Early Delivery

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In the 1940s British psychologist Robert H. Thouless set out to test the existence of life after death by publishing an enciphered message and then communicating the key to some living person after his own death. He published the following in the Proceedings of the Society for Psychical Research:

CBFTM HGRIO TSTAU FSBDN WGNIS BRVEF BQTAB QRPEF BKSDG MNRPS RFBSU TTDMF EMA BIM

He wrote that “it uses one of the well-known methods of encipherment with a key-word which I hope to be able to remember in the after life. I have not communicated and shall not communicate this key-word to any other person while I am still in this world, and I destroyed all papers used in enciphering as soon as I had finished.” He hoped that his message would be unsolvable without supernatural aid because the message was relatively short and the cipher wasn’t simple. To prevent an erroneous decipherment, he revealed that his passage was “an extract from one of Shakespeare’s plays.” And he left the solution in a sealed envelope with the Society for Psychical Research, to be opened if this finally proved necessary.

He needn’t have worried — an unidentified “cipher expert” took up the cipher as a challenge and solved it in two weeks, long before Thouless’ death. It was the last two lines of this quotation from Macbeth:

Sleep that knits up the ravelled sleave of care
The death of each day’s life, sore labour’s bath
Balm of hurt minds, great nature’s second course,
Chief nourisher in life’s feast.

(It’s a Playfair cipher — a full solution is given in Craig Bauer’s excellent Unsolved!, 2017.)

Interestingly, Thouless published two other encrypted ciphers before his death in 1984, and only one has been solved. If you can communicate with the dead perhaps you can still solve it — it’s given on Klaus Schmeh’s blog.

09/01/2019 UPDATE: The last one has been solved! (Thanks, Jason.)

The Jindo Sea Parting

Every year hundreds of thousands of people gather on Jindo Island at the southern tip of the Korean Peninsula to watch the sea part, revealing a 1.8-mile causeway that permits them to walk to the nearby island of Modo, where they dig for clams.

Legend tells that Yongwang, the ocean god, split the sea to permit an old woman to rejoin her family. But National Geographic explains that the truth lies in tidal harmonics.

Four of a Kind

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If squares are drawn on the sides of a triangle and external to it, then the areas of the triangles formed between the squares all equal the area of the triangle itself.

(Roger Webster, “Bride’s Chair Revisited,” Mathematical Gazette 78:483 [November 1994], 345-346.)

Not Dead Yet

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Image: Wikimedia Commons

The red-crested tree rat hadn’t been seen since 1898 when one turned up at the front door of a Colombian ecolodge in 2011. It posed for photos for two hours and then disappeared again. “He just shuffled up the handrail near where we were sitting and seemed totally unperturbed by all the excitement he was causing,” said volunteer researcher Lizzie Noble. “We are absolutely delighted to have rediscovered such a wonderful creature after just a month of volunteering with ProAves.”

The Bermuda land snail had been thought extinct for 40 years when it turned up in a Hamilton alleyway. “The fact that there was so much concrete around them probably saved them from the predators that we believe killed the vast majority of the population island-wide,” ecologist Mark Outerbridge told the Royal Gazette. “People have been looking for these snails for decades and here they are surrounded by concrete and air conditioners living in a 100-square-foot alleyway in Hamilton.”

The mountain pygmy possum was first identified in a fossil in New South Wales in 1895 and thought to be extinct. Seventy years later, in August 1966, a live pygmy possum was found by chance in a ski hut in the Snowy Mountains of Victoria. R.M. Warneke telegraphed W.D.L. Ride, “BURRAMYS EXTANT STOP NOT REPEAT NOT EXTINCT STOP LIVE MALE CAPTURED MOUNT HOTHAM STOP AM TRYING FOR FEMALE.” The lonely possum died before a companion could be found, but four isolated populations of pygmy possums are now known to persist in the Snowy Mountains.

(Joseph F. Merritt, The Biology of Small Mammals, 2010.)

Math Notes

26072323311568661931
43744839742282591947
118132654413675138222
186378732807587076747
519650114814905002347

Any three of these numbers add up to a perfect square.

(Discovered by Stan Wagon.)

Confession

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In 1662, while a student at Cambridge, 19-year-old Isaac Newton made a list of 57 sins he’d committed:

Before Whitsunday 1662

Using the word (God) openly
Eating an apple at Thy house
Making a feather while on Thy day
Denying that I made it.
Making a mousetrap on Thy day
Contriving of the chimes on Thy day
Squirting water on Thy day
Making pies on Sunday night
Swimming in a kimnel on Thy day
Putting a pin in Iohn Keys hat on Thy day to pick him.
Carelessly hearing and committing many sermons
Refusing to go to the close at my mothers command.
Threatning my father and mother Smith to burne them and the house over them
Wishing death and hoping it to some
Striking many
Having uncleane thoughts words and actions and dreamese.
Stealing cherry cobs from Eduard Storer
Denying that I did so
Denying a crossbow to my mother and grandmother though I knew of it
Setting my heart on money learning pleasure more than Thee
A relapse
A relapse
A breaking again of my covenant renued in the Lords Supper.
Punching my sister
Robbing my mothers box of plums and sugar
Calling Dorothy Rose a jade
Glutiny in my sickness.
Peevishness with my mother.
With my sister.
Falling out with the servants
Divers commissions of alle my duties
Idle discourse on Thy day and at other times
Not turning nearer to Thee for my affections
Not living according to my belief
Not loving Thee for Thy self
Not loving Thee for Thy goodness to us
Not desiring Thy ordinances
Not [longing] for Thee in [illegible]
Fearing man above Thee
Using unlawful means to bring us out of distresses
Caring for worldly things more than God
Not craving a blessing from God on our honest endeavors.
Missing chapel.
Beating Arthur Storer.
Peevishness at Master Clarks for a piece of bread and butter.
Striving to cheat with a brass halfe crowne.
Twisting a cord on Sunday morning
Reading the history of the Christian champions on Sunday

Since Whitsunday 1662

Glutony
Glutony
Using Wilfords towel to spare my own
Negligence at the chapel.
Sermons at Saint Marys (4)
Lying about a louse
Denying my chamberfellow of the knowledge of him that took him for a [illegible] sot.
Neglecting to pray 3
Helping Pettit to make his water watch at 12 of the clock on Saturday night

“We aren’t sure what prompted this confession,” writes Mitch Stokes in his 2010 biography of the physicist. “Some biographers think that it was in response to an inner crisis. Perhaps it was the occasion of his conversion, or at least of his ‘owning his faith.’ We simply don’t know.”

Cat Music

For what it’s worth: In 2015 University of Wisconsin psychologists Megan Savage and Charles Snowdon considered that music might appeal better to other species if it used the same tempos and frequency ranges that they use to communicate.

Accordingly they got musician David Teie to compose three songs that ought to appeal to felines and tried them out on 47 domestic cats, comparing their reactions to Bach’s “Air on a G String” and Fauré’s “Elegie.” The cat music was pitched about an octave higher than human voices, and its tempos replicated purring and suckling rather than a human heartbeat.

The cats showed no interest in the music intended for humans, but they showed a “significant preference for and interest in” Teie’s cat-targeted songs, approaching the speakers and often rubbing their scent glands on them. Also, for some reason young and old cats seemed to like the cat music better than middle-aged ones.

Savage and Snowdon conclude that these results “suggest novel and more appropriate ways for using music as auditory enrichment for nonhuman animals.” Here’s a sample to try on your own cat:

(Charles T. Snowdon, David Teie, and Megan Savage, “Cats Prefer Species-Appropriate Music,” Applied Animal Behaviour Science 166 [May 2015], 106–111.) (Thanks, Noah.)

Breakdown

borwein integrals

These are called Borwein integrals, after David and Jonathan Borwein, the father-and-son mathematicians who first presented them in 2001.

Engineer Hanspeter Schmid writes, “[W]hen this fact was recently verified by a researcher using a computer algebra package, he concluded that there must be a ‘bug’ in the software. It is not a bug, though; this series of integrals really only results in π/2 up to a certain point, and then breaks down. This astonishes most mathematically educated readers, as especially those readers mentally extrapolate the sequence shown above and find it surprising that something fundamental should change when the factor sinc(x/15) is introduced.” He gives a graphic explanation of what’s happening.

(David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 [March 2001], 73–89.) (Thanks, Dan.)

Sanity and Simpson

In his 2008 book Impossible?, Julian Havil presents an argument offered in Massachusetts in 1854 contending that foreigners were more likely to be insane than native-born Americans. These figures were offered:

Whole Population
Insane Not Insane Totals
Foreign-Born 625 229375 230000
Native-Born 2007 892669 894676
Totals 2632 1122044 1124676

The probability that a foreign-born person was deemed insane was 625/230000 = 2.7 × 10-3, and for a native-born person the probability was 2007/894676 = 2.2 × 10-3, which seems to support the claim.

But we get a different story when we divide the data by social hierarchy, into what were called the pauper and independent classes:

Pauper Class
Insane Not Insane Totals
Foreign-Born 182 9090 9272
Native-Born 250 12513 12763
Totals 432 21603 22035
Independent Class
Insane Not Insane Totals
Foreign-Born 443 220285 220728
Native-Born 1757 880156 881913
Totals 2200 1100441 1102641

In the pauper class the probability of a foreign-born person being deemed insane is 182/9272 = 0.02, which is the same as that for a native-born person (250/12763 = 0.02). And the same is true in the independent class, where both probabilities are 2.0 × 10-3. Havil writes, “So, if an adjustment is made for the status of the individuals we see that there is no relationship at all between sanity and origin” (an example of Simpson’s paradox).

Current Events

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Image: Wikimedia Commons

“Squaring the square” refers to tiling a square with other squares, each with sides of integer length.

In a “perfect” squared square, like the one above, each smaller square is of a different size. The Cambridge University team that first sought perfect squares found a novel way to go about it — they transformed the square tiling into an electrical circuit in which each square is a resistor that connects to its neighbors above and below, and then applied Kirchhoff’s circuit laws to that circuit.

The example below isn’t perfect, but the technique did succeed — the smallest perfect square they found is 69 units on a side.

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Image: Wikimedia Commons

(Rowland Leonard Brooks, et al., “The Dissection of Rectangles into Squares,” Duke Mathematical Journal 7:1 [1940], 312-340.)