This sentence is true.

Is that sentence true or false?

Draw a chord AB through a point P inside a circle, and the product PA × PB is constant — it has the same value for every chord through P.

“Again we have perfect democracy,” write Eli Maor in The Pythagorean Theorem. “Every chord has the same status in relation to P as any other.”

A stationmaster waves his flag, and a train begins to move. There is a last moment of rest, and a first moment of motion.

But this is a problem. If time is infinitely divisible, then there is a moment between these two moments. Is it a moment of rest or of motion?

(From Robin Le Poidevin, Travels in Four Dimensions, 2003.)

From Royal V. Heath in Scripta Mathematica, March 1955:

0264 + 4125 + 5610 = 0165 + 5214 + 4620

… remains valid if you split each term with a multiplication sign:

02 × 64 + 41 × 25 + 56 × 10 = 01 × 65 + 52 × 14 + 46 × 20

… or an addition sign:

02 + 64 + 41 + 25 + 56 + 10 = 01 + 65 + 52 + 14 + 46 + 20

Remarkably, everything above holds true if you square each term.

“If anything is possible, then it is possible to prove that something is impossible. And if it is possible to prove that something is impossible, then necessarily, something is impossible.”

— Roy Sorensen, Vagueness and Contradiction, 2001

Retired Pittsburgh math teacher Walter W. Horner devised this doubly magic square in 1955:

Each row, column, and long diagonal produces both a sum of 840 and a product of 2058068231856000.

And Rodolfo Marcelo Kurchan of Buenos Aires discovered this remarkable square in 1991:

Each number contains all 10 digits — and so does the magic sum, 4129607358.

In 1998, California physician Donald L. Unger wrote to the editors of Arthritis & Rheumatism to report a “50-year controlled study by one participant.” His mother had told him that cracking his knuckles would lead to arthritis, so for 50 years the science-minded Unger had cracked the knuckles of his left hand at least twice a day, more than 36,500 times in all, and left the right uncracked as a control. After 50 years he found no arthritis in either hand and no differences between the two hands.

“This result calls into question whether other parental beliefs, e.g., the importance of eating spinach, are also flawed,” Unger wrote. “Further investigation is likely warranted.”

The editors invited a response from Robert L. Swezey, who had published an earlier investigation in the Western Journal of Medicine. Swezey said that his own study had been inspired when his 12-year-old son’s grandmother had warned him that cracking his knuckles would cause arthritis. “It is now 22 years later and he continues to enjoy frequent KC without manifestations or evidence of arthritis.”

With motherly advice thrown into doubt, Swezey wondered whether knuckle cracking might even prevent osteoarthritis. “The possible utilization of KC by managed care providers as an economic, noninvasive, home preventative treatment for arthritis of the hands should be given further consideration,” he concluded. “A clear distinction between hand wringing related to managed care procedures and therapeutic KC will have to be made.”

(Does knuckle cracking lead to arthritis of the fingers? Unger DL. Arthritis Rheum. 1998 May;41(5):949-50.) (Thanks, Bob.)