A Banner Year

Next year’s date, 2025, is remarkable:

  • It’s a square (452), the sum of two squares (272 + 362), the product of two squares (92 × 52), and the sum of three squares (402 + 202 + 52).
  • It’s the sum of the cubes of the first nine positive integers (13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93).
  • Equivalently, it’s the square of the sum of those integers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2.
  • It’s the second in a trio of square numbers in arithmetic progression (81, 2025, 3969).
  • It’s one of only three four-digit numbers whose halves can be split, summed, and squared to produce the original number: (20 + 25)2 = 2025.
  • It’s the smallest square starting with 20 and the smallest number with exactly 15 odd factors (1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025).
  • It’s the sum of the entries in a 9×9 multiplication table.
  • July 24, 7/24/25, will be a “Pythagorean day,” because 72 + 242 = 252.

When asked his age, Augustus De Morgan used to say, “I was x years of age in the year x2.” (He was 43 in 1849.) People born in 1980 will be able to make the same cryptic response starting next year.

(Thanks to readers Chris Smith, Sam Householder, and Jim Howell.)

| Science & Math

Loaded

https://commons.wikimedia.org/wiki/File:Wreck_of_the_Central_America.jpg

“Lately in a wreck of a Californian ship, one of the passengers fastened a belt about him with two hundred pounds of gold in it, with which he was found afterwards at the bottom. Now, as he was sinking — had he the gold? or had the gold him?”

— John Ruskin, Unto This Last, 1860

| Society

Who You Know

https://commons.wikimedia.org/wiki/File:Cezanne_Ambroise_Vollard.jpg

Art dealer Ambroise Vollard was acquainted with many of the foremost artists of the early 20th century, and as a result he appears often in their work. Above are portraits by Cézanne, Renoir, and Bonnard, and he sat also for Rouault, Forain, Vallotton, Bernard, and Picasso.

Picasso wrote, “The most beautiful woman who ever lived never had her portrait painted, drawn, or engraved any oftener than Vollard.”

| Art

Checkless Chess

dawson checkless chess puzzle

In checkless chess it’s illegal to give check without giving checkmate. This changes the whole complexion of the game. T.R. Dawson published this example in Die Welt in 1951. White is to mate in two moves.

The answer, 1. f6, threatens 2. Qf5, which strangely is mate because the black king can’t move off the long light diagonal, since that would discover check by the black bishop without giving checkmate. Black can try to prevent this finish by playing 1. … Qc8, guarding f5 and thus making 1. Qf5 itself illegal. But this permits 2. Nxd6, which now is mate because the black king has nowhere to run on the long diagonal and 2. … cxd6 is illegal because this would give a mateless check.

Similarly, if Black tries to stop 1. Qf5# with 1. … Qxf6, then 2. Nc5 is mate because Black can’t capture the knight — again, this would expose White’s king to an illegal check. And if Black tries to answer White’s first move with 1. … d5, to block the long light diagonal and free his king to flee elsewhere, then White can play 2. Qe5#, an ordinary (and legal) mate.

| Puzzles

Lost Lessons

THOUGH I can never pay enough to your Grandfather’s Memory, for his tender care of my Education, yet I must observe in it this Mistake; That by keeping me at home, where I was one of my young Masters, I lost the advantage of my most docile time. For not undergoing the same Discipline, I must needs come short of their experience, that are bred up in Free Schools; who, by plotting to rob an Orchard, &c. run through all the Subtilties required in taking of a Town; being made, by use, familiar to Secresie, and Compliance with Opportunity; Qualities never after to be attained at cheaper rates than the hazard of all: whereas these see the danger of trusting others, and the Rocks they fall upon, by a too obstinate adhering to their own imprudent resolutions; and all this under no higher penalty than a Whipping: And ’tis possible this indulgence of my Father might be the cause I afforded him so poor a Return for all his Cost.

— Francis Osborne, Advice to a Son, 1656

| Society

Opaque Sets

https://commons.wikimedia.org/wiki/File:Unit_Square_Opaque_Forest_Solutions.svg

A creature living in the plane can’t see through a unit square — the square’s four line segments block its line of sight from any angle. Is there a way to achieve the same result using fewer building materials? Removing one of the square’s sides does the job — this requires only 3 units of line segment and still prevents anyone from seeing across the square’s area. The arrangement at lower left does better still, requiring only about 2.732 units. And the one at lower right requires only about 2.639 units.

Is that the shortest possible opaque set for the square? Possibly — but no one has been able to prove it.

| Science & Math

Brute Force

NBC’s Today Show had a surprising guest in 1959: G. Clifford Prout Jr., president of the Society for Indecency to Naked Animals, an activist group that hoped to clothe “any dog, cat, horse or cow that stands higher than 4 inches or longer than 6 inches.” Prout’s bizarre cause earned him regular media attention, and the society’s newsletter published this anthem:

High on the wings of SINA / we fight for the future now;
Let’s clothe every pet and animal / whether dog, cat, horse or cow!
G. Clifford Prout, our President / he works for you and me,
So clothe all your pets and join the march / for worldwide Decency!
S.I.N.A., that’s our call / all for one and one for all.
Hoist our flag for all to see / waving for Morality.
Onward we strive together / stronger in every way,
All mankind and his animal friends / for SINA, S-I-N-A!

Of course it didn’t last. When Walter Cronkite interviewed Prout in 1962, one of his staff realized he was really actor and screenwriter Buck Henry. The hoax had been masterminded by serial prankster Alan Abel. “When Cronkite eventually found out that he’d been conned, and I was the guy behind it, he called me up,” Abel recalled. “I’d never heard him that angry on TV — not about Hitler, Saddam Hussein, or Fidel Castro. He was furious with me.”

| Hoaxes

One Last Christmas Quiz

The Royal Statistical Society’s 2024 Christmas Quiz consists of two parts, a warmup section and the quiz itself, each consisting of 12 questions. Solvers (and teams) have until January 5 to submit their solutions, and the top three entrants will win a donation of £150, £120, or £90 to the charity or good cause of their choice. Updates and corrections will be posted on the competition website, and solutions will be published in January, together with the names of the winners.

| Puzzles