To find the least common multiple and the greatest common divisor of two natural numbers, construct a billiard table whose side lengths correspond to the two numbers (here, 15 and 40). Set a ball in one corner, fire it out at a 45-degree angle, and let it bounce around the table until it stops in a corner.

Now the least common multiple is the total number of unit squares traversed by the ball (here, 120).

And the greatest common divisor is the number of unit squares traversed by the ball before it reaches the first intersection (here, 5).

More details here.

A team of mathematicians in Canada and Japan discovered this remarkable polynomial in 1976 — let its 26 variables a, b, c, … z range over the non-negative integers and it will generate all prime numbers:

The snag is that it will sometimes produce negative numbers, which must be ignored. But every positive result will be prime, and every prime can be generated by some set of 26 non-negative integers.

(James P. Jones et al., “Diophantine Representation of the Set of Prime Numbers,” American Mathematical Monthly 83:6 [1976], 449-464.)

When a Western scrub jay discovers the body of a dead jay, it summons other birds to screech over the body for up to half an hour. It’s not clear why they do this — the birds are territorial and not normally social. Possibly it’s a way to share news of danger, concentrate attention to find a predator, or teach young about dangers in the environment.

The gatherings are sometimes called funerals, though we don’t know enough to understand the reasons behind them. But UC Davis student Teresa Iglesias said, “I think there’s a huge possibility that there is much more to learn about the social and emotional lives of birds.”

Three lines can be arranged to make one triangle. Four lines can make two, and five lines can make five.

But, generally, no one can say how many non-overlapping triangles can be formed by an arrangement of k lines — the problem remains unsolved.

In the 1850s, when a plague of spirits began to rotate tables at London séances, Michael Faraday devised a clever way to investigate: Two boards were lain one atop the other, with an upright haystalk inserted through the pair. The experimenters laid their hands on this. The apparatus gave a way to see “whether the table moved the hand, or the hand moved the table”: If the medium willed the table to move to the left, and it did so on its own, the haystalk would be seen to lean in one direction … but if the experimenters, even unconsciously, themselves pressed the table to turn, it would lean in the other.

Faraday wrote, “As soon as the index is placed before the most earnest, and they perceive — as in my presence they have always done — that it tells truly whether they are pressing downwards only or obliquely, then all effects of table-turning cease, even though the parties persevere, earnestly desiring motion, till they become weary and worn out. No prompting or checking of the hands is needed — the power is gone; and this only because the parties are made conscious of what they are really doing mechanically, and so are unable unwittingly to deceive themselves.”

(Michael Faraday, “On Table-Turning,” Times, June 30, 1853.)

Richard Feynman tangled regularly with military censors at Los Alamos. Playing one day with a computing machine, he discovered a pleasing little pattern:

1/243 = 0.004115226337448559670781893004115226337448559670781893004115226…

“It’s quite cute, and then it goes a little cockeyed when you’re carrying; confusion occurs for only about three numbers, and then you can see how the 10 10 13 is really equivalent to 114 again, or 115 again, and it keeps on going, and repeats itself nicely after a couple of cycles. I thought it was kind of amusing.”

Well, I put that in the mail, and it comes back to me. It doesn’t go through, and there’s a little note: ‘Look at Paragraph 17B.’ I look at Paragraph 17B. It says, ‘Letters are to be written only in English, Russian, Spanish, Portuguese, Latin, German, and so forth. Permission to use any other language must be obtained in writing.’ And then it said, ‘No codes.’

So I wrote back to the censor a little note included in my letter which said that I feel that of course this cannot be a code, because if you actually do divide 1 by 243 you do, in fact, get all that, and therefore there’s no more information in the number .004115226337… than there is in the number 243 — which is hardly any information at all. And so forth.

“I therefore asked for permission to use Arabic numerals in my letters. So, I got that through all right.”

(From his reminiscences.)