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”:
Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook! Ook. Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook. Ook. Ook! Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook. Ook! Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook. Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook! Ook. Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook! Ook! Ook? Ook! Ook? Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook! Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook. Ook. Ook? Ook. Ook? Ook. Ook. Ook! Ook.
“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.”
A charming little scene from mathematical history — in 1615 Gresham College geometry professor Henry Briggs rode the 300 miles from London to Edinburgh to meet John Napier, the discoverer of logarithms. A contemporary witnessed their meeting:
He brings Mr. Briggs up into My Lord’s chamber, where almost one quarter of an hour was spent, each beholding the other with admiration, before one word was spoke: at last Mr. Briggs began. ‘My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it before, when now being known it appears so easy.’
Their friendship was fast but short-lived: The first tables were published in 1614, and Napier died in 1617, perhaps due to overwork. In his last writings he notes that “owing to our bodily weakness we leave the actual computation of the new canon to others skilled in this kind of work, more particularly to that very learned scholar, my dear friend, Henry Briggs, public Professor of Geometry in London.”
A “self-interlocking” geomagic square by Lee Sallows. The 16 lettered pieces pave a single large square, and smaller squares can be produced by various groups of four pieces — those drawn from each row, column, and long diagonal, and 10 other symmetrically chosen quartets.
British inventor Sir Robert Watson-Watt pioneered the development of radar, a contribution that helped the Royal Air Force win the Battle of Britain. Ironically, after the war he was pulled over for speeding by a Canadian policeman wielding a radar gun. His wife tried to point out the absurdity of the situation, but the officer wasn’t interested, and the couple drove away with a $12.50 fine. Watson-Watt wrote this poem:
Pity Sir Robert Watson-Watt,
strange target of this radar plot
And thus, with others I can mention,
the victim of his own invention.
His magical all-seeing eye
enabled cloud-bound planes to fly
but now by some ironic twist
it spots the speeding motorist
and bites, no doubt with legal wit,
the hand that once created it.
Oh Frankenstein who lost control
of monsters man created whole,
with fondest sympathy regard
one more hoist with his petard.
As for you courageous boffins
who may be nailing up your coffins,
particularly those whose mission
deals in the realm of nuclear fission,
pause and contemplate fate’s counter plot
and learn with us what’s Watson-Watt.
Ohio State University philosopher Stewart Shapiro relates a puzzling experience that a friend once encountered in a physics lab. “The class was looking at an oscilloscope and a funny shape kept forming at the end of the screen. Although it had nothing to do with the lesson that day, my friend asked for an explanation. The lab instructor wrote something on the board (probably a differential equation) and said that the funny shape occurs because a function solving the equation has a zero at a particular value. My friend told me that he became even more puzzled that the occurrence of a zero in a function should count as an explanation of a physical event, but he did not feel up to pursuing the issue further at the time.
“This example indicates that much of the theoretical and practical work in a science consists of constructing or discovering mathematical models of physical phenomena. Many scientific and engineering problems are tasks of finding a differential equation, a formula, or a function associated with a class of phenomena. A scientific ‘explanation’ of a physical event often amounts to no more than a mathematical description of it, but what on earth can that mean? What is a mathematical description of a physical event?”
What right do we have to presume that the natural world will hew to mathematical laws? And why does the universe oblige us so graciously by doing so? Repeatedly, mathematicians have developed abstract structures and concepts that have later found unexpected applications in science. How can this happen?
“It is positively spooky how the physicist finds the mathematician has been there before him or her.” — Steven Weinberg
“I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.” — Richard Feynman
“One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.” — Heinrich Hertz
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner
(From Stewart Shapiro, Thinking About Mathematics, 2000; also his paper “Mathematics and Reality” in Philosophy of Science 50:4 [December 1983].)
- Denver International Airport is larger than Manhattan.
- C.S. Lewis, Aldous Huxley, and John F. Kennedy died on the same day.
- Shakespeare mentions America only once, in Act 3, Scene 2 of The Comedy of Errors.
- π4 + π5 ≈ e6
- “All styles are good except the boring kind.” — Voltaire
What’s unusual about this magic square?
It works just as well upside down:
From Royal V. Heath, Scripta Mathematica, March-June 1951. See Topsy Turvy.
To help interest young students in chemistry, James Tour of Rice University devised “NanoPutians,” organic molecules that take the form of stick figures. The body is a series of carbon atoms that join two benzene rings; the arms and legs are acetylene units, each terminating in an alkyl group; and the head is a 1,3-dioxolane ring.
This gets even better — by using microwave irradiation, Tour found a way to vary the heads, creating a range of NanoProfessionals:
The synthesis is detailed on the Wikipedia page.
I pass on to Eclipses. When the Moon (see above) gets between the Earth (see below) and the Sun (do what you like), the resulting phenomenon is called an Eclipse of the Sun. When the Sun gets between the Earth and the Moon there will be the devil to pay. It will be called the Eclipse of the Earth and is likely to be total.
— H.F. Ellis, So This Is Science!, 1932
Can two dice be weighted so that the probability of each of the numbers 2, 3, …, 12 is the same?
Sherlock Holmes is walking through the valley of Reichenbach Fall. On a clifftop overhead, Moriarty has perched a boulder. When he pushes it, it will have a 90 percent chance of killing Holmes. Just as he is about to send it over the edge, Watson arrives at the clifftop. Watson can’t see Holmes, so he’s not able to push the boulder safely clear, but he reasons that it’s better to push the boulder in a random direction than to let Moriarty aim it carefully. So he pushes the boulder off the cliff in such a way that Holmes’ chance of dying is reduced to 10 percent.
Unfortunately the boulder crushes Holmes anyway. Watson’s push decreased the chance of Holmes’ death, but it also caused it.
What are we to make of this? Generally speaking, it seems true to say that Pre-emptive pushing prevents death by crushing. That is, Watson’s push was of the sort that made it less likely that Holmes would die — if the scenario were re-enacted many times, with the boulder pushed sometimes by Watson, sometimes by Moriarty, Watson-type pushes would result in fewer deaths. But it also seems true to say that Watson’s pushing the rock caused Holmes to die. But cause and prevent are antonyms. How can both of these statements be true?
(Christopher Read Hitchcock, “The Mishap at Reichenbach Fall: Singular vs. General Causation,” Philosophical Studies, June 1995.)
A bisecting arc is one that bisects the area of a given region. “What is the shortest bisecting arc of a circle?” Murray Klamkin asked D.J. Newman. Newman supposed that it was a diameter. “What is the shortest bisecting arc of a square?” Newman answered that it was an altitude through the center. Finally Klamkin asked, “And what is the shortest bisecting arc of an equilateral triangle?”
“By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector,” Klamkin writes. “But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer.”
Was he right?
List the first 2N positive integers (here let N = 4):
1, 2, 3, 4, 5, 6, 7, 8
Divide them arbitrarily into two groups of N numbers:
1, 4, 6, 7
2, 3, 5, 8
Arrange one group in ascending order, the other in descending order:
1, 4, 6, 7
8, 5, 3, 2
Now the sum of the absolute differences of these pairs will always equal N2:
| 1 – 8 | + | 4 – 5 | + | 6 – 3 | + | 7 – 2 | = 16 = N2
(Presented by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads.)
From Lee Sallows:
In an electrical network, if resistors x and y are placed in series their total resistance is x + y; if they’re placed in parallel it’s 1/(1/x + 1/y).
This offers an intriguing opportunity for self-reference. Each of the networks above contains four resistors with values 1, 2, 3, and 4, and the total resistances of the networks themselves are 1, 2, 3, and 4. So any one of the numbered resistors in these networks can be replaced by one of the networks themselves.
The challenge was posed by Sallows and Stan Wagon as a Macalester College “problem of the week”; these examples were discovered by Brian Trial, an automotive electronics engineer from Ferndale, Mich. Sallows points out that any such solution has a dual that results from changing series connections to parallel, and vice versa, and then replacing all resistors values by their reciprocals.
This leads to a further idea: The two sets of resistors below are “co-replicating” — the four networks on the left can be used to replace the four resistors in any of the networks on the right, and vice versa.
Draw any triangle ABC and pick any point P on its circumcircle.
The closest points to P on lines AB, BC, and AC will be collinear.
The goal of the Shakespeare programming language is to create code that reads like a Shakespearean play: Variables are “characters” that interact through dialogue, constants are represented by nouns and adjectives, and if/then statements are phrased as questions. (Insulting Macbeth assigns him a negative value.) Act and scene numbers serve as GOTO labels, and characters can tell one another to “remember” or “recall” values. The phrases “Open your heart” and “Speak your mind” output a variable’s numerical value and the corresponding ASCII character, respectively.
This program prints the phrase HELLO WORLD:
Romeo, a young man with a remarkable patience. Juliet, a likewise young woman of remarkable grace. Ophelia, a remarkable woman much in dispute with Hamlet. Hamlet, the flatterer of Andersen Insulting A/S. Act I: Hamlet's insults and flattery. Scene I: The insulting of Romeo. [Enter Hamlet and Romeo] Hamlet: You lying stupid fatherless big smelly half-witted coward! You are as stupid as the difference between a handsome rich brave hero and thyself! Speak your mind! You are as brave as the sum of your fat little stuffed misused dusty old rotten codpiece and a beautiful fair warm peaceful sunny summer's day. You are as healthy as the difference between the sum of the sweetest reddest rose and my father and yourself! Speak your mind! You are as cowardly as the sum of yourself and the difference between a big mighty proud kingdom and a horse. Speak your mind. Speak your mind! [Exit Romeo] Scene II: The praising of Juliet. [Enter Juliet] Hamlet: Thou art as sweet as the sum of the sum of Romeo and his horse and his black cat! Speak thy mind! [Exit Juliet] Scene III: The praising of Ophelia. [Enter Ophelia] Hamlet: Thou art as lovely as the product of a large rural town and my amazing bottomless embroidered purse. Speak thy mind! Thou art as loving as the product of the bluest clearest sweetest sky and the sum of a squirrel and a white horse. Thou art as beautiful as the difference between Juliet and thyself. Speak thy mind! [Exeunt Ophelia and Hamlet] Act II: Behind Hamlet's back. Scene I: Romeo and Juliet's conversation. [Enter Romeo and Juliet] Romeo: Speak your mind. You are as worried as the sum of yourself and the difference between my small smooth hamster and my nose. Speak your mind! Juliet: Speak YOUR mind! You are as bad as Hamlet! You are as small as the difference between the square of the difference between my little pony and your big hairy hound and the cube of your sorry little codpiece. Speak your mind! [Exit Romeo] Scene II: Juliet and Ophelia's conversation. [Enter Ophelia] Juliet: Thou art as good as the quotient between Romeo and the sum of a small furry animal and a leech. Speak your mind! Ophelia: Thou art as disgusting as the quotient between Romeo and twice the difference between a mistletoe and an oozing infected blister! Speak your mind! [Exeunt]
Because it’s written as a play, a program can be performed by human actors, but the drama lacks a certain narrative drive: