Roy Sinclair posed this question in the November-December 2000 issue of MIT Technology Review. Bruce Layton offered a solution in the May 2001 issue: The values are 1, 2, 3, and 4. For n = 1 there are no distances to be unequal, and for n = 2 there’s only one distance, with nothing to be unequal to. The cases of n = 3 and n = 4 can be shown by choosing two points on the sphere’s surface and centering on each of them a sphere whose radius is the distance between the two points (so that now each of the two points is on the surface of one constructed sphere and at the center of the other). Any point that falls at the intersection of these two spheres and on the surface of the original sphere satisfies the requirement. There can be one or two such points, meaning n can be 3 or 4, but not higher. If it’s 4 then the four points are the vertices of a regular tetrahedron; if it’s 3 then the original two chosen points and one of the intersection points form an equilateral triangle (roughly, on Earth, reader Eugene Sard suggested that three such points fall at the South Pole, on Wake Island, and in southern Saudi Arabia).
In The Roots of Coincidence, Arthur Koestler mentions that the participants at a 1932 conference on nuclear physics put on a parody of Goethe’s Faust in which Wolfgang Pauli played Mephistopheles. “His Gretchen was the neutrino, whose existence Pauli had predicted, but which had not yet been discovered.”
MEPHISTOPHELES (to Faust):
Beware, beware, of Reason and of Science
Man’s highest powers, unholy in alliance.
You’ll let yourself, through dazzling witchcraft yield
To weird temptations of the quantum field.
Enter Gretchen; she sings to Faust. Melody: ‘Gretchen at the Spinning Wheel’ by Schubert.
My rest-mass is zero
My charge is the same
You are my hero Neutrino’s my name.
In 1778, shortly after Benjamin Franklin introduced the lightning rod, Paris saw a fad for umbrellas and hats that made use of the new technology. A chain ran from the accessory down to the ground and would (in principle) carry the electricity from a lightning strike harmlessly into the ground.
I can’t find any record that such a strike ever happened. Lightning rods didn’t become popular in the United States, even to protect structures, until the 19th century.
Each of the 52 integers can be expressed as 100a + b, where b can range from -49 to 50. That means that the absolute value of b can take only 51 possible values (0 to 50), so at least two of our 52 integers must share one such value. If these two integers have opposite signs associated with b, then the sum of those integers is divisible by 100; if they have the same sign, then their difference is divisible by 100.
I found this in “The Olympiad Corner” in the September 2003 issue of Crux Mathematicorum; reportedly it appeared originally in the Russian Mathematical Olympiad, but the date isn’t known.
In 1894 Francis Galton experimented with conducting addition and subtraction by smell. He designed an apparatus that would produce whiffs of scented air and then memorized their combinations: “I taught myself to associate two whiffs of peppermint with one whiff of camphor; three of peppermint with one of carbolic acid, and so on.”
After practicing sums using the scents themselves, he moved on to doing them entirely in his imagination. “There was not the slightest difficulty in banishing all visual and auditory images from the mind, leaving nothing in the consciousness besides real or imaginary scents. In this way, without, it is true, becoming very apt at the process, I convinced myself of the possibility of doing sums in simple addition with considerable speed and accuracy solely by means of imaginary scents.”
He had similar success with subtraction, but didn’t try multiplication. And some further experiments seemed to show that “arithmetic by taste was as feasible as arithmetic by smell.”
In the 19th century, an enormous hedge ran for more than a thousand miles across India, installed by the British to enforce a tax on salt. Though it took a Herculean effort to build, today it’s been almost completely forgotten. In this week’s episode of the Futility Closet podcast we’ll describe this strange project and reflect on its disappearance from history.
We’ll also exonerate a rooster and puzzle over a racing murderer.
Roy Moxham, “The Great Hedge of India,” in Jantine Schroeder, Radu Botez, and Marine Formentini, Radioactive Waste Management and Constructing Memory for Future Generations: Proceedings of the International Conference and Debate, September 15-17, 2014, Verdun, France, Organisation for Economic Co-Operation and Development, 2015.
Maurice Chittenden, “Great Hedge of India Defended the Empire,” Sunday Times, Dec. 10, 2000, 7.
Aneesh Gokhale, “Why British Built the Great Hedge of India,” DNA, Aug. 12 2018.
Roy Moxham, “The Great Hedge of India,” Sunday Telegraph, Jan. 7, 2001, 4.
Annabelle Quince, “Border Walls Around the World,” Rear Vision, ABC Premium News, May 17, 2017.
“Have You Heard of the Salt Hedge?” New Indian Express, March 16, 2015.
Roy Moxham, “Magnificent Obsession,” Weekend Australian, Oct. 5, 2002, B.26.
Matthew Wilson, “In the Thicket of It,” Financial Times, Nov. 12, 2016, 20.
Moxham writes, “My GPS reading at Pali Ghar was 26° 32.2’ N, 79° 09.2’ E. If this reading is put into Google Earth, the embankment of the Hedge is clearly visible – but only if you already know it is there.”
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Many thanks to Doug Ross for the music in this episode.