Inscribe a pentagon in a unit circle:
Now AB × AC × AD × AE = 5.
Pleasingly, this works with any regular n-gon: A hexagon yields 6, a heptagon 7, and so on. I think it was first noted by John Bull.
Take a whole stick and cut it in half. Half a minute later, cut each half in half. A quarter of a minute after that, cut each quarter in half, and so on ad infinitum.
What will remain at the end of a minute? An infinite number of infinitely thin pieces? Writes Oxford philosopher A.W. Moore, “Do we so much as understand this?”
Does each piece have any width? If so, couldn’t we reassemble them to form an infinitely long stick? If not, how can we assemble them to form anything at all?
Caltech number theorist Tom Apostol devised this elegant proof of the irrationality of the square root of 2.
Suppose the number is rational. Then there must be an isosceles right triangle with minimum integer sides (here, triangle ABC with sides n and hypotenuse m).
By drawing two arcs as shown, we can immediately establish triangle FDC — a smaller isosceles right triangle with integer sides.
This leads to an infinite descent. Hence n and m can’t both be integers, and the square root of 2 is irrational.
You’re presented with the four cards above. Each has a number on one side and a color on the other. Which card(s) must be turned over to test the idea that if a card shows an even number on one face, then its opposite face is red?
In a 1966 study by Peter Wason, fewer than 10 percent of respondents correctly indicated the 8 and brown cards.
Interestingly, respondents perform significantly better when they’re presented with the same task in the context of policing a social rule (e.g., the rule is “If you are drinking alcohol then you must be over 21″ and the cards are marked “27,” “16,” “drinking Coke,” “drinking beer”). About 90 percent of people perform this task correctly — supporting the idea that our facility for such tasks evolved to catch cheaters in a social environment.
123456789 = ((86 + 2 × 7)5 – 91) / 34
987654321 = (8 × (97 + 6/2)5 + 1) / 34
14459929 = 17 + 47 + 47 + 57 + 97 + 97 + 27 + 97
595968 = 45 + 49 + 45 + 49 + 46 + 48
397612 = 32 + 91 + 76 + 67 + 19 + 23
36428594490313158783584452532870892261556 = 342 + 642 + 442 + 242 + 842 + 542 + 942 + 442 + 442 + 942 + 042 + 342 + 142 + 342 + 142 + 542 + 842 + 742 + 842 + 342 + 542 + 842 + 442 + 442 + 542 + 242 + 542 + 342 + 242 + 842 + 742 + 042 + 842 + 942 + 242 + 242 + 642 + 142 + 542 + 542 + 642
A boy at the edge of a pond pulls a toy boat ashore. If he pulls in one yard of string, will the boat advance by more or less than one yard?
Surprisingly (to me), it will cover more than one yard. Because the boy is above the level of the water, he won’t pull in the entire length of string — length c will remain when the boat reaches shore. The length he’ll pull in, then, is a – c.
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side, so b + c > a and b > a – c — so the boat travels a greater distance than the length of string pulled in.
This magic word cube was devised by Jeremiah Farrell. Each cell contains a unique three-letter English word, and when the three layers are stacked, the words in each row and column can be anagrammed to spell MOUSETRAP.
Setting O=0, A=1, U=2, M=0, R=3, S=6, P=0, E=9, and T=18 produces a numerical magic cube (for example, MAE = 0 + 1 + 9 = 10).
Epimenides, a Cretan, says that all Cretans are liars. Is this a paradox? Not really: If we suppose that the statement is true then we’re led to a contradiction, but we can consistently suppose it to be false.
But, A.N. Prior writes, “We thus reach the peculiar conclusion that if any Cretan does assert that nothing asserted by a Cretan is true, then this cannot possibly be the only assertion made by a Cretan — there must also be, beside this false Cretan assertion, some true one. Yet how can there be a logical impossibility in supposing that some Cretan asserts that no Cretan ever says anything true, and that this is the only assertion ever made by a Cretan?”
Alonzo Church first raised this point in 1946: “Without factual information about other statements by Cretans, it has been proved by pure logic (so it seems) that some other statement by a Cretan, not the famous statement of Epimenides, must once have been true.”
The paradox, Prior writes, is that “such examination makes it seem possible to settle an empirical question on logical grounds.”
Draw two circles with three tangents as shown.
AB will always equal CD.
A psychologist at a girl’s college asked the members of his class to compliment any girl wearing red. Within a week the cafeteria was a blaze of red. None of the girls were aware of being influenced, although they did notice that the atmosphere was more friendly. A class at the University of Minnesota is reported to have conditioned their psychology professor a week after he told them about learning without awareness. Every time he moved toward the right side of the room, they paid more attention and laughed more uproariously at his jokes, until apparently they were able to condition him right out the door.
— W. Lambert Gardiner, Psychology: A Story of a Search, 1970
Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:
(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.
Smith’s evidence for (d) might be that the president of the company assured him that Jones would in the end be selected, and that he, Smith, had counted the coins in Jones’s pocket ten minutes ago. Proposition (d) entails:
(e) The man who will get the job has ten coins in his pocket.
Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the grounds of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.
But imagine, further, that unknown to Smith, he himself, not Jones, will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is then true, though proposition (d), from which Smith inferred (e), is false.
In our example, then, all of the following are true: (i) (e) is true, (ii) Smith believes that (e) is true, and (iii) Smith is justified in believing that (e) is true. But it is equally clear that Smith does not know that (e) is true; for (e) is true in virtue of the number of coins in Smith’s pocket, while Smith does not know how many coins are in Smith’s pocket, and bases his belief in (e) on a count of the coins in Jones’s pocket, whom he falsely believes to be the man who will get the job. [Does Smith know that the man who will get the job has ten coins in his pocket?]
— Edmund L. Gettier, “Is Justified True Belief Knowledge?”, Analysis, 1963
Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the Democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:
— If a Republican wins the election, then if it’s not Reagan who wins it will be Anderson.
— A Republican will win the election.
Yet they did not have reason to believe
— If it’s not Reagan who wins, it will be Anderson.
— Vann McGee, “A Counterexample to Modus Ponens,” Journal of Philosophy, September 1985
This curiosity was discovered by T.E. Lobeck. The square on the left is a conventional magic square — each row, column, and long diagonal totals 65. Replacing each number with the corresponding digit of pi (for example, replacing 17 with the 17th digit of pi, which is 2) yields the square on the right, in which the rows and columns yield equivalent sums.
Suppose [a] man who underwent [a] radical change of character — let us call him Charles — claimed, when he woke up, to remember witnessing certain events and doing certain actions which earlier he did not claim to remember, and that under questioning he could not remember witnessing other events and doing other actions which earlier he did remember. … [Suppose that] all the events he claims to have witnessed and all the actions he claims to have done point unanimously to the life-history of some one person in the past — for instance, Guy Fawkes. Not only do all Charles’ memory-claims that can be checked fit the pattern of Fawkes’ life as known to historians, but others that cannot be checked are plausible, provide explanations of unexplained facts, and so on. Are we to say that Charles is now Guy Fawkes, that Guy Fawkes has come to life again in Charles’ body, or some such thing?
— Bernard Williams, “Personal Identity and Individuation,” Proceedings of the Aristotelian Society, 1957
“If it is logically possible that Charles should undergo the changes described, then it is logically possible that some other man should simultaneously undergo the same changes: e.g., that both Charles and his brother Robert should be found in this condition. What should we say in that case? They cannot both be Guy Fawkes: if they were, Guy Fawkes would be in two places at once, which is absurd.”
The name of any integer can be transformed into a number by setting A=1, B=2, C=3, etc.: ONE = 15145, TWO = 202315, THREE = 2081855, and so on.
Because every English number name ends in D (4), E (5), L (12), N (14), O (15), R (18), T (20), X (24), or Y (25), no such transformation will produce a prime number.
But in Spanish, which uses 27 letters, both SESENTA (60) = 20520514211 and MIL SETENTA (1070) = 1391220521514211 yield primes.
59 + 39 + 49 + 49 + 99 + 49 + 89 + 39 + 69 = 534494836
A perpetual-motion scheme by William Congreve. A, B, and C are three horizontal rollers fixed in a frame. They’re surrounded by a continuous band of sponge, a, and that’s surrounded by a chain of weights, b. Immerse the whole thing partially in a cistern. The sponge on the left will absorb water by capillary action, say from x to y; the sponge on the right will not (because the weights squeeze it out!). “The band will begin to move in the direction A B; and as it moves downwards, the accumulation of water will continue to rise, and thereby carry on a constant motion.”
From Henry Dircks, Perpetuum Mobile; or, Search for Self-Motive Power, 1861.
In 1921, chemists at Arthur D. Little Inc. reduced 100 pounds of sows’ ears to glue, converted it to gelatin, forced it into fine strands, and wove these into a purse “of the sort which ladies of great estate carried in medieval days — their gold coin in one end and their silver coin in the other.”
“We made this silk purse from a sow’s ear because we wanted to, because it might serve as an example to clients who come to us with their ambitions or their troubles, and also as a contribution to philosophy,” they reported. “Things that everybody thinks he knows only because he has learned the words that say it, are poisons to progress.”
- Christopher Lee is Ian Fleming’s cousin.
- £12.12s.8d = 12128 farthings
- ii is real.
- Shouldn’t Juliet have asked, “Wherefore art thou Montague?”
- “Of soup and love, the first is the best.” — Thomas Fuller
The cheetah can reach speeds over 70 mph. In a dive, the peregrine falcon can reach 200 mph. But in 1927, entomologist Charles Townsend estimated that the deer botflies he’d observed in New Mexico surpassed both of these, reaching 400 yards per second. That’s 818 mph.
This claim stood for 11 years, until in 1938 chemist Irving Langmuir debunked it in Science:
- The power needed to maintain this speed amounts to 370 watts, or about half a horsepower. To deliver it, the fly would have to consume 1.5 times its own weight in food every second.
- Ballistics formulas show that the wind pressure on the fly’s head would amount to 8 pounds per square inch, probably enough to crush the fly.
- An 800 mph fly would strike the skin with a force of 310 pounds. “It is obvious that such a projectile would penetrate deeply into human flesh.”
- A supersonic fly would be invisible to the eye, not the “brownish blur” that Townsend had described.
Not to mention that an 800 mph fly would create its own sonic boom. After weighing the facts, Langmuir concluded, “The description given by Dr. Townsend of the appearance of the flies seems to correspond best with a speed in the neighborhood of 25 m/hr.”
When an explosion crippled Apollo 13’s command module, the crew used the lunar module as a “lifeboat.” The two modules had been built by different contractors, so when the mission was over Grumman sent a tongue-in-cheek bill to Rockwell for “towing” the ship to the moon and back:
The Associated Press reported that “North American Rockwell replied that the invoice had been examined by the company’s auditor, who pointed out that North American Rockwell had not yet received payment for ferrying LMs to the moon on previous missions.”
In 1939, University of Iowa graduate student Mary Tudor began an experiment with local orphans, warning them that they were showing signs of stuttering and lecturing them whenever they repeated a word. The children became acutely self-conscious, and many began to stutter, fulfilling the theory that “the affliction is caused by the diagnosis.”
Sixty years later, when Tudor was 84, she received a letter from one of the orphans. It was addressed to “Mary Tudor Jacobs The Monster.”
“You destroyed my life,” it ran. “I could have been a scientist, archaeologist or even president. Instead I became a pitiful stutterer. The kids made fun of me, my grades fell off, I felt stupid. Clear into my adulthood, I still want to avoid people to this day.”
“I didn’t like what I was doing to those children,” Tudor told the San Jose Mercury News in 2001. “It was a hard, terrible thing. Today, I probably would have challenged it. Back then you did what you were told. It was an assignment. And I did it.”
When entomologist Paul Marsh was given the chance to name two wasp species in the genus Heerz, he called them Heerz tooya and Heerz lukenatcha.
The International Commission on Zoological Nomenclature insists that “A zoologist should not propose a name that, when spoken, suggests a bizarre, comical, or otherwise objectionable meaning.” But a few get through. Examples:
- Vini vidivici (parrot)
- Apopyllus now (spider)
- Lalapa lusa (wasp)
- Agra vation (beetle)
- Phthiria relativitae (bee fly)
- Pison eyvae (wasp)
- Ba humbugi (snail)
- Eubetia bigaulae (“you betcha by golly”) (moth)
Three mythicomyiid flies are named Pieza kake, Pieza pi, and Pieza deresistans.
In 1912 the Zoological Society of London criticized entomologist George Kirkaldy for giving a series of hemipterans the generic names Polychisme, Elachisme, Marichisme, Dolichisme, Florichisme, and Ochisme (“Polly kiss me,” “Ella kiss me,” “Mary kiss me,” “Dolly kiss me,” “Flora kiss me,” “Oh, kiss me!”). In the same spirit, in 2002 a hopeful Neal Evenhuis named a fossil mythicomyiid Carmenelectra shechisme. “The offer’s still good,” he told the Chicago Tribune in 2008. “I’ll be willing to meet her.”