The Kolakoski Sequence

Write down the digit 1:

1

This can be seen as describing itself: It might denote the length of the string of identical digits at this point in the sequence. Well, in that case, if the length of this run is only one digit, then the next digit in the sequence can’t be another 1. So write 2:

1 2

Seen in the same light, the 2 would indicate that this second run of digits has length 2. So add a second 2 to the list to fulfill that description:

1 2 2

We can continue in this way, adding 1s and 2s so that the sequence becomes a recipe for writing itself:

https://commons.wikimedia.org/wiki/File:Kolakoski_animated.gif
Animation: Wikimedia Commons

This is a fractal, a mathematical object that encodes its own representation. It was described by William Kolakoski in 1965, and before him by Rufus Oldenburger in 1939. University of Evansville mathematician Clark Kimberling is offering a reward of $200 for the solution to five problems associated with the sequence:

  1. Is there a formula for the nth term?
  2. If a string occurs in the sequence, must it occur again?
  3. If a string occurs, must its reversal also occur?
  4. If a string occurs, and all its 1s and 2s are swapped, must the new string occur?
  5. Does the limiting frequency of 1s exist, and is it 1/2?

So far, no one has found the answers.

| Science & Math

Being There

https://commons.wikimedia.org/wiki/File:Church_Heart_of_the_Andes.jpg

Three meters wide, Frederic Edwin Church’s 1859 painting The Heart of the Andes was the IMAX feature of its day: On its debut in New York, 12,000 people waited in line for hours to pay 25 cents for a look at the canvas, which was displayed between theatrical curtains. One witness wrote, “Women felt faint. Both men and women succumb[ed] to the dizzying combination of terror and vertigo that they recognize[d] as the sublime. Many of them will later describe a sensation of becoming immersed in, or absorbed by, this painting, whose dimensions, presentation, and subject matter speak of the divine power of nature.” Mark Twain raved to his brother:

I have just returned from a visit to the most wonderfully beautiful painting which this city has ever seen — Church’s ‘Heart of the Andes’ … I have seen it several times, but it is always a new picture — totally new — you seem to see nothing the second time which you saw the first. We took the opera glass, and examined its beauties minutely, for the naked eye cannot discern the little wayside flowers, and soft shadows and patches of sunshine, and half-hidden bunches of grass and jets of water which form some of its most enchanting features. There is no slurring of perspective effect about it — the most distant — the minutest object in it has a marked and distinct personality — so that you may count the very leaves on the trees. When you first see the tame, ordinary-looking picture, your first impulse is to turn your back upon it, and say ‘Humbug’ — but your third visit will find your brain gasping and straining with futile efforts to take all the wonder in — and appreciate it in its fulness and understand how such a miracle could have been conceived and executed by human brain and human hands. You will never get tired of looking at the picture, but your reflections — your efforts to grasp an intelligible Something — you hardly know what — will grow so painful that you will have to go away from the thing, in order to obtain relief. You may find relief, but you cannot banish the picture — it remains with you still. It is in my mind now — and the smallest feature could not be removed without my detecting it.

Church had spent two years in South America retracing the steps of Alexander von Humboldt to create a composite of the continent’s topography. He hoped to share it with the explorer himself, but Humboldt died before the painting could reach Europe.

| Art

The Reminiscence Bump

https://en.wikipedia.org/wiki/File:Lifespan_Retrieval_Curve.jpg

If you seem to recall your adolescence and early adulthood years more clearly than your later life, that’s normal. Most of us can recall a disproportionate number of autobiographical memories made between ages 10 and 30, perhaps because of the important changes in identity, goals, attitudes, and beliefs that most of us went through in those years. (Also, that’s the span in which many of us have novel experiences such as graduation, marriage, and the birth of a child.)

Interestingly, this phenomenon extends to favorite books, movies, and records. In a 2007 study, psychologist Steve M.J. Janssen and his colleagues at the University of Amsterdam found that subjects best recorded memories of these things between 11 and 25. This is particularly true of music: Items that aren’t revisited frequently, such as books, are more likely to be forgotten, but records have a strong “reminiscence bump.”

“Books are read two or three times, movies are watched more frequently, whereas records are listened to numerous times. The results suggest that differential encoding initially causes the reminiscence bump and that re-sampling increases the bump further.” See the appendices for lists of favorite books, movies, and records and the average ages at which subjects first encountered them.

(Steve M.J. Janssen, Antonio G. Chessa, and Jaap M.J. Murre, “Temporal Distribution of Favourite Books, Movies, and Records: Differential Encoding and Re-Sampling,” Memory 15:7 [2007], 755-767.)

| Science & Math

“The Artist”

One evening there came into his soul the desire to fashion an image of ‘The Pleasure That Abideth For A Moment.’ And he went forth into the world to look for bronze. For he could only think in bronze.

But all the bronze in the whole world had disappeared; nor anywhere in the whole world was there any bronze to be found, save only the bronze of the image of ‘The Sorrow That Endureth For Ever.’

Now this image he had himself, and with his own hands, fashioned and had set it on the tomb of the one thing he had loved in life. On the tomb of the dead thing he had most loved had he set this image of his own fashioning, that it might serve as a sign of the love of man that dieth not, and a symbol of the sorrow of man that endureth for ever. And in the whole world there was no other bronze save the bronze of this image.

And he took the image he had fashioned, and set it in a great furnace, and gave it to the fire.

And out of the bronze of the image of ‘The Sorrow That Endureth For Ever’ he fashioned an image of ‘The Pleasure That Abideth For A Moment.’

— Oscar Wilde, Poems in Prose, 1894

| Art · Literature

The Dining Cryptographers Problem

https://commons.wikimedia.org/wiki/File:Dining_Cryptographers.svg
Image: Wikimedia Commons

Three cryptographers are having dinner at their favorite restaurant. The waiter informs them that arrangements have been made for their bill to be paid anonymously. It may be that the National Security Agency has picked up the tab, or it may be that one of the cryptographers himself has done so. The cryptographers respect each other’s right to pay the bill anonymously, but they want to know whether the NSA is paying. Happily, there is a way to determine this without forcing a generous cryptographer to reveal himself.

Each cryptographer flips a fair coin behind a menu between himself and his right-hand neighbor, so that only the two of them can see the outcome. Then each cryptographer announces aloud whether the two coins he can see — one to his right and one to his left — had the same outcome or different outcomes. If one of the cryptographers is the payer, he states the opposite of what he sees. If an even number of cryptographers say that they saw different outcomes, then the NSA paid; if an odd number say so, then one of the cryptographers paid the bill, but his anonymity is protected.

Computer scientist David Chaum offered this example in 1988 as the basis for an anonymous communication network; these networks are often referred to as DC-nets (for “dining cryptographers”).

(David Chaum, “The Dining Cryptographers Problem: Unconditional Sender and Recipient Untraceability,” Journal of Cryptology 1:1 [1988], 65-75.)

| Science & Math

Claque o’ Lanterns

Michigan art teacher Ray Villafane found enough success as a clay and wax sculptor to quit his job in 2006, but his career really took off when he changed media — the Wall Street Journal now calls him “the Picasso of pumpkin carving.”

More at his website.

(Thanks, Bill.)

| Art

“Summer”

Future poet laureate John Betjeman wrote this at age 13 as a “prep” exercise:

Whatever will rhyme with Summer?
There only is “plumber” and “drummer”:
Why! the cleverest bard
Would find it quite hard
To connect with the Summer — a plumber!

My Mind’s getting glummer and glummer
Hooray! there’s a word besides drummer;
Oh, I will think of some
Ere the prep’s end has come
But the rhymes will get rummer and rummer.

Ah! If the bee hums, it’s a hummer;
And the bee showeth signs of the Summer;
Also holiday babels
Make th’porter gum labels,
And whenever he gums, he’s a gummer!

The cuckoo’s a goer and comer
He goes in the hot days of Summer;
But he cucks ev’ry day
Till you plead and you pray
That his voice will get dumber and dumber!

| Poems

The Sincerest Form

https://commons.wikimedia.org/wiki/File:Tu4.jpg

The Soviet Tupolev Tu-4 strategic bomber of the 1950s was a reverse-engineered copy of the American Boeing B-29 Superfortress. Stalin wanted a strategic bomber, so when three B-29s were forced to land in Soviet territory in 1944, he ordered clones made, and 20 were ready by 1947, despite the engineering challenges caused by non-metric American specifications.

The Soviets revealed their coup during a Moscow parade in August 1947. When three aircraft flew overhead, Western analysts assumed they were the three captured B-29s. Then a fourth appeared.

(Thanks, Kevin.)

| History · Technology

Podcast Episode 219: The Greenbrier Ghost

https://commons.wikimedia.org/wiki/File:ZonaHeasterShue.jpg

In 1897, shortly after Zona Shue was found dead in her West Virginia home, her mother went to the county prosecutor with a bizarre story. She said that her daughter had been murdered — and that her ghost had revealed the killer’s identity. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Greenbrier Ghost, one of the strangest courtroom dramas of the 19th century.

We’ll also consider whether cats are controlling us and puzzle over a delightful oblivion.

See full show notes …

| Crime · Death · History · Oddities · Podcast