The Franklin Square

franklin magic square

In a 1769 letter, Ben Franklin describes a magic square he devised in his youth. The magic total of 260 can be reached by adding the numbers in each row or column, as in a normal magic square. But “bent rows” (shaded) produce the same total, even when “wrapped across” the border of the table. This works in all four directions.

Further: Half of each row or column sums to half of 260, as does any 2×2 subsquare. And the four corners and the four center squares sum to 260. (Alas, the main diagonals don’t, so this doesn’t strictly qualify as a magic square by the modern definition.)

Interestingly, no one knows how Franklin created the square. Many methods have been devised, but none apparently as quick as his, which he claimed could generate them “as fast as he could write.”

| Science & Math

“Singular Discovery at Aberdeen”

While in course of demolishing a block of old houses on the north side of Longacre, which requires to be removed for the extension of Marischal College buildings, the workmen made a curious discovery yesterday morning. About fifteen inches from the exterior of a wall composed of solid masonry they came across a couple of crabs, one being dead but still in a fresh state, and the other alive, although so attenuated as to be almost transparent. The crabs were handed over to Mr. Jones, assistant professor of chemistry at Marischal College. The live crab is preserved in a jar containing water. In size it is an inch long and a quarter broad, its dead companion being an inch and three-quarters in length and an inch in breadth. The house has been untenanted for six months, and it is a mystery how the creatures could have found their way into a mass of masonry twenty feet above the ground level of the outside, and three or four feet from the level of the floor.

— Newspaper paragraph quoted in Scottish Notes and Queries, February 1896

| Oddities

The Perfect Family

http://en.wikipedia.org/wiki/Platonic_solid
Image: Wikimedia Commons

Take any Platonic solid, join the centers of its faces, and, charmingly, you get another Platonic solid. The cube and the octahedron produce one another, as do the dodecahedron and the icosahedron, and the tetrahedron produces another tetrahedron.

Bonus factoid: If you inscribe a dodecahedron and an icosahedron in the same sphere, the dodecahedron will occupy more of the sphere’s volume. It has fewer faces than the icosahedron, but its faces are more nearly circular, so it fits the sphere more snugly.

See The Pup Tent Problem.

| Science & Math

All God’s Creatures

http://commons.wikimedia.org/wiki/File:Ernest_Thompson_Seton.jpg

Ernest Thompson Seton (1860-1946) loved nature and loved God — so in a 1907 book he tried to prove that animals follow the 10 commandments:

  • Thou shalt not steal: “A stick found in the woods is the property of the Rook that discovers it, and doubly his when he has labored to bring it to his nest. This is recognized law.”
  • Thou shalt not kill: “New born Rattlesnakes will strike instantly at a stranger of any other species, but never at one of their own.”
  • Honor thy father and mother: “A Hen sets out with her Chickens a-foraging; one loiters, does not hasten up at her ‘cluck cluck’ of invitation and command; consequently he gets lost and dies.”
  • Thou shalt not commit adultery: “The promiscuous animals to-day–the Northwestern Rabbit and the Voles–are high in the scale of fecundity, low in the scale of general development, and are periodically scourged by epidemic plagues.”
  • Thou shalt not bear false witness: “Oftentimes a very young Hound will jump at a conclusion, think, or hope, he has the trail, then allowing his enthusiasm to carry him away, give the first tongue, shouting in Hound language, ‘Trail!’ The other Hounds run to this, but if a careful examination shows that he was wrong, the announcer suffers in the opinion of the pack, and after a few such blunders, that individual is entirely discredited.”
  • Thou shalt not covet: “A Hen had made a nest in a certain place, and was already sitting. Later another Hen, desiring the same nest, took possession several times during the owner’s brief absence, adding some of her own eggs, and endeavoring to sit. The result was a state of war, and the eggs of both Hens were destroyed.”

Actually, he runs out of gas here — Seton was unable to convince even himself that animals avoid making graven images, swearing, or working on Sunday. So he concludes The Natural History of the Ten Commandments by deciding that “Man is concerned with all” the commandments, “the animals only with the last six.”

| Religion

“The Pessimist”

Nothing to do but work,
Nothing to eat but food,
Nothing to wear but clothes,
To keep one from going nude.

Nothing to breathe but air,
Quick as a flash ‘t is gone;
Nowhere to fall but off,
Nowhere to stand but on.

Nothing to comb but hair,
Nowhere to sleep but in bed,
Nothing to weep but tears,
Nothing to bury but dead.

Nothing to sing but songs,
Ah, well, alas! alack!
Nowhere to go but out,
Nowhere to come but back.

Nothing to see but sights,
Nothing to quench but thirst,
Nothing to have but what we’ve got;
Thus through life we are cursed.

Nothing to strike but a gait;
Everything moves that goes.
Nothing at all but common sense
Can ever withstand these woes.

— Ben King, collected in Joel Chandler Harris, ed., American Wit and Humor, 1907

| Poems

Mathematicians’ Graves

http://books.google.com/books?id=ssEKAAAAIAAJ&pg=PA103&dq=henry+perigal+pythagorean&hl=en&ei=twPNS9qlGoGB8gbJwbzSBA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDwQ6AEwAQ#v=onepage&q&f=false

Archimedes wanted no other epitaph than a sphere inscribed within a cylinder — he had determined the sphere’s relative volume and considered this his greatest achievement.

Henry Perigal’s tomb in Essex displays his graphic proof of the Pythagorean theorem (left).

Gauss wanted to be buried under a heptadecagon, which he’d shown can be constructed with compass and straightedge. (The stonemason demurred, fearing he’d produce only a circle.)

And Jakob Bernoulli opted for a logarithmic spiral and the words Eadem mutata resurgo—the motto means “I shall arise the same though changed.”

| Death · Science & Math

A Little Crusade

http://books.google.com/books?id=HRRBAAAAYAAJ&pg=PA134&dq=chess+curiosities&as_brr=4&ei=e_uzS8KSMY3mygTujrjfCw&cd=1#v=onepage&q=chess%20curiosities&f=false

White to move his king only and mate in six moves.

Click for Answer
| Puzzles

Unquote

“A great many people think they are thinking when they are merely rearranging their prejudices.” — William James

| Quotations

Managing the Managers

http://commons.wikimedia.org/wiki/File:The_Lighthouse_of_Alexandria_by_Magdalena_van_de_Pasee.jpg

Sostratos, architect of the famous light-house on the Island of Pharos, Alexandria, once numbered among the seven wonders of the world, engraved deeply on one of the stones the words, ‘Sostratos of Gnidos, son of Dexiphanos, to the Gods protecting those on the sea.’ Knowing very well that Ptolemy, his employer, would not be satisfied with this inscription, he covered it with a thin coating of plaster on which he inscribed the name of Ptolemy. In time the plaster disappeared, and with it the name of the king, so that in the end the architect had all the credit for the work.

— The Illustrated American, June 18, 1892

| History