Podcast Episode 148: The Perfect Murder
Insurance agent William Herbert Wallace had a terrible night in January 1931 — summoned to a nonexistent address in Liverpool, he returned home to find that his wife had been murdered in his absence. An investigation seemed to show a senseless crime with no weapon, no motive, and no likely suspects. In this week’s episode of the Futility Closet podcast we’ll revisit the slaying of Julia Wallace, which Raymond Chandler called “the impossible murder.”
We’ll also recount some wobbly oaths and puzzle over an eccentric golfer.
77777 …
A curious puzzle from the Penguin Problems Book, 1940:
A certain number consisting entirely of 7s is divisible by 199. Find the last four digits of the quotient, without finding the entire quotient.
Aronson’s Sequence
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:
1234567890 1234567890 1234567890 1234567890 1234567890 Tisthefirs tfourthele venthsixte enthtwenty fourthtwen tyninththi rtythirdth irtyfiftht hirtyninth fortyfifth fortyseven thfiftyfir stfiftysix thfiftyeig hthsixtyse condsixtyf ourthsixty ninthseven tythirdsev entyeighth eightiethe ightyfourt heightynin thninetyfo urthninety ninthonehu ndredfourt honehundre deleventho nehundreds ixteenthon ehundredtw entysecond onehundred twentysixt honehundre dthirtyfir stonehundr edthirtysi xthonehund redfortyse cond...
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!
The Lonely Runner
Suppose k runners are running around a circular track that’s 1 unit long. All the runners start at the same point, but they run at different speeds. A runner is said to be “lonely” if he’s at least distance 1/k along the track from every other runner. The lonely runner conjecture states that each of the runners will be lonely at some point.
This is obviously true for low values of k. If there’s a single runner, then he’s lonely before he even leaves the starting line. And if there are two runners, at some point they’ll occupy diametrically opposite points on the track, at which point both will be lonely.
But whether it’s always true, for any number of runners, remains an unsolved problem in mathematics.
Good Night Lights
Six years ago, Providence restaurant The Hot Club started the custom of flashing its lights each night at 8:30, as a way to say good night to the children in the six-story Hasbro Children’s Hospital across the Providence River. The club would flash its neon sign, and patrons would gather on the waterfront deck to wave flashlights and cell phones.
Now hotels, boat traffic, skyscrapers, and police cruisers have begun to join in, some with permanent signals but many using handheld lights. About two dozen children are in the hospital on any given night, and they’ve taken to flashing back with lights of their own.
“No one knows who’s on the other side of the gesture,” hospital cartoonist Steve Brosnihan told the Associated Press. “People often say, ‘I get goosebumps hearing about this.'”
“They don’t know me; they could skip the step of flicking the lights, but they do it anyway,” said 13-year-old lupus patient Olivia Stephenson. “I hope they saw the thank you.”
Conway’s RATS Sequence
Write down an integer. Remove any zeros and sort the digits in increasing order. Now add this number to its reversal to produce a new number, and perform the same operations on that:
In the example above, we’ve arrived at a pattern, alternating between 12333334444 and 5566667777 but adding a 3 and a 6 (respectively) with each iteration. (So the next two sums, after their digits are sorted, will be 123333334444 and 55666667777, and so on.)
Princeton mathematician John Horton Conway calls this the RATS sequence (for “reverse, add, then sort”) and in 1989 conjectured that no matter what number you start with (in base 10), you’ll either enter the divergent pattern above or find yourself in some cycle. For example, starting with 3 gives:
3, 6, 12, 33, 66, 123, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, …
… and now we’re in a loop — the last eight terms will just repeat forever.
Conway’s colleague at Princeton, Curt McMullen, showed that the conjecture is true for all numbers less than a hundred million, and himself conjectured that every RATS sequence in bases smaller than 10 is eventually periodic. Are they right? So far neither conjecture has been disproved.
(Richard K. Guy, “Conway’s RATS and Other Reversals,” American Mathematical Monthly 96:5 [May 1989], 425-428.)
Quickie
A puzzle by Frank Morgan: The meaning of a common English word becomes plural when an A is added at its start. What is the word?
Monte Kaolino
When Hirschau, Bavaria starting mining kaolinite a century ago, it faced a problem — one of the byproducts of kaolinite is quartz sand, which began piling up in enormous quantities. Fortunately sand itself has multiple uses — in the early 1950s an enterprising skier tried slaloming down the mountain, and soon the dune had its own ski club.
Today the 35-million-tonne “Monte Kaolino” even hosts the Sandboarding World Championships. And, unlike other ski resorts, it’s open in summer.
Breath Control
In 1999 the German state of Mecklenburg-Vorpommern passed a law governing the labeling of beef; its short title was Rinderkennzeichnungs- und Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz (PDF). The hyphen indicates that the first word would have the same ending as the second; thus the two words are Rinderkennzeichnungsüberwachungsaufgabenübertragungsgesetz and Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz. (The law’s formal long title is Gesetz zur Übertragung der Aufgaben für die Überwachung der Rinderkennzeichnung und Rindfleischetikettierung, or “Law on Delegation of Duties for Supervision of Cattle Marking and Beef Labeling.”)
Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz was nominated for Word of the Year by the German Language Society. Here it is sung by the Australian National University’s UniLodge Choir:
There’s more: In 2003 a decree was passed modifying real estate regulations; its short title was Grundstücksverkehrsgenehmigungszuständigkeitsübertragungsverordnung. Can someone sing that?