“Standards for inconsequential trivia,” offered by Philip A. Simpson in the NBS Standard, Jan. 1, 1970:

10^-15 bismols = 1 femto-bismol
10^-12 boos = 1 picoboo
1 boo² = 1 boo-boo
10^-18 boys = 1 attoboy
10¹² bulls = 1 terabull
10¹ cards = 1 decacards
10^-9 goats = 1 nanogoat
2 gorics = 1 paregoric
10^-3 ink machines = 1 millink machine
10⁹ los = 1 gigalos
10^-1 mate = 1 decimate
10^-2 mentals = 1 centimental
10^-2 pedes = 1 centipede
10⁶ phones = 1 megaphone
10^-6 phones = 1 microphone
10¹² pins = 1 terapin

A puzzle from the Middle Ages, adapted by A.N. Prior:

Four people, on a certain occasion, say one thing each.

A says that 1 + 1 = 2.

B says that 2 + 2 = 4.

C says that 2 + 2 = 5.

Can D now say that exactly as many truths as falsehoods are uttered on this occasion?

“If what D says is true,” Prior writes, “that makes 3 truths to 1 falsehood, so that it is false; while if it is false, that makes two truths and two falsehoods, and it is true.”

Choose a prime number p, draw a p×p array, and fill it with integers like so:

Now: Can we always find p cells that contain prime numbers such that no two occupy the same row or column? (This is somewhat like arranging rooks on a chessboard so that every rank and file is occupied but no rook attacks another.)

The example above shows one solution for p=11. Does a solution exist for every prime number? No one knows.

What is the smallest integer that’s not named on this blog?

Suppose that the smallest integer that’s not named (explicitly or by reference) elsewhere on the blog is 257. But now the phrase above refers to that number. And that instantly means that it doesn’t refer to 257, but presumably to 258.

But if it refers to 258 then actually it refers to 257 again. “If it ‘names’ 257 it doesn’t, so it doesn’t,” writes J.L. Mackie, “but if it doesn’t, then it does, so it does.”

(Adduced by Max Black of Cornell.)

Mr. X, who thinks Mr. Y a complete idiot, walks along a corridor with Mr. Y just before 6 p.m. on a certain evening, and they separate into two adjacent rooms. Mr. X thinks that Mr. Y has gone into Room 7 and himself into Room 8, but owing to some piece of absent-mindedness Mr. Y has in fact entered Room 6 and Mr. X Room 7. Alone in Room 7 just before 6, Mr. X thinks of Mr. Y in Room 7 and of Mr. Y‘s idiocy, and at precisely 6 o’clock reflects that nothing that is thought by anyone in Room 7 at 6 o’clock is actually the case. But it has been rigorously proved, using only the most general and certain principles of logic, that under the circumstances supposed Mr. X just cannot be thinking anything of the sort.

— A.N. Prior, “On a Family of Paradoxes,” Notre Dame Journal of Formal Logic, 1961

Dick and Jane are playing a game. Each holds up one or two fingers. If the total number of fingers is odd, then Dick pays Jane that number of dollars. If it’s even, then Jane pays Dick:

At first blush this looks fair, but in fact it’s distinctly favorable for Jane. Let p be the proportion of times that Jane holds up one finger. Her average winnings when Dick holds up one finger are -2p + 3(1 – p), and her average winnings when he holds up two fingers are 3p – 4(1 – p). If she sets those equal to one another she gets p = 7/12. This means that if she raises one finger with probability 7/12, then on average she’ll win -2(7/12) + 3(5/12) = 1/12 dollar every round, no matter what Dick does. Dick’s best strategy is also to raise one finger 7/12 of the time, but the best this can do is to restrict his loss to 1/12 dollar on average. It’s not a fair game.

Why do mathematicians confuse Halloween with Christmas?

Because 31 Oct = 25 Dec.

(Thanks, Jan.)

In 1976, Queen Elizabeth College chemist Leslie Hough asked graduate researcher Shashikant Phadnis to test a certain chlorinated sugar compound. Phadnis, whose English was imperfect, “thought I needed to taste it! … So I took a small quantity of the sample on a spatula and tasted it with the tip of my tongue.”

To his surprise, Phadnis found the compound intensely and pleasantly sweet. When he reported his discovery to Hough, “‘Are you crazy or what?’ he asked me. ‘How could you taste compounds without knowing anything about their toxicity?'” After some further cautious tasting, Hough dubbed the compound Serendipitose. It became the artificial sweetener Splenda.

“Later on, Les even had a cup of coffee sweetened with a few particles of Serendipitose. When I reminded him that it could be toxic (as it contained a high proportion of chlorine), he simply said, ‘Oh, forget it, we’ll survive!'”