The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.
Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.
Thomas Huxley’s Evolution and Ethics took China by storm — phrases such as the strong are victorious and the weak perish resonated in the national consciousness and “spread like a prairie fire, setting ablaze the hearts and blood of many young people,” noted philosopher Hu Shih.
People even adopted Darwin’s ideas as names. “The once famous General Chen Chiung-ming called himself ‘Ching-tsun’ or ‘Struggling for Existence.’ Two of my schoolmates bore the names ‘Natural Selection Yang’ and ‘Struggle for Existence Sun.’
“Even my own name bears witness to the great vogue of evolutionism in China. I remember distinctly the morning when I asked my second brother to suggest a literary name for me. After only a moment’s reflection, he said, ‘How about the word shih [fitness] in the phrase “Survival of the Fittest”?’ I agreed and, first using it as a nom de plume, finally adopted it in 1910 as my name.”
(Hu Shih, Living Philosophies, 1931.)
Draw a right triangle whose legs a and b each measure 1. Draw d and e to complete a unit square. Clearly d + e = 2.
Now if we cut a “step” into the square as shown, then f + h = 1 and g + i = 1, so the total length of the “staircase” is still 2. Cut still finer steps and j + k + l + m + n + o + p + q is likewise 2.
And so on: The more finely we cut the steps, the more closely their shape approximates that of the original triangle’s diagonal. Yet the total length of the stairstep shape remains 2, the sum of its horizontal and vertical elements. At the limit, then, it would seem that c must measure 2 … but we know that the length of a unit square’s diagonal is the square root of 2. Where is the error?
(Thanks, Alex.)
A curiosity attributed to a Professor E. Ducci in the 1930s:
Arrange four nonnegative integers in a circle, as above. Now construct further “cyclic quadruples” of integers by subtracting consecutive pairs, always subtracting the smaller number from the larger. So the quadruple above would produce 22, 8, 38, 8, then 14, 30, 30, 14, and so on.
Ducci found that eventually four equal numbers will occur.
A proof appears in Ross Honsberger’s Ingenuity in Mathematics (1970).
The Hoover Dam contains a star map depicting the sky of the Northern Hemisphere as it appeared at the moment that Franklin Roosevelt dedicated the dam. Artist Oskar Hansen imagined that the massive structure might outlive our civilization, and that the map could help future astronomers to calculate the date of its creation. The center star on the map, Alcyone, is the brightest star in the Pleiades, and our sun occupies a position at the center of a flagpole. The whole map traces a complete sidereal revolution of the equinox, a period of 25,694 of our years, and marks the point of the dam’s dedication in that period.
“Man has always sought to express and preserve the magnitude of his exploits in symbols,” Hansen said in 1935. “The written words are symbols arranged so as to preserve in objectified form the thought of man and to record his variant states, both mental and physical. All other arts are similar as to their symbolic significance. They take their place among the category of human endeavor simply as the interpreter of life to itself. They serve as an outer object typifying the inner process. They form the connecting link between the spiritual and the material world. They are the shadows cast by the realities of the soul.”
No one knows whether Andrew Jackson was born in North Carolina or South Carolina. The border hadn’t been surveyed well at the time.
When Glenn Seaborg appeared as a guest scientist on the children’s radio show Quiz Kids in 1945, one of the children asked whether any new elements, other than plutonium and neptunium, had been discovered at the Metallurgical Laboratory in Chicago during the war.
In fact two had — Seaborg announced for the first time anywhere that two new elements, with atomic numbers 95 and 96 (americium and curium), had been discovered. He said, “So now you’ll have to tell your teachers to change the 92 elements in your schoolbook to 96 elements.”
In his 1979 Priestley Medal address, Seaborg recalled that many students apparently did bring this knowledge to school. And “judging from some of the letters I received from such youngsters, they were not entirely successful in convincing their teachers.”
A pretty new theorem by Lee Sallows: Connect each vertex of a triangle to the midpoint of the opposite side, and place a hinge at that point. Now rotate the smaller triangles about these hinges and you’ll produce three congruent triangles.
If the original triangle is isosceles (or equilateral), then the three resulting triangles will be too.
The theorem appears in the December 2014 issue of Mathematics Magazine.
Achilles overtakes the tortoise and runs on into the sunset, exulting. As he does so, a fly leaves the tortoise’s back, flies to Achilles, then returns to the tortoise, and continues to oscillate between the two as the distance between them grows, changing direction instantaneously each time. Suppose the tortoise travels at 1 mph, Achilles at 5 mph, and the fly at 10 mph. An hour later, where is the fly, and which way is it facing?
Strangely, the fly can be anywhere between the two, facing in either direction. We can find the answer by running the scenario backward, letting the three participants reverse their motions until all three are again abreast. The right answer is the one that returns the fly to the tortoise’s back just as Achilles passes it. But all solutions do this: Place the fly anywhere between Achilles and the tortoise, run the race backward, and the fly will arrive satisfactorily on the tortoise’s back at just the right moment.
This is puzzling. The conditions of the problem allow us to predict exactly where Achilles and the tortoise will be after an hour’s running. But the fly’s position admits of an infinite number of solutions. Why?
(From University of Arizona philosopher Wesley Salmon’s Space, Time, and Motion, after an idea by A.K. Austin.)
In 2008, physicist Yuki Sugiyama of the University of Nagoya demonstrated why traffic jams sometimes form in the absence of a bottleneck. He spaced 22 drivers around a 230-meter track and asked them to proceed as steadily as possible at 30 kph, each maintaining a safe distance from the car ahead of it. Because the cars were packed quite densely, irregularities began to appear within a couple of laps. When drivers were forced to brake, they would sometimes overcompensate slightly, forcing the drivers behind them to overcompensate as well. A “stop-and-go wave” developed: A car arriving at the back of the jam was forced to slow down, and one reaching the front could accelerate again to normal speed, producing a living wave that crept backward around the track.
Interestingly, Sugiyama found that this phenomenon arises predictably in the real world. Measurements on various motorways in Germany and Japan have shown that free-flowing traffic becomes congested when the density of cars reaches 40 vehicles per mile. Beyond that point, the flow becomes unstable and stop-and-go waves appear. Because it’s founded in human reaction times, this happens regardless of the country or the speed limit. And as long as the total number of cars on the motorway doesn’t change, the wave rolls backward at a predictable 12 mph.
“Understanding things like traffic jams from a physical point of view is a totally new, emerging field of physics,” Sugiyama told Gavin Pretor-Pinney for The Wavewatcher’s Companion. “While the phenomenon of a jam is so familiar to us, it is still too difficult to truly understand why it happens.”
