How many distinct knots have exactly 10 crossings? By the late 20th century, mathematicians believed the number to be 166.
Then, in 1973, New York attorney and part-time mathematician Kenneth A. Perko Jr. discovered that two of these were essentially the same knot.
The correspondence had gone unnoticed for 75 years.
Origami can solve general cubic equations! The method was developed by Italian mathematician Margherita Piazzolla Beloch, who in 1936 found a way to use paper folding to construct the common tangents to two parabolas.
Given two points p1 and p2 and two lines l1 and l2, we can, whenever possible, make a single fold (dashed line) that puts p1 onto l1 and p2 onto l2 simultaneously. This fold finds a common tangent to two parabolas: one with focus p1 and directrix l1, the other with focus p2 and directrix l2.
“Now, two parabolas drawn in the plane can have at most three different common tangents, suggesting that this origami fold is equivalent to solving a cubic equation,” writes Western New England College mathematician Thomas C. Hull. “Straightedge and compass constructions, on the other hand, can only solve general quadratic equations.”
Beloch’s contribution went uncredited for decades, but it’s now receiving a fuller appreciation. See the 2011 paper below for more details.
(Thomas C. Hull, “Solving Cubics With Creases: The Work of Beloch and Lill,” American Mathematical Monthly 118:4 [April 2011], 307-315. More here.)
The Javan cucumber, Alsomitra macrocarpa, broadcasts its seeds on papery wings that can glide long distances. Some have been found on the decks of ships.
The unique design inspired aviation pioneer Igo Etrich to build an artificial flying wing, which he adapted into Germany’s first mass-produced military aeroplane.
The 1961 GCE O-level exam included this question:
If one square yard of material costs 18 pence, what is the price of one square foot?
One student considered:
1 square yard costs 18 pence.
Therefore 1 yard costs 
Therefore 1 foot costs 4.243 ÷ 3 = 1.414 pence.
Therefore 1 square foot costs 1.4142 = 2 pence.
(Via Eureka.)
05/26/2026 UPDATE: Reader Catalin Voinescu adds:
For more dimensional fun, check out ‘ohms per square’ (symbol: capital omega divided by a literal square). The resistance of sheet material depends only on the shape of the object, not on the scale (assuming the thickness of the sheet stays the same). Any square of a given material, of any size, has the same resistance when measured between opposite edges. Longer, narrower shapes have higher resistance, and shorter, wider ones have lower resistance, but only the aspect ratio matters, not the actual dimensions. So the resistivity of conductive sheet is expressed as the resistance of a square piece of that material: ohms per square. This unit is used in electrical engineering, where thin conductive layers and foils are common, most obviously in PCB manufacturing, but also in the manufacturing of resistors, capacitors, semiconductors, batteries and solar panels.
(Thanks, Catalin.)
This is the most efficient way to pack 10 unit squares into a surrounding square.
It looks like something I would turn in after running out of time.
Cut the decimal expansion of π into parcels of 10 digits:
Mathematician John Conway points out that the chance is only about 1 in 40,000 that all of the digits 1234567890 will appear in any given parcel … and yet there they all are in the seventh.
(Via Alfred Posamentier and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number, 2004.)
This baffling message illustrates a cipher adopted by the Union Army in 1862:
TO GEORGE C. MAYNARD, Washington
Regulars ordered of my to public out suspending received 1862 spoiled thirty I dispatch command of continue of best otherwise worst Arabia my command discharge duty of my last for Lincoln September period your from sense shall duties the until Seward ability to the I a removal evening Adam herald tribune.
PHILIP BRUNER
The address and signature are “covers” that don’t enter into the cipher. The first word, Regulars, is a code indicating that the original message had been written in five columns of nine words each. Tribune, herald, spoiled, Seward, for, and worst are null words; Lincoln is code for Louisville, Kentucky; Adam means General Henry Wager Halleck; and Arabia is code for Major General Don Carlos Buell. The word Period indicates a full stop. This had been the original message:
Louisville, Kentucky September thirty 1862 General Halleck: (Adam) (period) I received last evening your dispatch suspending my removal from command. Out of a sense of public duty, I shall continue to discharge the duties of my command to the best of my ability until otherwise ordered. D.C. Buell, Major General
This message had been enciphered by reading up the fourth column, down the third, up the fifth, down the second, and up the first; inserting the null words; and encoding the most sensitive particulars. The system worked well until July 1864, when Union cipher operator Stephen L. Robinson was captured by Confederate guerrillas and the key seized.
(John Laffin, Codes and Ciphers Secret Writing Through the Ages, 1964.)
When a container of granular material is shaken, we might expect the largest particles to make their way to the bottom. Instead the opposite often happens: Vibrating a container of mixed nuts, muesli, or raisin bran often brings the largest (and presumably heaviest) items to the top.
Precisely why this happens is unclear. An irregularly shaped Brazil nut might “shoulder” its way above smaller nuts as it turns among them; the rising of large particles might help to lower the center of mass of the aggregate; or perhaps the size of the largest particles prevents them from descending in a container’s natural convection flow once they reach the surface. For now it’s an unsolved problem in physics.
J.M. Roberts’ 1987 Hutchinson History of the World contains this arresting sentence:
At one site in Spain the mind of what one scholar called a ‘primitive Archimedes’ has been seen at work three hundred thousand years ago, directing the removal and use of the tusks of slaughtered elephants as levers to shift the carcasses for cutting up.
The scholar seems to be archaeologist François Bordes, who had written in his 1968 book The Old Stone Age that the Acheuleans of Torralba-Ambrona had killed elephants half engulfed in mud, “and that a primitive Archimedes had the idea of using their tusks as levers for shifting their enormous bulk and making it easier to cut them up.”
From what I can understand, the evidence for butchery at these sites is now thought to be ambiguous, but it’s a striking image nonetheless.
Completely unrelated, but similarly notable: In Days With Bernard Shaw, his 1948 memoir of his friendship with George Bernard Shaw, Stephen Winsten remembers Shaw remarking, “Leonardo da Vinci ruled his notebooks in columns headed fox, wolf, bear and monkey and made notes of human faces by ticking them off in these columns.” I can’t confirm this either, but it seems worth recording.
In this network of 9 points, any two points that are linked have 1 linked point in common, forming a triangle. Any two points that aren’t linked have 2 linked points in common, forming a quadrilateral. Is such a pattern possible in a network of 99 points? In 2014 Princeton mathematician John Horton Conway offered $1000 for the answer to this question; so far the prize is unclaimed.
