March 26, 2009

The Ladies' Diary

Odds bobs, ladies, what am I?
I'm at a distance, yet am nigh;
I'm high and low, round, short, and long,
I'm very weak, and very strong.
Sometimes gentle, sometimes raging.
Now disgusting, now engaging.
I'm sometimes ugly, sometimes handsome. . . .
I'm very dirty, very clean,
I'm very fat, and very lean,
I'm very thick, and very thin,
Can lift a stone, tho' not a pin. . . .

This riddle, written entirely in verse, goes on for about 30 lines. Submitted by a reader, it starts off the riddle section of the 1808 edition of The Ladies' Diary: or, Woman's Almanack. The Ladies' Diary was published annually, starting in 1704. Printed in London, it featured the usual stuff of almanacs: calendar material, phases of the moon, sunrise and sunset times, important dates (eclipses, holidays, school terms, etc.), and a chronology of remarkable events.

The little book's subtitle provides a glimpse of its purpose: "Containing New Improvements in ARTS and SCIENCES, and many entertaining PARTICULARS: Designed for the USE AND DIVERSION OF THE FAIR SEX."

Among those diversions were sections devoted to riddles (called enigmas), rebuses, charades, scientific queries, and mathematical questions. A typical volume in the series included answers submitted by readers to problems posed the previous year and a set of new problems, nearly all proposed by readers. Both the puzzle and the answer (revealed the following year) were often in verse.

In a paper just published in Historia Mathematica, Joe Albree and Scott H. Brown of Auburn University review the mathematical contribution of The Ladies' Diary, particularly the leading role it played in the early development of British mathematical periodicals.

"From the first decade of the 18th century and for many decades following, The Diary was a pioneer in more than one respect," Albree and Brown write. "With unfailing regularity, starting in 1708, it presented an array of mathematical problems and their solutions to a wide range of readers."

"From the very start women were encouraged to participate fully in The Diary's mathematical program," they add.

The Ladies' Diary was founded by John Tipper (before 1680–1713), in part to advertise the Bablake School in Coventry, England, where he was master, and to help support his family financially. The publication also represented his recognition of a need for an almanac that catered to women.

"Tipper's attitude toward women was often remarkably progressive," Albree and Brown remark. The cover of each issue featured a picture of a prominent English woman, beginning with Queen Anne in 1704.

The publication was also a success. In its second year, about 4,000 copies of The Diary were sold; circulation reached about 7,000 in 1718 and, in the middle of the 18th century, amounted to around 30,000 copies a year.

The readership was diverse, and the publication attracted a wide range of contributors of both problems and solutions. However, because many contributor used pseudonyms, it's difficult to determine how many were female.

"The earliest mathematical questions in The Diary were puzzles, enigmas solved by numbers, and such contrived problems continued to appear through the whole life of The Diary," Albree and Brown say. "However, very quickly, the level of difficulty and the seriousness of many of the questions in The Diary increased, and The Diary became one of the participants in the popularization of mathematics and the Newtonian sciences in the 18th century."

Note that The Ladies' Diary had been founded while Isaac Newton (1643–1727) was still alive.

By the 19th century, some of the proposed problems were quite sophisticated and occasionally called for knowledge of contemporary issues and advances in mathematics or physics. The Diary even attracted some foreign contributions. Here's a geometry question submitted in 1830 by French mathematician C.J. Brianchon (1783–1864).

The six edges of any irregular tetrahedron are opposed two by two, and the nearest distance of the two opposite sides is called breadth; so that the tetrahedron has three breadths and four heights. It is required to demonstrate that, in every tetrahedron, the sum of the reciprocals of the squares of the breadths is equal to the sum of the reciprocals of the squares of the heights.

Some questions were open-ended. Here's one from 1822:

Required a better method than has yet been published of finding x in the equation xx = a.

I came across the 1808 and 1809 editions of The Ladies' Diary among the volumes in a remarkable collection of books at the library of the University of Calgary in Alberta. The Eugène Strens Recreational Mathematics Collection contains more than 6,000 items, including books, periodicals, newspaper clippings, and manuscripts devoted to recreational mathematics (in a very broad sense) and its history.

"The bulk of mathematics has really always been recreational," says mathematician Richard K. Guy, who was instrumental in bringing to the University of Calgary material collected by the late Eugène Strens, an engineer, amateur mathematician, and friend of the artist M.C. Escher. "Only a tiny fraction of all mathematics is actually applied or used."

Here's a riddle from the 1809 edition of The Ladies' Diary:

I'm a singular creature, pray tell me my name,
I partake of an Englishman's freedom and fame;
I daily am old, and I daily am new,
I am praised, I am blam'd, I am false, I am true;
I'm the talk of the nation, while still in my prime,
But forgotten when once I've outlasted my time.
In the morning no Miss is more coveted than I,
In the evening no toy thrown more carelessly by.
Take warning, ye fair! I, like you, have my day,
And, alas! You, like me, must grow old and decay.

A typical rebus provided clues for a starting word, which was then modified step by step to produce the final word:

My whole's a small but luscious fruit;
Take off my head and then you'll see
What sinful men sometimes commit,
That brings them to the fateful tree.
Two letters now you may transpose,
But place them both with care,
Another luscious fruit will then,
Most plainly soon appear.


Take two-fifths of what, seen on Delia's fair face,
Always adds to her charms an ineffable grace;
These selected with care, and with art combined,
Will produce a Diarian of genius refin'd.

Scientific queries covered a wide range of subjects:

Query: A drop of oil let fall on a wasp, kills it in less than a minute; query: how is this effected?

Query: Thirteen years have elapsed since the Northern Lights have made their appearance. How is their absence accounted for?

The mathematical questions tended to focus on geometry, and they were reminiscent of the types of word problems found in math textbooks of a few generations ago.

A circular vessel, whose top and bottom diameter are 70 and 92, and perpendicular depth 60 inches, is so elevated on side that the other becomes perpendicular to the horizon; required what quantity of liquor, ale measure, will just cover the bottom when in that position.

The book didn't provide a diagram.

The math questions featured in The Ladies' Diary were of sufficient interest that collections of them were published in four volumes covering the years 1704 to 1817.

I wonder if Jane Austen (1775–1817) or Ada Lovelace (1815–1852) ever pondered the puzzles posed by The Ladies' Diary?

Riddles: water; newspaper.
Rebuses: grape, rape, pear; smile, smart.
Math question: Very nearly 189 gallons.


Albree, J., and S.H. Brown. 2009. "A valuable monument of mathematical genius": The Ladies' Diary (1704-1840). Historia Mathematica 36(February):10-47.

Almkvist, G., and B. Berndt. 1988. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi, and The Ladies' Diary. American Mathematical Monthly 95(August-September):585-608.

Guy, R.K., and R.E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Mathematical Association of America.

March 12, 2009

Rock-Paper-Scissors for Winners

The simple game of rock-paper-scissors has a long history as a way of settling disputes, whether in the playground or at the conference table.

In the playground version of rock-paper-scissors, each of two players makes a fist. On a count of three, each player simultaneously puts down a fist, which means rock, a flat hand, which means paper, or two fingers in a "V," which means scissors. The following non-transitive rules decide the winner: Rock beats scissors, scissors cut paper, and paper wraps rock. If both players make the same gesture, the bout ends in a tie.

Is there a winning strategy for this game? It doesn't make sense to show the same configuration each time. An alert opponent would quickly learn to anticipate your move, make the appropriate response, and always win. A similar danger lies in following any other sort of pattern. Thus, unless you can find a flaw in your opponent's play, your best bet is to mix the three choices randomly.

But without the help of a randomizer of some sort, people aren't very good at making random choices on their own. In the article "Winning at Rock-Paper-Scissors" in the March 2009 College Mathematics Journal, Derek Eyler, Zachary Shalla, Andrew Doumaux, and Tim McDevitt present strategies for defeating people who are poor generators of random sequences.

Suppose, for example, that a player tends to choose one option (symbol) more often than the others. According to McDevitt and his coauthors, winning play generalizes to the following strategy:

* Never choose the symbol that loses to the most likely symbol.
* Choose the most likely symbol if the symbol that it beats has probability greater than 1/3.
* Otherwise, choose the symbol that beats the most likely symbol.

To see what happens in practice, McDevitt and his associates collected data from 119 people who played 50 games of rock-paper-scissors against a computer using this optimal strategy. Their subjects had a strong preference for rock: 55.5% started with rock, 32.8% with paper, and 11.8% with scissors.

"Symbol choices beyond the first seem to depend on previous choices," the authors note. "Players seem to have a distinct preference for repeating plays."

For example, a player who chose paper on one trial had a .421 probability of repeating this play on the next trial, well above 1/3.

So, when playing against such a player, your best bet is a strategy of always playing the symbol that defeats the symbol that your opponent previously played. But that only works in the short term. In the long run, you would end up choosing the same symbol every time, and your opponent might catch on.

"A more sophisticated approach analyzes an opponent's previous two choices," McDevitt and his colleagues write. Experimental results suggest that people have a tendency to repeat symbols or to cycle through the symbols.

The researchers developed a computer program that bases its choice of symbol adaptively on its human opponent's previous two choices, updating the probabilities of different pairs of choices every time its human opponent chooses a symbol.

In one test that involved 241 participants each playing 100 games against the computer, the computer won 42.1%, lost 27.7%, and tied 30.2% of individual games. That's significantly better than playing randomly would achieve.

"In closing, we note that deviating from the optimal strategy is always dangerous because a superior strategy always exists," the researchers warn.

In the test, for example, two players defeated the program 100 times out of 100. "Conversations with those two players revealed that one of them stopped and started the match repeatedly until he found the perfect strategy through trial and error," McDevitt and his colleagues note, "and the second accessed and analyzed the program to determine the computer's strategy."

You can try beating a computer implementation of the strategy described above at McDevitt's "Rock-Paper-Scissors" Web page.


Beasley, J. 1990. The Mathematics of Games. Oxford University Press.

Eyler, D., Z. Shalla, A. Doumaux, and T. McDevitt. 2009. Winning at rock-paper-scissors. College Mathematics Journal 40(March):125-128.

Peterson, I. 2002. Mating games and lizards. In Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.