are the colors on the second die?
Discovered by R.V. Heath in 1950:
Think of two positive integers.
Add them to get a third number.
Add the second number and the third number to get a fourth number.
Continue in this way until you have 10 numbers.
The sum of the 10 numbers is 11 times the seventh number.
Mathematician Yutaka Nishiyama of the Osaka University of Economics has designed a nifty paper boomerang that you can use indoors. A free PDF template (with instructions in 70 languages!) is here.
Hold it vertically, like a paper airplane, and throw it straight ahead at eye level, snapping your wrist as you release it. The greater the spin, the better the performance. It should travel 3-4 meters in a circle and return in 1-2 seconds. Catch it between your palms.
By 1958 many of the attributes of living things could be found in our technology: locomotion (cars), metabolism (steam engines), energy storage (batteries), perception of stimuli (iconoscopes), and nervous or cerebral activity (computers). The missing element was reproduction: We hadn’t yet created a nonliving artifact that could make copies of itself.
So Brooklyn College chemistry professor Homer Jacobson built one. Using an HO gauge model railroad, he designed an “organism” made of boxcars that could use sensors to select other cars on the track and assemble them on a siding into models of itself. “Head” cars have “brains,” and “tail” cars have “muscles” and “eyes”; together, a head and a tail make an organism in which the head directs the tail to watch for available cars elsewhere on the track and shunt them appropriately onto a siding to create a new organism.
“Any new ‘organisms’ formed continue the propagation in a linear fashion,” Jacobson wrote, “until the environment runs out of parts, or there are no more sidings available, or a mistake is made somewhere in the operation of a cycle, i.e., a ‘mutation.’ Such an effect, like that with living beings, is usually fatal.”
(Homer Jacobson, “On Models of Reproduction,” American Scientist, September 1958.)
A conventional balloon rises because its airbag displaces a large volume of air. But the gas that fills the bag has some weight; it, along with the weight of the gondola, reduces the balloon’s total lift.
Realizing this, Italian monk Francesco Lana de Terzi in 1670 proposed a “vacuum airship,” a balloon whose airbag was filled with nothing at all. Since a vacuum weighs nothing, this should maximize the vehicle’s lift — the vacuum could displace a large volume of air without itself adding any weight.
In principle this might work; the problem is that the vacuum would tend to collapse its container, and building a shell sturdy enough to withstand it would leave us with a ship too heavy to lift. It’s not clear whether any material or structure could overcome this problem.
Lee Sallows tells me that the postal system of Macau is releasing a new series of stamps based on magic squares. The full set will touch on everything from the Roman SATOR square to Dürer’s Melencolia. Details are here.
Charmingly, the values of the stamps will be 1, 2, …, 9 Macau patacas, so that the sheet of the nine stamps will itself form a classic Lo Shu magic square. Lee’s contribution, above, is a Nasik 2D geomagic square of order 3 — not only are all the rows and columns magic, but so are all six diagonals, including the four “broken” diagonals.
Somewhat related: In 2000 Finland issued seven stamps in classic tangram shapes, featuring images of science and education. (One of the small triangles, barely visible here, is a Sierpinski gasket.) Only three of the seven shapes are denominated postage, but I should think the temptation is overwhelming to arrange all seven on an envelope in the shape of a little man or a fish or something. I wonder what the post office makes of that.
Two circles intersect. A line AC is drawn through one of the intersection points, B. AC can pivot around point B — what position will maximize its length?
Howard C. Saar of Albion, Mich., pointed out an innovative solution to this problem in Recreational Mathematics Magazine, April 1962:
log(3x + 2) + log(4x – 1) = 2log11
Divide each side of the equation by the word “log”:
(3x + 2) + (4x – 1) = (2)(11)
7x = 21
x = 3
… which is correct.
Claude Shannon, the father of information theory, took an active interest in juggling. He used to juggle balls while riding a unicycle through the halls of Bell Laboratories, and he built the first juggling robot from an Erector set in the 1970s. (The machine above mimics W.C. Fields, who himself juggled in vaudeville before turning to comedy.)
Noting that juggling seems to appeal to mathematics-minded people, Shannon offered the following theorem:
F is flight time, the time the ball spends in the air
D is “dwell time,” the time it spends in the hand
V is vacancy, the time a hand spends empty
B is the number of balls
H is the number of hands
“Theorem 1 allows one to calculate the range of possible periods (time between hand throws) for a given type of uniform juggle and a given time of flight,” he wrote. “A juggler can change this period, while keeping the height of his throws fixed, by increasing dwell time (to increase the period) or reducing dwell time to reduce the period. The total mathematical range available for a given flight time can be obtained by setting D = 0 for minimum range and V = 0 for maximum range in Theorem 1. The ratio of these two extremes is independent of the flight time and dependent only on the number of balls and hands.”
To measure dwell times, Shannon actually created a “jugglometer” in which a juggler wore copper mesh over his fingers and juggled foil-covered lacrosse balls; catching the ball closed a connection between the fingers and started a clock. “Preliminary results from testing a few jugglers indicate that, with ball juggling, vacant time is normally less than dwell time, V ranging in our measurements from fifty to seventy per cent of D.”
Shannon noted that juggling gets dramatically harder as the number of balls increases. He worked out a foolproof solution, at least in theory. A light ray that starts at one focus of an ellipse will be reflected to the other focus. If the ellipse is rotated around its major axis, it will create an egglike shell with two foci. Now if a juggler stands with a hand at each focus, then a ball thrown from either hand, in any direction, will bounce off the shell and arrive at the other hand!
(“Scientific Aspects of Juggling,” in Claude Elwood Shannon: Collected Papers, 1993.)
In 1980 Philip K. Dick was asked to forecast some significant events in the coming years. Among his predictions:
1983: The Soviet Union will develop an operational particle-beam accelerator, making missile attack against that country impossible. At the same time the U.S.S.R. will deploy this weapon as a satellite killer. The U.S. will turn, then, to nerve gas.
1989: The U.S. and the Soviet Union will agree to set up one vast metacomputer as a central source for information available to the entire world; this will be essential due to the huge amount of information coming into existence.
1993: An artificial life form will be created in a lab, probably in the U.S.S.R., thus reducing our interest in locating life forms on other planets.
1997: The first closed-dome colonies will be successfully established on Luna and on Mars. Through DNA modification, quasi-mutant humans will be created who can survive under non-Terran conditions, i.e., alien environments.
1998: The Soviet Union will test a propulsion drive that moves a starship at the velocity of light; a pilot ship will set out for Proxima Centaurus, soon to be followed by an American ship.
2000: An alien virus, brought back by an interplanetary ship, will decimate the population of Earth, but leave the colonies on Luna and Mars intact.
2010: Using tachyons (particles that move backward in time) as a carrier, the Soviet Union will attempt to alter the past with scientific information.
Also: “Computer use by ordinary citizens (already available in 1980) will transform the public from passive viewers of TV into mentally alert, highly trained, information-processing experts.”
(From David Wallechinsky, Amy Wallace, and Irving Wallace, The Book of Predictions, 1980.)
The editors of the Journal of Organic Chemistry received a novel submission in 1970 — Brown University chemists J.F. Bunnett and Francis Kearsley wrote their paper “Comparative Mobility of Halogens in Reactions of Dihalobenzenes With Potassium Amide in Ammonia” in blank verse:
Reactions of potassium amide
With halobenzenes in ammonia
Via benzyne intermediates occur.
Bergstrom and associates did report,
Based on two-component competition runs,
Bromobenzene the fastest to react,
By iodobenzene closely followed,
The chloro compound lagging far behind,
And flurobenzene to be quite inert
At reflux (-33°).
This goes on for three pages. The journal published it with a note: “Although we are open to new styles and formats for scientific publication, we must admit to surprise upon receiving this paper. However, we find the paper to be novel in its chemistry, and readable in its verse. Because of the somewhat increased space requirements and possible difficulty to some of our nonpoetically inclined readers, manuscripts in this format face an uncertain future in this office.”
French civil engineer Henri Genaille introduced these “rulers” in 1891 as a way to perform simple multiplication problems directly, without mental calculation.
A set consists of 10 numbered rulers and an “index.” To multiply 52749 by 4, arrange rulers 5, 2, 7, 4, and 9 side by side next to the index ruler. We’re multiplying by 4, so go to the 4th row and start at the top of the rightmost column:
Now just follow the gray triangles from right to left:
The answer is 210996. “[Édouard] Lucas gave these rulers enough publicity that they became quite popular for a number of years,” writes Michael R. Williams in William Aspray’s Computing Before Computers. “Unfortunately he never lived to see their popularity grow, for he died, aged 49, shortly after Genaille’s demonstration.”
Ook! is a programming language designed to be understood by orangutans. According to the design specification, the language has only three syntax elements (“Ook.” “Ook?” “Ook!”), and it “has no need of comments. The code itself serves perfectly well to describe in detail what it does and how it does it. Provided you are an orang-utan.”
This example prints the phrase “Hello world”:
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“Um, that’s it. That’s the whole language. What do you expect for something usable by orang-utans?”
Tangrams can demonstrate the Pythagorean theorem. The yellow figure in the diagram above is a right triangle; the seven pieces that make up the square on the hypotenuse can be rearranged to form squares on the other two sides.
The third-century mathematician Liu Hui used to explain the theorem by dissecting and rearranging squares. Proper tangrams did not appear until centuries later, but modern Chinese mathematician Liu Dun writes, “We can hypothesize that the inventor of the Tangram, if not a mathematician, was at least inspired or enlightened by” this practice.
(From Jerry Slocum, The Tangram Book, 2003.)
Smallpox ravaged the New World for centuries after the Spanish conquest. In 1797 Edward Jenner showed that exposure to the cowpox virus could protect one against the disease, but the problem remained how to transport cowpox across the sea. In 1802 Charles IV of Spain announced a bold plan — 22 orphaned children would be sent by ship; after the first child was inoculated, his skin would exude fluid that could be passed to the next child. By passing the live virus from arm to arm, the children formed a transmission chain that could transport the vaccine in an era before refrigeration and other modern technology was available.
It worked. Over the next 10 years Spain spread the vaccine throughout the New World and to the Philippines, Macao, and China. Oklahoma State University historian Michael M. Smith writes, “These twenty-two innocents formed the most vital element of the most ambitious medical enterprise any government had ever undertaken.” Jenner himself wrote, “I don’t imagine the annals of history furnish an example of philanthropy so noble, so extensive as this.”