The Parallel Climbers Problem

Two climbers stand on opposite sides of a two-dimensional mountain range. Is it always possible for both of them to make their way through the mountains, remaining constantly at the same altitude as one another, and arrive together at the top of the tallest peak?

The example shown here looks relatively straightforward, but that doesn’t prove that it’s possible in every mountain range. Each time either climber reaches a peak or a valley, she must decide whether to go forward or back, and in a complex range it’s not always clear whether there’s a series of choices that will lead both climbers to the goal.

As it turns out, though, the answer is yes. Alan Tucker gave an accessible explanation, using graph theory, in the November 1995 issue of Math Horizons.

All’s Well That Ends Well

In 2003 Carl Libis of Assumption College in Worcester, Mass., received this solution from a student in an algebra course:

\begin{aligned} \frac{1}{x+1} + \frac{1}{x-2} &= \frac{x+3}{x^2-x-2} \\ \frac{x+1}{1} + \frac{x-2}{1} &= \frac{x^2-x-2}{x+3} \\ x+1+x-2 &= \frac{x^2 - \frac{x}{x}-2}{3} \\ 2x-1 &= \frac{x^2 - 1 - 2}{3} \\ 3(2x-1) &= x^2 - 3 \\ 6x &= x^2 - 2 \\ \frac{6x}{2x} &= \frac{x^2 - 2}{2x} \\ 3 &= \frac{x^2}{x} - \frac{2}{2} \\ 3 &= x-1 \\ 4 &= x \end{aligned}

(Via Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 34:1 [January 2003], 50-54.)

A Late Carroll Game

Two years before he died, Lewis Carroll came up with a remarkable number-guessing game in which he’d send a volunteer through a long series of arithmetic manipulations, even allowing her to make some private decisions as to how to proceed. Along the way Carroll asked only three questions:

“Is the result odd or even?”

“Is the result odd or even?”

“How often does it go?”

And yet he would always be able to find the original number quickly. Here’s the procedure:

Think of a number (a positive integer).

Multiply by 3.

• If the result is odd, then add either 5 or 9 (whichever you like), then divide by 2, then add 1.
• If the result is even, then subtract either 2 or 6 (whichever you like), then divide by 2, then add 29 or 33 or 37 (whichever you like).

Multiply by 3.

• If the result is odd, then add either 5 or 9 (whichever you like), then divide by 2, then add 1.
• If the result is even, then subtract either 2 or 6 (whichever you like), then divide by 2, then add 29 or 33 or 37 (whichever you like).

Add 19 to original number you chose and append any digit, 0-9, to this number.

Divide by 7 and drop any remainder.

Divide by 7 again and drop any remainder, and tell me what result you get. (“How often does it go?”)

Here’s how to derive the number that was chosen originally:

Multiply the final answer by 4 and subtract 15. If the first answer was “even,” subtract 3 more, and if the second answer was “even,” subtract 2 more.

(Note: Carroll’s version contained an unfortunate flaw; the improvement given here was devised by Richard F. McCoart and includes a faster way to get the answer. See the article cited below. Carroll’s original appears in Morton N. Cohen’s Lewis Carroll, A Biography, 1995. Note too that the two “parity checks” above are identical, so the whole setup is easy to memorize and less bewilderingly complex than it’s intended to appear.)

(Richard F. McCoart, “Lewis Carroll’s Amazing Number-Guessing Game,” College Mathematics Journal 33:5 [November 2002], 378-383.)

Impression

This is remarkable — in 2014 ophthalmological neurologist Frederick Lepore showed this image to 100 migraine patients who experience a visual aura. (Click to enlarge.) 48 recognized it instantly.

“People are astonished,” Lepore told National Geographic. “They say, ‘Where did you get that?'”

He got it from English physician Hubert Airy, who had drawn his own aura in 1870, before the phenomenon was even understood.

(Via MetaFilter.)

Neat

The radius of a circle inscribed in a 3-4-5 triangle is 1.

(In fact, the inradius of any Pythagorean triangle is an integer.)

Theory and Practice

All new ventures have their detractors, and James had his full share with the Cavendish project. One diminishing but still powerful school of critics held that, while experiments were necessary in research, they brought no benefit to teaching. A typical member was Isaac Todhunter, the celebrated mathematical tutor, who argued that the only evidence a student needed of a scientific truth was the word of his teacher, who was ‘probably a clergyman of mature knowledge, recognised ability, and blameless character’. One afternoon James bumped into Todhunter on King’s Parade and invited him to pop into the Cavendish to see a demonstration of conical refraction. Horrified, Todhunter replied: ‘No, I have been teaching it all my life and don’t want my ideas upset by seeing it now!’

— Basil Mahon, The Life of James Clerk Maxwell, 2004

The Motorist’s Complaint

There are 8 ways to fold a 2 × 2 map along its creases:

But no one knows how to calculate the number of ways that a larger map might be folded. Amazingly, the number of ways to fold an n × n map has been worked out only as high as 7 × 7:

1, 8, 1368, 300608, 186086600, 123912532224, 129950723279272

Via Fermat’s Library.

The Cute Response

“Humans feel affection for animals with juvenile features,” noted Konrad Lorenz. “Large eyes, bulging craniums, retreating chins. Small-eyed, long-snouted animals do not elicit the same response.”

This induces people to care for small, cuddly animals. “And this has led some experts to argue that the entire phenomenon of pet-keeping is nothing more nor less than an elaborate case of social parasitism,” writes zoologist James Serpell. “Needless to say, this idea has done little to promote a positive view of pets or their owners. Rather, it creates the impression that pet-owners are the victims of some kind of bizarre affliction, and that dogs, cats and budgerigars are little different from body lice, fleas or tapeworms or, indeed, any other sort of parasitic organism.”

(From James Serpell, In the Company of Animals, 1986.)

Good Faith

“Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.” — Gabriel Lippmann

(Thanks, Tom.)

Nosedive

I just noticed this last night in Joseph Wood Krutch’s Treasury of Bird Lore — in 1832 ornithologist Alexander Wilson encountered a flock of passenger pigeons near Frankfort, Kentucky, that he estimated at 2,230,270,000 birds. If each bird ate only a pint of beech nuts in the course of a day, the flock would consume nearly 35 million bushels a day. A century and a half earlier, in 1687, Louis Armand, Baron de Lahontan, reported that pigeons had “so swarmed and ravaged the colonists’ crop near Montreal that a bishop was constrained to exorcise them with holy-water, as if they had been demons.”

Yet by 1914 human rapacity had reduced the species to a single bird, Martha, who died that year at the Cincinnati Zoo.