Each of the first 18 multiples of 526315789473684210 contains all 10 digits:

526315789473684210 x  1 =  526315789473684210
526315789473684210 x  2 = 1052631578947368420
526315789473684210 x  3 = 1578947368421052630
526315789473684210 x  4 = 2105263157894736840
526315789473684210 x  5 = 2631578947368421050
526315789473684210 x  6 = 3157894736842105260
526315789473684210 x  7 = 3684210526315789470
526315789473684210 x  8 = 4210526315789473680
526315789473684210 x  9 = 4736842105263157890
526315789473684210 x 10 = 5263157894736842100
526315789473684210 x 11 = 5789473684210526310
526315789473684210 x 12 = 6315789473684210520
526315789473684210 x 13 = 6842105263157894730
526315789473684210 x 14 = 7368421052631578940
526315789473684210 x 15 = 7894736842105263150
526315789473684210 x 16 = 8421052631578947360
526315789473684210 x 17 = 8947368421052631570
526315789473684210 x 18 = 9473684210526315780

The 19th multiple, alas, is 9999999999999999990.

Is every integer the sum of four perfect cubes?

Interestingly, no one knows. It’s conjectured that the answer is yes, but no one yet has been able to prove this.

“Good tests kill flawed theories; we remain alive to guess again.” — Karl Popper

“There are two possible outcomes: if the result confirms the hypothesis, then you’ve made a measurement. If the result is contrary to the hypothesis, then you’ve made a discovery.” — Enrico Fermi

“This particular thesis was addressed to me a quarter of a century ago by John Campbell, who … told me that all theories are proven wrong in time. … My answer to him was, ‘John, when people thought the Earth was flat, they were wrong. When people thought the Earth was [perfectly] spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.’ The basic trouble, you see, is that people think that ‘right’ and ‘wrong’ are absolute; that everything that isn’t perfectly and completely right is totally and equally wrong. However, I don’t think that’s so. It seems to me that right and wrong are fuzzy concepts.” — Isaac Asimov, The Relativity of Wrong, 1988

By Wikimedia user Cmglee, a visual proof that 3³ + 4³ + 5³ = 6³:

This value, 216, is sometimes called Plato’s number because it seems to correspond to this enigmatic passage in the Republic:

Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births.

Unfortunately, Plato’s meaning is far from certain. Other values that have been proposed (with various rationales) include 1,728, 3,600, 5,040, 8,128, 17,500, 760,000, and 12,960,000. Cicero politely described the question as “obscure”; it remains open.

In the December 2024 issue of Recreational Mathematics Magazine, Illinois State University mathematician Sunil Chebolu demonstrates that the digits of π can be generated using the positions of stars.

The probability P(x, y) that two randomly chosen positive integers, x and y, are relatively prime is 6/π². The stars in the night sky appear to be distributed randomly on the celestial sphere, and this suggests a novel experiment: Use the pairwise angular distances between stars to simulate a large random sample of numbers, then pick random pairs and determine the proportion of relatively prime pairs. Equating that with the theoretical probability above should generate an approximation of π.

In Nature in 1995, University of Aston mathematician Robert Matthews used the positions of 100 stars to compute π to within 0.5% of its correct value (3.12772). By expanding the sample to 1,000 stars, Chebolu got an average estimate of 3.141540567, an error of less than 0.002%.

“The above method can also be viewed as a statistical hypothesis test,” Chebolu notes. “Since the error in the estimate for π obtained using this method is pretty low, one may also argue that it supports the hypothesis that the bright stars are randomly distributed on the celestial sphere.”

(Sunil K. Chebolu, “Baking Star π,” Recreational Mathematics Magazine 11:19 [December 2024], 67-70.)

From Lee Sallows:

Thanks, Lee!

In 1940, George Gamow published Mr Tompkins in Wonderland, in which a bank clerk attends a lecture on relativity and then finds that Einstein’s principles have become apparent in the ordinary street scenes before him:

A single cyclist was coming slowly down the street and, as he approached, Mr Tompkins’s eyes opened wide with astonishment. For the bicycle and the young man on it were unbelievably shortened in the direction of the motion, as if seen through a cylindrical lens. The clock on the tower struck five, and the cyclist, evidently in a hurry, stepped harder on the pedals. Mr Tompkins did not notice that he gained much in speed, but, as the result of his effort, he shortened still more and went down the street looking exactly like a picture cut out of cardboard.

In later adventures Tompkins explores cosmology and quantum physics, again in an exaggerated world in which extreme effects become observable. Gamow called this a “fantastic but scientifically correct dream.”