An industrious ant sets out to travel the length of a rubber rope 1 kilometer long. Just as it begins, the rope starts to stretch uniformly at a constant rate of 1 kilometer per second, so that after 1 second the rope is 2 kilometers long, after 2 seconds it’s 3 kilometers long, and so on. The ant advances heroically at 1 centimeter per second relative to the rubber it’s crawling on. Will it ever reach the end of the rope?

This seems hopeless, but the answer is yes. Because the rope’s stretch carries the ant forward, it never loses ground, and because its proportional speed is inversely proportional to the length of the rope, the distance it can travel is unbounded. But it will take a stupendously long time — 8.9 × 10⁴³⁴²¹ years — to reach the far end.

By Vincent Pantaloni, a wordless proof that the sum of the first n odd integers is n².

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.)

By Basile Morin, a very thorough demonstration of the commutativity of addition!

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.

A visual proof that a² – b² = (a + b)(a – b).

Sophie Germain wrote, “It has been said that algebra is but written geometry and geometry is but diagrammatic algebra.”

For any positive integer n, there exists a circle that passes through exactly n lattice points in the Euclidean plane.

This was first proven by Polish mathematician Andrzej Schinzel in 1958. It’s known as Schinzel’s theorem.

From Lee Sallows:

(Thanks, Lee.)