The most ancient problem connected with combinatorics may be the house-cat-mice-wheat problem of the Rhind Papyrus (Problem 79), which occurs in a similar form in a problem of Fibonacci'sLiber Abaci and in an English nursery rhyme. All are concerned with successive powers of 7.

The first occurrence of combinatorics per se may be in the 64 hexagrams of theI Ching. (However, the more modern binary ordering of these is first seen in China in the 10th century.)

A Chinese monk in the 700s may have had a rule for the number of configurations of a board game similar to go.

In Greece, one of the very few references to combinatorics is a statement by Plutarch about the number of compound statements from 10 simple propositions; Plutarch quotes Chrysippus, Hipparchus, and Xenocrates on the subject, so all apparently had some interest in the subject.

Boethius apparently had a rule for the number of combinations ofnthings taken two at a time.

The author discusses interest in combinatorics in the Hindu world, by the Jainas, Varahamihira, and Bhaskara (the latter in theLilavati). The work of Brahmagupta should be relevant, but is not currently available in English.

The Arabs seem to have adopted their combinatorics from the Hindus.

The author also briefly discusses some interest in combinatorics in the Jewish mathematical tradition; two examples are Rabbi ben Ezra and Levi ben Gerson.

Magic squares may first occur in thelo shudiagram, which is often linked with theI Ching. The author discusses how the idea of magic squares may have entered the Islamic world, was then improved, appeared in the work of Manuel Moschopoulos, and possibly through him entered the Western world.

What happened in China is less clear. As the author suggests, the work of Yang Hui suggests that there had been a Chinese tradition of work in magic squares, already dead by Yang Hui's time. For example, the squares Yang Hui gives are not of types found elsewhere. In addition, Yang Hui seems unclear on the techniques for construction.

It is interesting that De la Loubère learned of a simple method for constructing magic squares in Siam.

The author also discusses the possibility of a Hindu study of magic squares; the presumably Arab source of Western magic square mysticism and later developments, such as Euler's questions on orthogonal Latin squares.

The author discusses how questions in partitions arose in gambling, such as the throwing ofastrogali (huckle bones, which can land 4 ways) or dice (which can land in 6 ways). An early systematic study is in the late Medieval Latin poemDe Vetula, which gives the number of ways you can obtain any given total from a throw of 3 dice. Cardano and Galileo examined the subject in more depth.

Combinatorial thinking in games and puzzles. Discusses the wolf-goat-cabbage, attributed to Alcuin. [Similar puzzles also occur in a variety of other cultures, but are not discussed in this article.]

Also discusses theJosephus problem, based on a process similar to the childhood process of "counting-out". The Josephus problem is named for the Jewish historian Josephus of the 1st century AD, who supposedly saved his life with a correct solution. This problem unexpectedly turned up in Japan.

The author discusses how "Pascal's" triangle was possibly known to Omar Khayyam in the context of taking roots. The Hindu scholar Pingala may have known a method, but the case is more cryptic. At any rate, it was known by the time of Halayudha, who may have lived in the 900s AD. A more clear-cut reference occurs in the work of Nasir al-Din al-Tusi in 1265.

In China, the triangle appears in the work of Chu Shih-Chieh (1303), but may have been very ancient by then.

The triangle was used by Pascal and Fermat to resolve the "problem of points". This problem had the goal of determining how to distribute stakes when a game ends early.

from a précis by J. Dieudonné of The roots of combinatorics ,Biggs, N. L.,Historia Math.6(1979), no. 2.

As always, a perverse calculus refreshed and redefined the world.J.G. Ballard: Millenium People

## Sunday, May 22, 2005

### a sunday afternoon conjunction

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