Climbing Chains

Princeton mathematician John Horton Conway investigated this curious permutation:

3n ↔ 2n

3n ± 1 ↔ 4n ± 1

It’s a simple set of rules for creating a sequence of numbers. In the words of University of Calgary mathematician Richard Guy, “Forwards: if it divides by 3, take off a third; if it doesn’t, add a third (to the nearest whole number). Backwards: if it’s even, add 50%; if it’s odd, take off a quarter.”

If we start with 1, we get a string of 1s: 1, 1, 1, 1, 1, …

If we start with 2 or 3 we get an alternating sequence: 2, 3, 2, 3, 2, 3, …

If we start with 4 we get a longer cycle that repeats: 4, 5, 7, 9, 6, 4, 5, 7, 9, 6, …

And if we start with 44 we get an even longer repeating cycle: 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66, 44, …

But, curiously, these four are the only loops that anyone has found — start with any other number and it appears you can build the sequence indefinitely in either direction without re-encountering the original number. Try starting with 8:

…, 72, 48, 32, 43, 57, 38, 51, 34, 45, 30, 20, 27, 18, 12, 8, 11, 15, 10, 13, 17, 23, 31, 41, 55, 73, 97, …

Paradoxically, the sequence climbs in both directions: Going forward we multiply by 2/3 a third of the time and by roughly 4/3 two-thirds of the time, so on average in three steps we’re multiplying by 32/27. Going backward we multiply by 3/2 half the time and by roughly 3/4 half the time, so on average in two steps we’re multiplying by 9/8. And every even number is preceded by a multiple of three — half the numbers are multiples of three!

What happens to these chains? Will the sequence above ever encounter another 8 and close up to form a loop? What about the sequences based on 14, 40, 64, 80, 82 … ? “Again,” writes Guy, “there are many more questions than answers.”

(Richard K. Guy, “What’s Left?”, Math Horizons 5:4 [April 1998], 5-7; and Richard K. Guy, Unsolved Problems in Number Theory, 2004.)

A Disjoint Map

When Danish naval officer Gustav Holm was exploring the eastern coast of Greenland in 1885, an Inuit named Kunit gave him this three-dimensional wooden map.

The two parts form one whole: The bottom carving represents the coast from Sermiligak to Kangerdlugsuatsiak, and the top is an island offshore. The Inuit would carry these maps in their kayaks to navigate the waters between the two landmasses.

(From David Turnbull, Maps Are Territories, 1989.)


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Hear Hear

Where is a sound? If I play a note at the piano, you and I both seem to locate it at the instrument. But we both also know that we perceive the note because the piano sends waves through the air that strike our ears. That would mean that most of our auditory perception is illusion. Is that what we want to say?

Philosopher George Berkeley wrote, “When I hear a coach drive along the streets, immediately I perceive only the sound, but from experience I have had that such a sound is connected with a coach, I am said to hear the coach.” Perhaps the sound lies at our ears, or at our sensation of it, and it’s only our experience of the world that leads us to attribute it to some remote source. But that raises problems of its own. If sound is sensation, then can a sound occur if no one is present to hear it?

Perhaps the answer lies in between: Acoustics tells us that sounds are vibrations transmitted by the air. But vibrations of very high or low pitch aren’t perceptible to human ears. Are these still sounds?

A similar puzzle concerns smells.

A Long Swim

Just an oddity — in Nature, June 5, 1919, Cambridge zoologist John Stanley Gardiner notes that a Fiji harbormaster had informed him of a saltwater crocodile that had landed alive on Rotuma, 260 miles to the north and 600 miles east of the New Hebrides.

“It certainly did not come from Fiji or any lands to the east, as crocodiles do not now exist on them,” Gardiner wrote. “It must indeed have crossed from the west, and covered at least 600 miles of open, landless sea.

“This occurrence is sufficiently remarkable to be placed on permanent record.”


wile e coyote, super genius

Cartoon laws of physics:

  1. Any body suspended in space will remain in space until made aware of its situation. Daffy Duck steps off a cliff, expecting further pastureland. He loiters in midair, soliloquizing flippantly, until he chances to look down. At this point, the familiar principle of 32 feet per second per second takes over.
  2. Any body in motion will tend to remain in motion until solid matter intervenes suddenly. Whether shot from a cannon or in hot pursuit on foot, cartoon characters are so absolute in their momentum that only a telephone pole or an outsize boulder retards their forward motion absolutely. Sir Isaac Newton called this sudden termination of motion the stooge’s surcease.
  3. Any body passing through solid matter will leave a perforation conforming to its perimeter. Also called the silhouette of passage, this phenomenon is the specialty of victims of directed-pressure explosions and of reckless cowards who are so eager to escape that they exit directly through the wall of a house, leaving a cookie-cutout-perfect hole. The threat of skunks or matrimony often catalyzes this reaction.
  4. The time required for an object to fall twenty stories is greater than or equal to the time it takes for whoever knocked it off the ledge to spiral down twenty flights to attempt to capture it unbroken. Such an object is inevitably priceless, the attempt to capture it inevitably unsuccessful.

There are 10 laws altogether, including “9. Everything falls faster than an anvil.” As early as 1956 Walt Disney was describing the “plausible impossible.” In Who Framed Roger Rabbit, Eddie Valiant says, “Do you mean to tell me you could’ve taken your hand out of that cuff at any time?” Roger answers, “Not at any time! Only when it was funny!”


In 1942, uncertain whether one of its spies had been replaced by a German impersonator, Britain’s Special Operations Executive hit on a clever plan: After a regular radio communication, the British radio operator signed off with HH, short for “Heil Hitler,” the standard farewell among Nazi operators. His counterpart, “Netball,” responded HH automatically, giving himself away.

They confirmed this at the next session:

Netball was several minutes late for his sked (not significant) and signalled ‘q r u’ (‘I have no traffic for London’). Howell replied ‘q t c’ (‘We have a message for you’), and proceeded to transmit it (the message warned Netball never to send less than 150 letters). Howell then signalled ‘HH’, and Netball immediately replied ‘HH’.

‘Right,’ Nick was heard to say to his companion, ‘that’s it then.’

(From Leo Marks, Between Silk and Cyanide, 2001.)