How many pairs of prime numbers are there whose sum is 999?
How many pairs of prime numbers are there whose sum is 999?
The International Statistical Classification of Diseases and Related Health Problems (ICD) is a list of more than 10,000 diseases and maladies that patients might present. The medical community uses it for recordkeeping — for example, a patient admitted to the hospital with whooping cough would be logged in the database with code A37. Reader Will Beattie sent me a list of some of the stranger complaints on the list:
Will says his favorite so far is “Burn due to water skis on fire – V9107.” It’s a dangerous world,” he writes. “Be safe out there.”
Related: Each year the Occupational Safety and Health Administration publishes a list of workplace deaths, with a brief description of each incident:
It’s hard to pick the worst one. “Worker was operating a skid-steer cleaning out a dairy cattle barn near an outdoor manure slurry pit. The skid-steer and the worker fell off the end of the push-off platform into the manure slurry pit, trapping the worker in the vehicle. Worker died of suffocation due to inhalation of manure.”
If 6 cats can kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?
It’s easy enough to work out that the answer is 12, but consider what this means. “When we come to trace the history of this sanguinary scene through all its horrid details, we find that at the end of 48 minutes 96 rats are dead, and that there remain 4 live rats and 2 minutes to kill them in,” observed Lewis Carroll in the Monthly Packet in February 1880. “The question is, can this be done?”
Consider the original statement: 6 cats can kill 6 rats in 6 minutes. What can this actually mean? Carroll counts at least four possibilities:
A. “All 6 cats are needed to kill a rat; and this they do in one minute, the other rats standing meekly by, waiting for their turn.”
B. “3 cats are needed to kill a rat, and they do it in 2 minutes.”
C. “2 cats are needed, and they do it in 3 minutes.”
D. “Each cat kills a rat all by itself, and takes 6 minutes to do it.”
Now try to apply these to our conclusion that 12 cats can kill 100 rats in 50 minutes. Cases A and B work out, but Case C can work only if we understand that fractional deaths are possible: that 2 cats could kill two-thirds of a rat in 2 minutes. Similarly, Case D works only if a cat can kill one-third of a rat in 2 minutes.
The only way to resolve this absurdity, it seems, is to supply extra cats. “In case C less than 2 extra cats would be of no use. If 2 were supplied, and if they began killing their 4 rats at the beginning of the time, they would finish them in 12 minutes, and have 36 minutes to spare, during which they might weep, like Alexander, because there were not 12 more rats to kill. In case D, one extra cat would suffice; it would kill its 4 rats in 24 minutes, and have 24 minutes to spare, during which it could have killed another 4. But in neither case could any use be made of the last 2 minutes, except to half-kill rats — a barbarity we need not take into consideration.”
“To sum up our results: If the 6 cats kill the 6 rats by method A or B, the answer is ’12’; if by method C, ’14’; if by method D, ’13’.”
(Another problem: “If a cat can kill a rat in a minute, how long would it be killing 60,000 rats? Ah, how long, indeed! My private opinion is that the rats would kill the cat.”)
During World War II, Alan Turing enrolled in the infantry section of the Home Guard so that he could learn to shoot a rifle. After completing this section of his training he stopped attending parades, as he had no further use for the service. Summoned to account for this, he explained that he was now an excellent shot and this was why he had joined.
“But it is not up to you whether to attend parades or not,” said Colonel Fillingham. “When you are called on parade, it is your duty as a soldier to attend.”
“But I am not a soldier.”
“What do you mean, you are not a soldier! You are under military law!”
“You know, I rather thought this sort of situation could arise,” Turing said. “I don’t know I am under military law. If you look at my form you will see that I protected myself against this situation.”
It was true. On his application form Turing had encountered the question “Do you understand that by enrolling in the Home Guard you place yourself liable to military law?” He could see no advantage in answering yes, so he answered no, and the clerk had filed the form without looking at it.
“So all they could do was to declare that he was not a member of the Home Guard,” remembered Peter Hilton. “Of course that suited him perfectly. It was quite characteristic of him. And it was not being clever. It was just taking this form, taking it at its face value and deciding what was the optimal strategy if you had to complete a form of this kind. So much like the man all the way through.”
(From Andrew Hodges, Alan Turing: The Enigma, 1992.)
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.)
Cartoon laws of physics:
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!”
About 150,000 years ago, a Neanderthal man was exploring the Lamalunga Cave in southern Italy when he fell into a sinkhole. Too badly injured to climb out again, he died of dehydration or starvation. Over the ensuing centuries, water running down the cave walls gradually incorporated the man’s bones into concretions of calcium carbonate. Undisturbed by predators or weather, they lay in an immaculate state of preservation until cave researchers finally discovered them in 1993.
This is a great boon for paleoanthropologists — “Altamura Man” is one of the most complete Paleolithic skeletons ever discovered in Europe — but there’s a downside: The bones have become so deeply involved in their matrix of limestone that no one has found a way to remove them without destroying them. So, for now, all research must be carried out in the cave.
In order to get a license, London taxicab drivers must pass a punishing exam testing their memory of 25,000 streets and every significant business and landmark on them. “The Knowledge” has been called the hardest test of any kind in the world; applicants must put in thousands of hours of study to pass a series of progressively difficult oral exams that take, on average, four years to complete. The guidebook for prospective cabbies says:
To achieve the required standard to be licensed as an ‘All London’ taxi driver you will need a thorough knowledge, primarily, of the area within a six-mile radius of Charing Cross. You will need to know: all the streets; housing estates; parks and open spaces; government offices and departments; financial and commercial centres; diplomatic premises; town halls; registry offices; hospitals; places of worship; sports stadiums and leisure centres; airline offices; stations; hotels; clubs; theatres; cinemas; museums; art galleries; schools; colleges and universities; police stations and headquarters buildings; civil, criminal and coroner’s courts; prisons; and places of interest to tourists. In fact, anywhere a taxi passenger might ask to be taken.
Interestingly, licensed London cabbies show a significantly larger posterior hippocampus than non-taxi drivers. Psychologist Hugo J. Spiers writes, “Current evidence suggests that it is the acquisition of this spatial knowledge and its use on the job that causes the taxi driver’s posterior hippocampus to grow larger.” Apparently it’s not actually driving the streets, or learning the information alone, that causes the change — London bus drivers don’t show the same effect; nor do doctors, who must also acquire vast knowledge; nor do cabbies who fail the exam. Rather it seems to be the regular use of the knowledge that causes the change: Retired cabbies tend to have a smaller hippocampus than current drivers.
While driving virtual routes in fMRI studies, cabbies showed the most hippocampal activity at the moment a customer requested a destination. One cabbie said, “I’ve got an over-patched picture of Peter Street. It sounds daft, but I don’t view it from ground level, it was slightly up and I could see the whole area as though I was about 50 foot up. And I saw Peter Street, I saw the market and I knew I had to get down to Peter Street.” Non-cabbie volunteers also showed the most activity when they were planning a route. “Thus,” writes Spiers, “the engagement of the hippocampus appears to depend on the extent to which someone thinks about what the possible streets they might want to take during navigation.”
(Hugo J. Spiers, “Will Self and His Inner Seahorse,” in Sebastian Groes, ed., Memory in the Twenty-First Century, 2016.)
Multiply any two of these numbers together and add 1 and you’ll always get a perfect square:
1 + 1 × 3 = 4
1 + 1 × 8 = 9
1 + 1 × 120 = 121
1 + 3 × 8 = 25
1 + 3 × 120 = 361
1 + 8 × 120 = 961
In 1932, at the end of a 60-year career studying hydrodynamics, Sir Horace Lamb addressed the British Association for the Advancement of Science.
“I am an old man now,” he said, “and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather more optimistic.”