In relation to the discovery of Permutations and Combinations is the chessboard problem. It is generally viewed that the chessboard problem reached Europe from China. The invention of chess in its modern form involves work with series, permutations and combinations and probability.
One of the best known European medieval problems related to the Chinese chessboard is the problem involving the grains. The problem involves placing a number of grains on the chessboard i.e. one being put on the first square, two on the second, four on the third and so on in a geometrical progression. In the end the total amounts to:-
Another famous problem is associated with a Thang monk by the name of I-Hsing. Shen Kua the author of Meng Chhi Pi Than discusses I-Hsing's method. Infact, according to Needham J (1959), Shen Kua says:-
"The story tellers say that I-Hsing once calculated
the total number of possible situations in chess and
found he could exhaust them all. I thought about this
a good deal and came to the conclusion that it is quite
easy. But the number's involved cannot be expressed in
the commonly used terms for number's. I will only briefly
mention the large number's which have to be used. With
two row (fang erh lu) and four pieces (tzu) the number
of probable situations will be of 81 different kinds.
With three rows and nine pieces the number will be 19,683.
Using four rows and sixteen pieces the number will be
43,046,721 ... When the whole 361 places are used, the
number will come to some figure (of the order of)
Shen Kua then goes on further to discuss I-Hsing's method explaining that it is capable to enumerate all possible changes and transformations occurring on the chessboard. It looks as if the name for these calculations was Shang chhu, Tayin, Chhung yin meaning driving upwards, overlapping factors and repeating roots.
When discussing the study of permutations and combinations in China one immediately thinks of the 'I Ching' which is The Book of Changes.
There is a query whether or not this origin of the chessboard problem had any valuable early contributions to the research of permutations and combinations. This is an interesting question as the discovery of this subject appeared in Europe about 1140 AD and in India by Bhaskara in 1115AD.
However, this subject did not really progress until the end of the 15th century approximately about the time of Pacioli. The first book discussing permutations and combinations appeared in 1713 by Bernoulli and was called 'Ars Conjectandi'.
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