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.”

A flock of starlings masses near sunset over Gretna Green in Scotland, preparatory to roosting after a day’s foraging. The flock’s shape has a mesmerisingly fluid quality, flowing, stretching, rippling, and merging with itself. Similarly massive flocks form over Rome and over the marshlands of western Denmark, where more than a million migrating starlings form an enormous display known as the “black sun.”

What rules produce this behavior? In the 1970s scientists thought that the birds might be following an electrostatic field produced by the leader. Earler, in the 1930s, one paper even suggested that they use thought transference.

But in 1986 computer graphics expert Craig Reynolds found that he could create a lifelike virtual flock (below) using a surprisingly simple set of rules: direct each bird to avoid crowding nearby flockmates, steer toward the average heading of nearby flockmates, and move toward the center of mass of nearby flockmates.

Studies with real birds seem to bear this out: Under rules like these a flock can react sensitively to a change in direction by any of its members, permitting the whole group to respond efficiently as one organism. “News of a predator’s approach can be communicated rapidly through the flock by whichever of the hundreds of birds on the outside notice it first,” writes Gavin Pretor-Pinney in The Wavewatcher’s Companion. “When under attack by a peregrine falcon, for instance, starling flocks will contract into a ball and then peel away in a ribbon to distract and confuse the predator.”

Utica College mathematician Hossein Behforooz devised this “permutation-free” magic square in 2007:

Each row, column, and long diagonal totals 2775, and this remains true if the digits within all 25 cells are permuted in the same way — for example, if we exchange the first two digits of each number, changing 231 to 321, etc., the square retains its magic sum of 2775. Further:

231 + 659 + 973 + 344 + 568 = 2775
979 + 234 + 653 + 341 + 568 = 2775
231 + 343 + 568 + 654 + 979 = 2775
564 + 979 + 233 + 348 + 651 = 2775
231 + 654 + 563 + 978 + 349 = 2775
231 + 348 + 654 + 979 + 563 = 2775

And these combinations of cells maintain their magic totals when their contents are permuted in the same way.

(Hossein Behforooz, “Mirror Magic Squares From Latin Squares,” Mathematical Gazette, July 2007.)

1!, 22!, 23!, and 24! contain 1, 22, 23, and 24 digits, respectively.

266!, 267!, and 268! contain 2 × 266, 2 × 267, and 2 × 268 digits, respectively.

2,712! and 2,713! contain 3 × 2,712 and 3 × 2,713 digits, respectively.

27,175! and 27,176! contain 4 × 27,175 and 4 × 27,176 digits, respectively.

271,819!, 271,820!, and 271,821! contain 5 × 271,819, 5 × 271,820, and 5 × 271,821 digits, respectively.

2,718,272! and 2,718,273! contain 6 × 2,718,272, and 6 × 2,718,273 digits, respectively.

27,182,807! and 27,182,808! contain 7 × 27,182,807, and 7 × 27,182,808 digits, respectively.

271,828,170! 271,828,171!, and 271,828,172! contain 8 × 271,828,170, 8 × 271,828,171, and 8 × 271,828,172 digits, respectively.

2,718,281,815! and 2,718,281,816! contain 9 × 2,718,281,815, and 9 × 2,718,281,816 digits, respectively.

27,182,818,270! and 27,182,818,271! contain 10 × 27,182,818,270 and 10 × 27,182,818,271 digits, respectively.

271,828,182,830! and 271,828,182,831! contain 11 × 271,828,182,830, and 11 × 271,828,182,831 digits, respectively.

The pattern continues at least this far:

271,828,182,845,904,523,536,028,747,135,266,249,775,724,655!, 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656!, and 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657! contain 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,655, 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656, and 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657 digits, respectively.

(By Robert G. Wilson. More at the Online Encyclopedia of Integer Sequences. Thanks, David.)