For the writer of fantastic stories to help the reader to play the game properly, he must help him in every possible unobtrusive way to domesticate the impossible hypothesis. He must trick him into an unwary concession to some plausible assumption and get on with his story while the illusion holds. And that is where there was a certain slight novelty in my stories when first they appeared. Hitherto, except in exploration fantasies, the fantastic element was brought in by magic. Frankenstein even, used some jiggery-pokery magic to animate his artificial monster. There was trouble about the thing’s soul. But by the end of last century it had become difficult to squeeze even a momentary belief out of magic any longer. It occurred to me that instead of the usual interview with the devil or a magician, an ingenious use of scientific patter might with advantage be substituted. That was no great discovery. I simply brought the fetish stuff up to date, and made it as near actual theory as possible.
— H.G. Wells, June 1934 (from the H.G. Wells Scrapbook)
So, if we include repeated instances of a given factor:
1, 2, 3, 5, and 7 have 0 proper prime divisors
4, 6, and 9 have 2 proper prime divisors
8 has 3 proper prime divisors
Mathematicians Ana Luzón and Manuel A. Morón of Universidad Politecnica de Madrid point out a coincidence: The numerals in each of these groups have the same basic shape — within each group it’s possible to transform one numeral into another by bending, shrinking, and expanding. So, for example, it’s possible to bend a numeral 1 made of clay into a 2 or a 7, but not into a 9 — we’re not allowed to poke a new hole in the clay or to affix one part of it to another.
Luzón and Morón write that if two of these nine numerals have the same number of proper prime divisors, then those two will “cut a sheet in the same number of pieces if you write them down with a scalpel.” And if the scalpel doesn’t cut the sheet into multiple pieces, then the number you’re writing is prime (except for 1).
Note: This works only if the numeral 4 is “closed” at the top, not open. So this post will make sense if you’re reading it on Futility Closet (which uses the “closed” font Georgia), but possibly not if you’re reading it in a different font elsewhere. Maybe this tells us how 4 “ought” to be written!
(Ana Luzón and Manuel A. Morón, “4 or 4? Mathematics or Accident?” Mathematics Magazine 75:4 [October 2002], 274.)
If you apply one straight cut to a pancake, pretty clearly you’ll get 2 pieces. With two cuts, the most you can get is 4. What’s the greatest number you can produce with three cuts? If the cuts meet neatly in the center, you’ll get 6 pieces, but if you’re artfully sloppy you can make 7 (above). Charmingly, this leads us into the “lazy caterer’s sequence” — the maximum number of pieces you can produce with n straight cuts:
Generally it turns out that the maximum number for n cuts is given by the formula
each number equals 1 plus a triangular number.
A related question is the pancake flipping problem. You’re presented with a spatula and an untidy stack of pancakes of varying sizes. You can insert the spatula at any point in the stack and flip all the pancakes above it. What’s the least number of flips required to sort the pancakes in order of size? Interestingly, no one has found a general answer. It’s possible to work out the solution for relatively small stacks (in which the number of pancakes is 1, 2, 3, …):
0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, …
But no one has found a formula that will tell how many flips will get the job done for a stack of any given size.
The problem has an interesting pedigree. Bill Gates worked on it at Harvard (PDF), and David X. Cohen, who went on to write for The Simpsons and Futurama, worked on a related problem at Berkeley in which the bottom of each pancake is burnt and the sort must be completed with the burnt sides facing down.
In 2014, Michigan Technological University physicists Robert Nemiroff and Teresa Wilson thought up a novel way to detect time travelers: Search the Internet.
They searched for mentions of “Comet ISON,” a sun-grazing comet discovered in September 2012, and for “Pope Francis,” whose papacy began in March 2013 and who is the first of his name. Both of these subjects are historically momentous enough that they might be known even to people in the far future; if those people travel into our past, then they might mention them inadvertently in, say, 2011, before we could plausibly have done so ourselves.
“Given the current prevalence of the Internet … this search might be considered the most sensitive and comprehensive search yet for time travel from the future,” they reported, acknowledging that “technically, what was searched for here was not physical time travellers themselves, but rather informational traces left by them.”
And they note that our failure to detect travelers doesn’t mean they’re not there. “First, it may be physically impossible for time travellers to leave any lasting remnants of their stay in the past, including even non-corporeal informational remnants on the Internet. Next, it may be physically impossible for us to find such information as that would violate some yet-unknown law of physics. … Furthermore, time travellers may not want to be found, and may be good at covering their tracks.”
In 1978, inspired by this Peanuts cartoon, Nathaniel Hellerstein invented the Linus sequence, a sequence of 1s and 2s in which each new entry is chosen the prevent the longest possible pattern from emerging at the end of the line. Start with 1:
Now if the second digit were also a 1 then we’d have a repeating pattern. So enter a 2:
If we choose 2 for the third entry we’ll have “2 2” at the end of the line, another emerging pattern. Prevent that by choosing 1:
1 2 1
Now what? Choosing 2 would give us the disastrously tidy 1 2 1 2, so choose 1 again:
1 2 1 1
But now the 1s at the end are looking rather pleased with themselves, so choose 2:
1 2 1 1 2
And so on. The rule is to avoid the longest possible “doubled suffix,” the longest possible repeated string of digits at the end of the sequence. For example, choosing 1 at this point would give us 1 2 1 1 2 1, in which the end of the sequence (indeed, the entire sequence) is a repeated string of three digits. Choosing 2 avoids this, so we choose that.
In February 1966, the Soviet Union’s Luna 9 landed safely on the moon and became the first spacecraft to transmit photographs of the moon seen from surface level.
The Soviets didn’t release the photos immediately, but scientists at England’s Jodrell Bank Observatory, who were observing the mission, realized that the signal format was the same as the Radiofax system that newspapers used to transmit pictures. So they just borrowed a receiver from the Daily Express, decoded the images, and published them.
The BBC observes, “It is thought that Russian scientists had deliberately fitted the probe with the standard television equipment, either to ensure that they would get the higher-quality pictures from Jodrell Bank without having the political embarrassment of asking for them, or to prevent the Soviet authorities from making political capital out of the achievement.”
On April 12, 1961, witnesses saw a spaceship enter Earth’s atmosphere and descend to the ground in a ploughed field in the Leninsky Put Collective Farm near the Soviet village of Smelovka. At a height of 7 kilometers, a spaceman left the ship and drifted to earth on a parachute. The spaceman later reported:
As I stepped on the firm soil, I saw a woman and a girl. They were standing beside a spotted calf and gazing at me with curiosity. I started walking towards them and they began walking towards me. But the nearer they got to me the slower their steps became. I was still wearing my flaming orange spacesuit and they were probably frightened by it. They had never seen anything like it before.
‘I’m a Russian, comrades. I’m a Russian,’ I shouted, taking off my helmet.
The woman was Anna Takhtarova, wife of the local forester, and the girl, Rita, was her granddaughter.
‘Have you really come from outer space?’ she asked a little uncertainly.
‘Just imagine, I certainly have,’ I replied.
He was Yuri Gagarin, and the site would soon receive a permanent monument marking the landing place of Vostok-1.
In 1982, J.K. Aronson of Oxford, England, sent this mysterious fragment to Douglas Hofstadter:
‘T’ is the first, fourth, eleventh, sixteenth, twenty-fourth, twenty-ninth, thirty-third …
The context of their discussion was self-reference, so presumably the intended conclusion of Aronson’s sentence was … letter in this sentence. If one ignores spaces and punctuation, then T does indeed occupy those positions in Aronson’s fragment; the next few terms would be 35, 39, 45, 47, 51, 56, 58, 62, and 64. The Online Encyclopedia of Integer Sequences gives a picture:
But there’s a catch: In English, most ordinal adjectives (FIRST, FOURTH, etc.) themselves contain at least one T, so the sentence continually creates more work for itself even as it lists the locations of its Ts. There are a few T-less ordinals (NINE BILLION ONE MILLION SECOND), but these don’t arrange themselves to mop up all the incoming Ts. This means that the sentence must be infinitely long.
And, strangely, that throws our initial presumption into confusion. We had supposed that the sentence would end with … letter in this sentence. But an infinite sentence has no end — so it’s not clear whether we ought to be counting Ts at all!