3² + 4² = 5²

21² + 22² + 23² + 24² = 25² + 26² + 27²

The first LSD trip took place on April 19, 1943, when Swiss chemist Albert Hofmann ingested 250 micrograms and tried to go home:

I had to struggle to speak intelligibly. I asked my laboratory assistant, who was informed of the self-experiment, to escort me home. We went by bicycle, no automobile being available because of wartime restrictions on their use. On the way home, my condition began to assume threatening forms. Everything in my field of vision wavered and was distorted as if seen in a curved mirror. I also had the sensation of being unable to move from the spot. Nevertheless, my assistant later told me we had traveled very rapidly.

It’s remembered as “Bicycle Day.”

135 = 1 + 3² + 5³
175 = 1 + 7² + 5³
518 = 5 + 1² + 8³
598 = 5 + 9² + 8³

Once upon a time, there lived a rich farmer who had 30 children, 15 by his first wife who was dead, and 15 by his second wife. The latter woman was eager that her eldest son should inherit the property. Accordingly one day she said to him, “Dear Husband, you are getting old. We ought to settle who shall be your heir. Let us arrange our 30 children in circle, and counting from one of them, remove every tenth child until there remains but one, who shall succeed to your estate.”

The proposal seemed reasonable. As the process of selection went on, the farmer grew more and more astonished as he noticed that the first 14 to disappear were children by his first wife, and he observed that the next to go would be the last remaining member of that family. So he suggested that they should see what would happen if they began to count backwards from this lad. She, forced to make an immediate decision, and reflecting that the odds were now 15 to 1 in favour of her family, readily assented. Who became the heir?

— W.W. Rouse Ball, Mathematical Recreations & Essays, 1892

Clearly there are integers so huge they can’t be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It’s “the smallest integer that can’t be described in fewer than 22 syllables.”

That phrase has 21 syllables.

8⁴ + 2⁴ + 0⁴ + 8⁴ = 8208

Remarkably, you can estimate π by dropping needles onto a flat surface. If the surface is ruled with lines that are separated by the length of a needle, then:

drops is the number of needles dropped. hits is the number of needles that touch a line. The method combines probability with trigonometry; a needle’s chance of touching a line is related to the angle at which it comes to rest. It was discovered by the French naturalist Georges-Louis Leclerc in 1777.

1³ + 3³ + 6³ = 244
2³ + 4³ + 4³ = 136