## The Six Circles Theorem

Fit a circle into one corner of a triangle. Now fit a second circle into a second corner so that it’s tangent to the first circle. Then fit a third circle into the third corner so that it’s tangent to the second circle.

Keep this up, cycling among the three corners, and the sixth circle will be tangent to the first one.

## Dog Tired

Maybe figures can’t lie, but liars can certainly figure, and that is why statistics can be made to ‘prove’ almost anything. Consider a group of ten girls, nine of them virgins, one pregnant. On the ‘average’ each of the nine virgins is ten per cent pregnant, while the girl who is about to have a baby is ninety per cent a virgin. Or, assuming that a fox terrier two feet long, with a tail an inch and a half high, can dig a hole three feet deep in ten minutes, to dig the Panama Canal in a single year would require only one fox terrier fifteen miles long, with a tail a mile and a half high.

— Stuart Cloete, *The Third Way*, 1947

## Never the Twain

A paradox attributed to Proclus Lycaeus (412-485):

Consider two nonparallel lines, AQ and BP. BP is perpendicular to AB; AQ isn’t. Find the midpoint of AB and mark AC = BD = AB/2. Now if AQ and BP are going to intersect, it can’t happen on AC or BD; if it did, say at a point R, then that would give us a triangle ARB where the sum AR + RB < AB, which is impossible.

But now we can connect CD and follow the same process: CE and DF can't intersect for the same reason. EG and FH are likewise ruled out, and so on up the line forever.

This seems to mean that two nonparallel lines will *never* intersect. That can’t be right, but where is the error?

(From Alfred Posamentier, *Magnificent Mistakes in Mathematics*, 2013.)

## Non-Fiction

Sherlock Holmes is an honorary fellow of the Royal Society of Chemistry.

“Holmes did not exist, but he should have existed,” society chief David Giachardi said in bestowing the award in 2002. “That is how important he is to our culture. We contend that the Sherlock Holmes myth is now so deeply rooted in the national and international psyche through books, films, radio and television that he has almost transcended fictional boundaries.”

## The Wheel Cipher

Thomas Jefferson, already absurdly accomplished by 1795, somehow found time to delve into cryptography, where he devised this cipher system. The letters of the alphabet are printed along the rim of each of 36 disks, which are stacked on an axle. One party rotates the disks until his message can be read along one of the 26 rows of letters, somewhat like a modern cylindrical bike lock. Now he can record the letters in any one of the other 25 rows and send that string safely to another party, who decodes it by reversing this procedure. If the message is intercepted, it’s useless even to someone who has the disks, because he must also know the order in which to stack them, and 36 disks can be stacked in 371,993,326,789,901,217,467,999,448,150,835, 200,000,000 different ways.

This is pretty robust. The cipher below, created in 1915 by U.S. Army cryptographer Joseph Mauborgne, has never been solved. “The known systems from this year (or earlier) shouldn’t be too hard to crack with modern attacks and technology,” writes NSA cryptologist Craig P. Bauer. “So, why don’t we have a plaintext yet? My best guess is that it used a cipher wheel” like Jefferson’s.

(L. Kruh, “A 77-year-old challenge cipher,” *Cryptologia*, 17(2), 172-174, 1993, quoted in Bauer’s *Secret History: The Story of Cryptology*, 2013.)

## Accord

Squeeze six circles into a larger circle so that each is tangent to its two neighbors. Now the three lines drawn through opposite points of tangency will pass through the same point.

Remarkably, this wasn’t discovered until 1974.

## The Pitot Theorem

If a quadrilateral circumscribes a circle, then the sums of its opposite sides are equal.

Above, *a* + *c* = *b* + *d*.

## The Traveler’s Dilemma

Two travelers are transporting identical antiques. Unfortunately, the airline smashes both of them. The airline manager proposes that each traveler write down the cost of his antique, any value from $2 to $100. If both write the same number, the airline will pay this amount to both travelers. If they write different numbers, the airline will assume that the lower number is the accurate price; the low bidder will receive this amount plus $2, and the high bidder will receive this amount minus $2. If they can’t confer, what strategy should the travelers take in deciding how to bid?

At first Traveler A might like to bid $100, the maximum allowed. If his opponent does the same then they’ll both net $100. But A can do better than this: If B bids $100 and A bids $99 then A will come away with $101.

Unfortunately if B is rational then he’ll have the same insight and also bid $99. So A had better undercut him again and bid $98.

This chain leads all the way down to $2. If both travelers are perfectly rational then they’ll both bid (and make) $2, the minimum price.

But this seems very unlikely to happen in actual practice — in real life both travelers would likely make high bids and get high (though perhaps unequal) payoffs.

“All intuition seems to militate against all formal reasoning in the traveler’s dilemma,” wrote economist Kaushik Basu in propounding the problem in 1994. “There is something very rational about rejecting (2, 2) and expecting your opponent to do the same. … The aim is to explain why, despite rationality being common knowledge, players would reject (2, 2), as intuitively seems to be the case.”

## Orderly

A Japanese geometry theorem from the Edo period: If the blue circles are equal, the green circles will be equal too.

This can be extended: Circles spanning three of these triangles will also be equal, and so on.

## Disc World

Jump into the sea and look up. The surface above you is dark except for a bright circle that follows you around like a portable skylight. This is Snell’s window: Because light is refracted as it enters the water, the 180-degree world above you is compressed into a tight 97 degrees.

Physicist Robert W. Wood was thinking of this effect when he created a new wide-angle lens in 1906. Fittingly, he called it the fisheye.

## Required Reading

In 1990, Spanish philosopher Jon Perez Laraudogoitia submitted an article to *Mind* entitled “This Article Should Not Be Rejected by *Mind*.” In it, he argued:

- If statement 1 in this argument is trivially true, then this article should be accepted.
- If statement 1 were false, then its antecedent (“statement 1 in this argument is trivially true”) would be true, which means that statement 1 itself would be true, a contradiction. So statement 1 must be true.
- But that seems wrong, since
*Mind*is a serious journal and shouldn’t publish trivial truths. - That means statement 1 must be either false or a non-trivial truth. We know it can’t be false (#2), so it must be a non-trivial truth, and its antecedent (“statement 1 in this argument is trivially true”) is false.
- What then is the truth value of its consequent, “this article should be accepted”? If this were false then
*Mind*shouldn’t publish the article; that can’t be right, since the article consists of a non-trivial truth and its justification. - So the consequent must be true, and
*Mind*should publish the article.

They published it. “This is, I believe, the first article in the whole history of philosophy the content of which is concerned exclusively with its own self, or, in other words, which is totally self-referential,” Laraudogoitia wrote. “The reason why it is published is because in it there is a proof that it should not be rejected and that is all.”

## Leave-Taking

In 1964 Canadian writer Graeme Gibson bought a parrot in Mexico. The bird, which Gibson named Harold Wilson, was bright and affectionate at first, but he seemed to grow lonely in the dark Canadian winter, so in the spring Gibson made arrangements to donate him to the Toronto Zoo. At the aviary Gibson carried Harold into the cage that had been prepared for him, placed him on a perch, said his goodbyes, and turned to go.

“Then Harold did something that astonished me. For the very first time, and in exactly the voice my kids might have used, he called out ‘Daddy!’ When I turned to look at him he was leaning towards me expectantly. ‘Daddy’, he repeated.

“I don’t remember what I said to him. Something about him being happier there, that he’d soon make friends. The kind of things you say to kids when you abandon them at camp. But outside the aviary I could still hear him calling ‘Daddy! Daddy!’ as we walked away. I was shattered to discover that Harold knew my name, and that he did so because he’d identified himself with my children.

“I now believe he’d known it all along, but was using it — for the first time — out of desperation. Both Konrad Lorenz and Bernd Heinrich mention instances of birds calling out the private names of intimates when threatened by serious danger. I am no longer surprised by such information. We think of our captive birds as our pets, but perhaps we are theirs as well.”

(From Gibson’s *Perpetual Motion*, 1982.)

## Pure-Hearted

Inscribe a hexagon in a unit circle such that AB = CD = EF = 1.

Now the midpoints of BC, DE, and FA form an equilateral triangle.

See A Better Nature.

## Rolling

Arthur W.J.G. Ord-Hume calls this “the most graceful and simple perpetual motion machine of all time.” It was offered by American inventor F.G. Woodward in the 19th century. A heavy wheel is mounted between two rollers so that the wheel’s weight causes it to roll continuously in the direction of the arrow.

Or so Woodward hoped. Ord-Hume notes that the principle required the left half of the wheel always to be heavier than the right half. “Sadly, it wasn’t.”

## Author!

How do I know that I’m not just a fictional character in some imagined story? What could I learn about myself that would prove that I’m real? “I am human, male, brunette, etc., but none of that helps,” writes UCLA philosopher Terence Parsons. “I see people, talk to them, etc., but so did Sherlock Holmes.”

Descartes would say that the very fact that I’m thinking about this shows that I exist: *cogito ergo sum*. But a fictional character could make the same argument. “Hamlet did think a great many things,” writes Jaakko Hintikka. “Does it follow that he existed?” Robert Nozick adds, “Could not *any* proof be written into a work of fiction and be presented by one of the characters, perhaps one named ‘Descartes’?”

Tweedledee tells Alice that she’s only a figment of the Red King’s dream. “If that there King was to wake,” adds Tweedledum, “you’d go out — bang! — just like a candle!”

Alice says, “Hush! You’ll be waking him, I’m afraid, if you make so much noise.”

“Well, it’s no use YOUR talking about waking him,” replies Tweedledum, “when you’re only one of the things in his dream. You know very well you’re not real.”

“It seems to me that this is a philosophical problem that deserves to be treated seriously on a par with issues like the reality of the external world and the existence of other minds,” Parsons writes. “I don’t know how to solve it.”

(Terence Parsons, *Nonexistent Objects*, 1980; Charles Crittenden, *Unreality*, 1991; Robert Nozick, “Fiction,” *Ploughshares* 6:3 (1980), pp. 74-78; Jaakko Hintikka, “Cogito, Ergo Sum: Inference or Performance?”, *The Philosophical Review*, 71:1 (January 1962), pp. 3-32.)

## Scattered Objects

If a flock of birds disperses gradually, at what point does it cease to be a flock?

“There is at the moment a pipe on my desk,” wrote MIT philosopher Richard Cartwright in 1987. “Its stem has been removed, but it remains a pipe for all that; otherwise no pipe could survive a thorough cleaning.”

But he also owned a two-volume set of John McTaggart’s *The Nature of Existence*, one volume of which was in Cambridge and the other in Boston. Do those two volumes still make one thing? If so, is there a “thing” composed of the Eiffel Tower and the Old North Church? Why not?

(From Cartwright’s *Philosophical Essays*.)

## A Golden March

Draw a circle whose circumference is the golden mean. Choose a point and label it 1, then move clockwise around the circle in steps of arc length 1, labeling the points 2, 3, and so on. At each step, the difference between each pair of adjacent numbers on the circle is a Fibonacci number.

## Misc

- What time is it at the North Pole?
- The shortest three-syllable word in English is W.
- After the revolution, the French frigate
*Carmagnole*used a guillotine as its figurehead. - 82350
^{2}+ 38125^{2}= 8235038125 - PRICES: CRIPES!
- “Conceal a flaw, and the world will imagine the worst.” — Martial

When Montenegro declared independence from Yugoslavia, its top-level domain changed from .yu to .me.

## A Drugstore Puzzle

If I buy two toothbrushes in a “buy one, get one free” offer … which one did I buy, and which was free?

(From philosopher Roy Sorensen.)

## The Richardson Effect

How long is a coastline? If we measure with a long yardstick, we get one answer, but as we shorten the scale the total length goes up. For certain mathematical shapes, indeed, it goes up without limit.

English mathematician Lewis Fry Richardson discovered this perplexing result in the early 20th century while examining the relationship between the lengths of national boundaries and the likelihood of war. If the Spanish claim that the length of their border with Portugal is 987 km, and the Portuguese say it’s 1,214 km, who’s right? The ambiguity arises because a wiggly boundary occupies a fractional dimension — it’s something between a line and a surface.

“At one extreme, *D* = 1.00 for a frontier that looks straight on the map,” Richardson wrote. “For the other extreme, the west coast of Britain was selected because it looks like one of the most irregular in the world; it was found to give *D* = 1.25.”

This is a mathematical notion, but it’s also a practical problem. On the fjord-addled panhandle of Alaska, the boundary with British Columbia was originally defined as “formed by a line parallel to the winding of the coast.” Who gets to define that? On the map below, the United States claimed the blue border, Canada wanted the red one, and British Columbia claimed the green. The yellow border was arbitrated in 1903.

## Two Lists

Write out the positive powers of 10 in both base 2 and base 5:

Now for any integer *n* > 1, we’ll find exactly one number of length *n* somewhere on the two lists. They contain one 3-digit number, one 4-digit number, and so on forever — if *n* = 100 we find a 100-digit number in the 30th position on the base 2 list.

(This result first appeared in the 1994 Asian Pacific Mathematics Olympiad. I found it in Ravi Vakil’s *A Mathematical Mosaic*.)

Two further curious lists: If we write out the triangular numbers, those in positions 3, 33, etc. show a pattern:

T(3) = 6

T(33) = 561

T(333) = 55611

T(3,333) = 5556111

T(33,333) = 555561111

T(333,333) = 55555611111

Similarly:

T(6) = 21

T(66) = 2211

T(666) = 222111

T(6,666) = 22221111

T(66,666) = 2222211111

T(666,666) = 222222111111

(Thanks, Larry.)

## Some “Odd” Theorems

Draw any triangle and divide each leg into three equal segments. Connect each vertex to one of the trisection points on the opposite leg, as shown, and the triangle formed in the center will have 1/7 the area of the original triangle.

A square inscribed in a semicircle has 2/5 the area of a square inscribed in a circle of the same radius.

Draw a square and connect each vertex to the midpoint of an opposite side, as shown. The square formed in the center will have 1/5 the area of the original square.

A “proof without words”:

## Naturally

Steven Bartlett and Peter Suber’s *Self-Reference: Reflections on Reflexivity* contains a bibliography of works on reflexivity.

It includes an entry for Steven Bartlett and Peter Suber’s *Self-Reference: Reflections on Reflexivity*.