Further to my March post “A Lucrative Loop,” reader Snehal Shekatkar of S.P. Pune University notes a similar discovery of iterates leading to strange cycles among natural numbers.

Here is a simple example. Take a natural number and factorize it (12 = 2 * 2 * 3), then add all the prime factors (2 + 2 + 3 = 7). If the answer is prime, add 1 and then factorize again (7 + 1 = 8 = 2 * 2 * 2) and repeat (2 + 2 + 2 = 6). Eventually ALL the natural numbers greater than 4 eventually get trapped in cycle (5 -> 6 -> 5). Instead of adding 1 after hitting a prime, if you add some other natural number A, then depending upon A, numbers may get trapped in a different cycle. For example, for A = 19, they eventually get trapped in cycle (5 -> 24 -> 9 -> 6 -> 5).

For some values of A, several cycles exist. For example, when A = 3, some numbers get trapped in cycle (5 -> 8 -> 6 -> 5) while others get trapped in the cycle (7 -> 10 -> 7).

(Made with Tian An Wong of Michigan University.) (Thanks, Snehal.)

In the College Mathematics Journal in 2001, Rick Mabry published this “proof without words” that

He gives a charming explanation here.

(Rick Mabry, “Mathematics Without Words,” College Mathematics Journal 32:1 [January 2001], 19.)

It is a traditional axiom of medicine that health is the absence of disease. What is a disease? Anything that is inconsistent with health. If the axiom has any content, a better answer can be given. The most fundamental problem in the philosophy of medicine is, I think, to break the circle with a substantive analysis of either health or disease.

— Christopher Boorse, “Health as a Theoretical Concept,” Philosophy of Science 44:4 (1977), 542-573

Statistics textbooks sometimes ask: Suppose you’re driving on the highway and adjust your speed so that the number of cars you pass is equal to the number that pass you. Is your speed the median or the mean speed of the cars on the highway?

The expected answer is that it’s the median speed, since the number of cars traveling more slowly than you is equal to the number traveling faster. But California State University mathematician Larry Clevenson and his colleagues wrote in 2001, “This certainly is true of the cars that you see, but that isn’t what the problem asks, and it isn’t the correct answer.”

Surprisingly, they found that the correct answer is the mean. “If you adjust your speed so that as many cars pass you as you pass, then your speed is the mean speed of all the other cars on the highway.” Details at the link below.

(Larry Clevenson et al., “The Average Speed on the Highway,” College Mathematics Journal 32:3 [2001], 169-171.)

In The Art of Computer Programming, Donald Knuth notes an interesting pattern in the common units of liquid measure in 13th-century England:

2 gills = 1 chopin
2 chopins = 1 pint
2 pints = 1 quart
2 quarts = 1 pottle
2 pottles = 1 gallon
2 gallons = 1 peck
2 pecks = 1 demibushel
2 demibushels = 1 bushel or firkin
2 firkins = 1 kilderkin
2 kilderkins = 1 barrel
2 barrels = 1 hogshead
2 hogsheads = 1 pipe
2 pipes = 1 tun

“Quantities of liquid expressed in gallons, pottles, quarts, pints, etc. were essentially written in binary notation,” Knuth writes. “Perhaps the true inventors of binary arithmetic were British wine merchants!”

(Thanks, Colin.)

What most clinicians do when they receive a laboratory report is, of course, to look up the normal range for the tests in question. … Traditionally, a normal range is calculated in such a way that it includes 95% of the results found in a group of normal or healthy persons, and, consequently, there is a 5% risk that a healthy person will present with an abnormal laboratory result. Then, imagine that you do ten tests on a normal person. In that case the risk that at least one of these tests is abnormal is (1 – 0.95¹⁰) which amounts to 0.40 or 40%. If you do twenty-five tests (and that is not uusual in clinical practice), this chance is 72%! As Edmond A. Murphy puts it so aptly, ‘Therefore, a normal person is anyone who has not been sufficiently investigated.’

— Henrik R. Wulff, Stig Andur Pedersen, and Raben Rosenberg, Philosophy of Medicine: An Introduction, 1990, citing Murphy’s The Logic of Medicine, 1976