Science & Math

Twin Cities

Young Pierre has never left France, but he hears that London is an attractive city. He adopts the belief “Londres est jolie” (“London is pretty”).

Now Pierre’s family moves to another country, whose language he learns directly, without translation into French. He learns that his new city is called London, and eventually forms the belief that “London is not pretty.” He doesn’t know that Londres and “London” refer to the same city.

What are we to make of this? We can’t deny the sincerity of Pierre’s original belief, which he still holds; nor of his new belief; and it seems absurd to claim that he holds both or neither. “Each possibility seems to lead us to say something either plainly false or downright contradictory,” writes Princeton philosopher Saul Kripke. “Yet the possibilities appear to be logically exhaustive.”

“This, then, is the paradox. I have no firm belief as to how to solve it.”

Math Notes

3435= 33 + 44 + 33 + 55

Stops and Starts

Charade sentences devised by Howard Bergerson:





In the same spirit: 1! 10! 22! 1! = 11! 0! 2! 21!

Faith and Reason

Jean Buridan presented a logical proof of the existence of God:

  1. God exists.
  2. Neither of these sentences is true.

The two statements can be reconciled only if God exists.

But see Cloudy, Kangaroo Court, and Powerless.

Blowing in the Wind

A dust storm approaches Spearman, Texas, April 1935. Vance Johnson, a resident of western Kansas in the 1930s, described similar storms there:

The darkness was dust. The windows turned solid pitch; even flower boxes six inches beyond the pane were shut from view. … Dust sifted into houses, through cracks around the doors and windows — so thick even in well-built homes a man in a chair across the room became a blurred outline. Sparks flew between pieces of metal, and men got a shock when they touched the plumbing. It hurt to breath[e], but a damp cloth held over mouth and nose helped for a while. Food on tables freshly set for dinner ruined. Milk turned black. Bed, rugs, furniture, clothes in the closets, and food in the refrigerator were covered with a film of dust. Its acrid odor came out of pillows for days afterward.

During a dust storm in January 1895, J.C. Neal of Oklahoma A&M College reported “flashes of light that apparently started from no particular place, but prevaded [sic] the dust everywhere. As long as the wind blew, till about 2 a.m., January 21, this free lightning was everywhere but there was no noise whatever. It was a silent electrical storm.”

Inside Straight

Draw any two lines, pick three points on each, and lace them all together like so:

pappus theorem

The crossings of the laces will always form a straight line.

The Value of Tardiness

One day in 1939, Berkeley doctoral candidate George Dantzig arrived late for a statistics class taught by Jerzy Neyman. He copied down the two problems on the blackboard and turned them in a few days later, apologizing for the delay — he’d found them unusually difficult. Distracted, Neyman told him to leave his homework on the desk.

On a Sunday morning six weeks later, Neyman knocked on Dantzig’s door. The problems that Dantzig had assumed were homework were actually unproved statistical theorems that Neyman had been discussing with the class — and Dantzig had proved both of them. Both were eventually published, with Dantzig as coauthor.

“When I began to worry about a thesis topic,” he recalled later, “Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis.”

An Alphamagic Square

alphamagic square

Each row and column in this magic square totals 170.

If you update the contents of each cell by spelling out its number in English, counting the letters and recording the result (26 = TWENTY-SIX = nine letters = 9), you’ll produce another magic square.

Inventory Trouble

  1. This sentence contains four words.
  2. This sentence contains five words.
  3. Exactly one sentence in this list is true.


“Sometimes I think we’re alone in the universe, and sometimes I think we’re not. In either case the idea is quite staggering.” — Arthur C. Clarke

Chisholm’s Paradox

Adam and Noah exist here now. Adam lives to be 930, Noah 950. We can imagine a possible world in which they swap ages and yet retain their essential identities. But we can also imagine a further world in which they swap ages and names; another in which they swap ages, names, and hat sizes; and finally a world in which Adam and Noah have swapped all their qualitative properties.

Now isn’t Adam in our world identical with Noah in the other world? For how are they distinct? If a thing can retain its essential identity when a single property is changed, then, writes Willard Van Orman Quine, “You can change anything to anything by easy stages through some connecting series of possible worlds.”

See Spare Parts.

Thrice Sure

Three patricians of the coal yards fared forth on mercy bent, each in his great black chariot. Their overlord, the yard superintendent, had bade them deliver to seven families a total of twenty-eight tons of coal equally divided.

Well out of the yards, each with his first load, Kelly and Burke and Shea paused to discuss the problem of equal distribution — how much coal should each family get?

”Tis this way,’ argued Burke. ”Tis but a bit of mathematics. If there are 7 families an’ 28 tons o’ coal ye divide by 7, which is done as follows: Seven into 8 is 1, 7 into 21 is 3, which makes 13.’ He triumphantly exhibited his figures made with a stubby pencil on a bit of grimy paper:

thrice sure - 1

The figures were impressive but Shea was not wholly convinced. ‘There’s a easy way o’ provin’ that,’ he declared. ‘Ye add 13 seven times,’ and he made his column of figures according to his own formula. Then, starting from the bottom of the 3 column, he reached the top with a total of 21 and climbed down the column of 1’s, thus; ‘3, 6, 9, 12, 15, 18, 21, 22, 23, 24, 25, 26, 27, 28.’ ‘Burke is right,’ he announced with finality.

This was Shea’s exhibit:

thrice sure - 2

‘There is still some doubt in me mind,’ said Kelly. ‘Let me demonstrate in me own way. If ye multiply the 13 by 7 and get 28, then 13 is right.’ He produced a bit of stubby pencil and a sheet of paper. ”Tis done in this way,’ he said. ‘Seven times 3 is 21; 7 times 1 is 7, which makes 28. ‘Tis thus shown that 13 is the right figure and ye’re both right. Would ye see the figures?’

Kelly’s feat in mathematics was displayed as follows;

thrice sure - 3

‘There is no more argyment,’ the three agreed, so they delivered thirteen tons of coal to each family.

— Irvin S. Cobb, A Laugh a Day Keeps the Doctor Away, 1923

Napoleon’s Theorem

Construct equilateral triangles on the sides of any triangle, and their centers will form an equilateral triangle.

This discovery is traditionally credited to Napoleon, but there’s no evidence supporting that contention. Indeed, this theorem is said to be one of the most frequently rediscovered results in mathematics.

See A Better Nature.

Rational Self-Denial

You are to choose exactly one of two opaque boxes, A and B. A mean demon has put $1,000 in the box he predicted you would not take and nothing in the other. Since you know that the predictions are quite reliable, you can be sure you will pick the wrong box. … [Now suppose] we add a small bonus for taking box B. Some of us are now inclined to say that this modification renders the A option irrational. For it seems that the bonus tips the balance that previously existed between two equally good choices. If taking box A is as rational as taking box B, then the package deal of taking B plus the bonus must be more rational than taking Box A. Yet … if the bonus makes taking B the uniquely rational choice, then you would know that the money was in box A. This knowledge would force you to change your mind in favour of taking box A.

— Roy A. Sorensen, Blindspots, 1988, after Brian Skyrms

Sorensen adds: “Perhaps this reply has some persuasiveness when the bonus is small. But now suppose that the bonus is almost as great as the prize itself, say $900. Wouldn’t it be irrational to forgo a sure $900 by taking box A?”

See Newcomb’s Paradox.

Partly Cloudy

But how are we to figure the change from ‘undecided’ to ‘true’? Is it sudden or gradual? At what moment does the statement ‘it will rain tomorrow’ begin to be true? When the first drop falls to the ground? And supposing that it will not rain, when will the statement begin to be false? Just at the end of the day, 12 p.m. sharp? … We wouldn’t know how to answer these questions; this is due not to any particular ignorance or stupidity on our part but to the fact that something has gone wrong with the way the words ‘true’ and ‘false’ are applied here.

— F. Waismann, “How I See Philosophy,” in H.D. Lewis, ed., Contemporary British Philosophy, 1956

Mirror Years

If you’re over 18, you’ve lived through two years whose dates are palindromes: 1991 and 2002. That’s a rare privilege. Since 1001, the normal gap between palindromic years has been 110 years (e.g., 1661-1771). The 11-year gap 1991-2002 has been the only exception, and we’ll wait a millennium for the next such gap, 2992-3003. Until then we’re back to 110-year intervals, and most people will see only one palindrome in a lifetime.

See Two Milestones.

Conway’s Prime-Producing Machine

Here’s something amazing — a machine made of fractions:

conway's prime-producing machine

Start with the number 2 as your seed. Multiply it by each of the fractions above, in order, until you find one that produces an integer. (It’s 15/2.) Now adopt that integer (15) as the new seed, and multiply that by each of the fractions until you produce another integer. Keep this up, making a note whenever you produce a power of 2.

The first such power (4, or 22) appears after 19 steps. Fifty steps later, 23 turns up. Then 25 appears about 200 steps further on. A pattern emerges: the exponents are 2, 3, 5 …

It turns out that “these fourteen fractions alone have it in them to produce an infinity of primes, even those that no one yet knows about,” writes Dominic Olivastro. “There is something enormously magical about it.” John Horton Conway devised the technique; it’s an instance of his Fractran computing algorithm.

The Breaks

I once had a friend who objected to assigning chores by lot on the grounds that random selection was biased in favour of lucky people. He claimed to be serious and went on to compare unlucky people with … groups he took to be victims of discrimination. Sincere or not, wherein lies the absurdity of my friend’s objection?

— Roy A. Sorensen, Blindspots, 1988

On the House

The thirsty but impecunious soul approaches the bar-tender with a request for brandy, or what not. He takes a sip, pronounces it detestable, and offers to change it for a glass of whiskey. The obliging bar-tender substitutes the whiskey. The customer drinks, smacks his lips, and prepares to depart. ‘Here,’ says the bar-tender, ‘you haven’t paid for your whiskey.’ ‘No,’ is the innocent response; ‘I gave you the brandy in exchange for it.’ ‘But you didn’t pay for the brandy.’ ‘But I didn’t drink it.’ And while the publican intellect is vainly struggling with the mathematical puzzle involved, the puzzler makes good his escape.

— William Shepard Walsh, Handy-Book of Literary Curiosities, 1892


  • EPISCOPAL is an anagram of PEPSI COLA.
  • Only a perfect square has an odd number of divisors.
  • Makes no sense makes no sense” makes no sense.
  • The grounds of the Oklahoma state capitol include working oil rigs.
  • “Time is the only critic without ambition.” — John Steinbeck

The Loneliest Number

A woman approached Bertrand Russell after a lecture.

“I’m so happy to find that you’re a solipsist,” she told him. “I’ve been one all my life, and I’m surprised there aren’t more of us.”

“Boat Moved by a Rope”

There is a form of boat-racing, occasionally used at regattas, which affords a somewhat curious illustration of certain mechanical principles. The only thing supplied to the crew is a coil of rope, and they have, without leaving the boat, to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. I am told that in still water a pace of two or three miles an hour can be thus attained.

The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but it is not great enough to affect materially motion caused by a sufficiently big impulse. Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow n times more rapidly than it returns after the jerk, then the boat is impelled forwards at least n times more than backwards: hence on the whole the motion is forwards.

— W.W. Rouse Ball, Mathematical Recreations and Essays, 1905

Math Notes

2 × 5 × 27 = 1 × 15 × 18
2 + 5 + 27 = 1 + 15 + 18

213 × 624 = 312 × 426
102 × 402 = 201 × 204
936 × 213 = 639 × 312

Viviani’s Theorem

Pick any point inside an equilateral triangle and measure the distances to its sides.

The sum of those distances is the altitude of the triangle.