“The Pyramid,” by A.C. White, from the *British Chess Magazine*. White to mate in two moves.

# Locke’s Index

Like many thinkers of his age, John Locke maintained a commonplace book, an intellectual scrapbook of ideas and quotations he’d found in his readings. In order to be useful, such a book needs an index, and Locke’s method is both concise (occupying only two pages) and flexible (accommodating new topics as they come up, without wasting pages in trying to anticipate them).

The index lists the letters of the alphabet, each accompanied by the five vowels. Then:

When I meet with any thing, that I think fit to put into my common-place-book, I first find a proper head. Suppose for example that the head be EPISTOLA. I look unto the index for the first letter and the following vowel which in this instance are E. i. If in the space marked E. i. there is any number that directs me to the page designed for words that begin with an E and whose first vowel after the initial letter is I, I must then write under the word Epistola in that page what I have to remark.

The result is a useful compromise: Each of the book’s pages is put to productive use without any need for an overarching plan, and the contents are kept accessible through a simple, expanding index that occupies only two pages. The whole project can grow in almost any direction, and when the pages are full then a new volume can be begun.

(Via the Public Domain Review.)

# All Together Now

In 1833, Heinrich Scherk conjectured that every prime of odd rank (accepting 1 as prime) can be composed by adding and subtracting all the smaller primes, each taken once. For instance, 13 is the 7th prime and 13 = 1 + 2 – 3 – 5 + 7 + 11.

In 1967 J.L. Brown Jr. proved that this is true.

# Podcast Episode 330: The Abernathy Boys

In 1909, Oklahoma brothers Bud and Temple Abernathy rode alone to New Mexico and back, though they were just 9 and 5 years old. In the years that followed they would become famous for cross-country trips totaling 10,000 miles. In this week’s episode of the Futility Closet podcast we’ll trace the journeys of the Abernathy brothers across a rapidly evolving nation.

We’ll also try to figure out whether we’re in Belgium or the Netherlands and puzzle over an outstretched hand.

# Looking Up

Two perplexing roofs, by Kokichi Sugihara of Japan’s Meiji Institute for Advanced Study of Mathematical Sciences.

I suppose these could be designed at scale!

# Unquote

“A multitude of books confuses the mind. Accordingly, since you cannot read all the books which you may possess, it is enough to possess only as many books as you can read.” — Seneca, *Moral Letters to Lucilius*

# One Nation, Indivisible

The second professor of mathematics in the American colonies suggested reckoning coins, weights, and measures in base 8.

Arguing that ordinary arithmetic had already become “mysterious to Women and Youths and often troublesome to the best Artists,” the Rev. Hugh Jones of the College of William and Mary wrote that his proposal was “only to divide every integer in each *species* into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho’ all nations count universally by *tens* (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision *ten* is uncapable.”

Successive powers of 8 would be called *ers*, *ests*, *thousets*, *millets*, and *billets*; *cash*, *casher*, and *cashest* would be used in counting money, *ounce*, *ouncer*, and *ouncest* in weighing, and *yard*, *yarder*, and *yardest* in measuring distance (so “352 yardest” would signify 3 × 8^{2} + 5 × 8 + 2 yards).

Jones pressed this system zealously, arguing that “Arithmetic by *Octaves* seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic.” But he seems to have had no illusions about its prospects, acknowledging that “there seems no Probability that this will be soon, if ever, universally complied with.”

(H.R. Phalen, “Hugh Jones and Octave Computation,” *American Mathematical Monthly* 56:7 [August-September 1949), 461-465.)

# A Folded Cube

A simple and surprisingly effective illusion by Lisbon anamorphosis specialists Sonhos com Dimensão.

# Calendar Boy

Gary Foshee presented this puzzle at the 2010 Gathering for Gardner:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

The first thing you think is ‘What has Tuesday got to do with it?’ Well, it has everything to do with it.

He proposed the answer 13/27, with this reasoning:

There are 14 equally likely possibilities for a single birth — (boy, Tuesday), (girl, Sunday), and so on.

If all we knew were that Foshee had two children, then it would seem that there are 14^{2} = 196 equally likely possibilities as to their births.

But we know that at least one of his children is a (boy, Tuesday), and only 27 of the 196 outcomes meet this criterion. (There are 14 cases in which the (boy, Tuesday) is the firstborn child and 14 in which he’s born second, and we must remove the single case in which he’s counted twice.)

Of those 27 possibilities, 13 include two boys — 7 with (boy, Tuesday) as the first child and 7 with (boy, Tuesday) as the second child, and we subtract the one in which he’s counted twice. That, Foshee says, gives the answer 13/27.

This generated a lot of discussion when it appeared — unfortunately because the meaning of Foshee’s question is open to interpretation. See the end of this *New Scientist* article and the comments on Columbia statistician Andrew Gelman’s blog.

# Three in One

*Dark Matter* (2014), by the artistic collaborative Troika, manages to be a circle, a hexagon, and a square all at once.

The same group had created *Squaring the Circle* a year earlier.