Vu Pham

Presented 6/15/95
Received 6/5/95



In the Middle Ages there developed a new form of puzzle problem, one suggested by the later Greek writers and modified by oriental influences. Problem of Metrodorus.

So far as the Greek were concerned the source book for this material is the Greek Anthology. This contains the arithmetical puzzle, supposed to be due to Metrodorus about the year 500.

Polycrates speaks : " Blessed Pythagoras ( c 540BC), Heliconian scoin of the Muses, answer my question: How many in thy house are engaged in the contest for wisdom performing excellently?"

Pythagoras answers: " I will tell thee, then, Polycrates. Half of them are occupied with belles lettres; a quarter apply themselves to studying immortal nature; a seventh are all intent on silence and the eternal discourse of their hearts. There are also three women, and above the rest is Theano. That is the number of interpreters of the Muses I gather round me."

The following problem relates to a statue of Pallas ( about the time of such Christian scholars as Capella and Cassiodorus, 460 AD and 502AD) lusty poets. Christians gave half of gold, Thespis one eighth, one tenth, and one twentieth, but the remaining nine talents and the workmanship are the gift of."

The following relates to the finding of the hour indicated on a sundial and still appears in many, modified to refer to modern clocks:
" Best of clock, how much of the day is past?"
" There remain twice two thirds of what is gone."
A type that has long been familiar in its general nature is seen in the following:
A. " Where are thy apples gone, my child?"
B. " Ino has two sixths, and Semele one eighth, and Autonoe went off with one fourth, while Agave snatched from my bosom and carried away a fifth. For thee ten apples are left, but I, yes I swear it by dear Cypris, have only this one."

The following problem has more of an Oriental atmosphere (unknown author and time):
" After staining the holy chaplet of fair-eyed Justice that I might see thee, all- subduing gold, grow so much, I have nothing; For I gave forty talents under evil auspices to my friends in vain, while, O ye varies mischances of men, I see my enemy in possession of the half, third, and eighth of my fortune."

One of the remote ancestors of a type frequently found in our algebras appears in the following form:
" Brick-maker, I am in a great hurry to erect this house. Today is cloudless, and I do not require many more brick, but I have all I want but three hundred. Thou alone in one day couldst make as many, but thy son left off working when he had finished two hundred, and thy son-in-law when he had made two hundred and fifty. Working all together, in how many days can you make these?"

Such problems seem more Oriental than Greek in their general form, but if we could ascertain the facts we should probably find that every people cultivated the somewhat poetic style in the recreations of mathematics.

The Chessboard Problem

One of the best- known problems of the Middle Ages is that relating to the grain being put on the first square, two on the second, four on the third, and so on.

In geometric progression, the total number being

264 - 1 = 18, 446, 744, 073, 709, 551, 615
This problem is Oriental.

A Dutch arithmetician, Wilkens ( 1669), in Europe, takes the ratio in the chessboard problems as three instead of two, and considers not only the number of grains but also the number of ships necessary to carry the total amount, the value of the cargoes, and the impossibility that all the countries of the world should produce such an amount of wheat.

2) Commercial Problems Equation of Payments
Interest: The taking of interest is very old custom, going back long before the invention of coins, to the period in which values were expressed by the weight of metal or by the quantity of produce. The custom of paying interest was well known in ancient Babylon ( before 2000 BC). In general, in the later Babylonian records, the rate ran from 5(1/2) per cent to 20 per cent on money and from 20 per cent to 33(1/3) per cent on produce, although not expressed in per cents. For one of the tablets relates the following:
"Twenty manehs of silver, the price of wool, the property of Belshazzar, the son of the king.... All the property of nadin Merodach in town and country shall be the security of belshazzar, the son of the king, until belshazzar shall receive in full the money as well as the interest upon it."

Tablets of Nineveh as old as the 7th century BC. have the following records:
" The interest [ may be computed ] by the year. The interest may be computed by the month. The interest on ten drachmas is two drachmas. Four manehs of silver...produce five drachmas of silver per month."

Interest in Ancient India Bahaskara (1150 AD.)

If the interest of a hundred for a month be five, say what is the interest of sixteen for a year.

If the interest of a hundred for a month and one third be five and one fifth, say what is the interest of sixty- two and a half for three month and one fifth.

3) Applications of Algebra
General nature
The first applications of algebra were in the nature of number puzzles. Such was the first algebraic problem of Ahmes (1550 BC ):
" mass, its whole, its seventh, it makes 19"

The second general application is geometric, and this characterizes the works of the Greek writers, with the exception of Diophantus.

The third general application is to fanciful problems relating to human affairs, and this is essentially Oriental in spirit.

The fourth type is characterized by the attempt to relate algebra actually to the affairs of life. Ahmes ( 1550 B.C) gave numerous problems like :
" Mass, its 2/3, its 1/2, its 1/7, its whole, it makes 33."
Modern symbol: x + (2/3)x + (1/2)x + (1/7)x = 33
In the Middle Ages in Europe, from the De Numeris Datis of jordanus Nemorarius ( 1225 AD) as follow:
" If there should be four munbers in proportion and three of them should be given, the fourth would be given."

Simultaneous Linear Equations. Problems involving simultaneous linear equations were more numerous in the Orient in early times than they were in Europe.

For example in India, Mahavira ( 850 AD) as follows:
"The mix price of 9 citrons and fragrant wood apples is 107; again, the mixed price of 7 citrons and 9 fragrant wood-apple is 101. O you arithmetician, tell me quickly the price of a citron and of a wood-apple here, having distinctly separated those prices well."

His rule for the solution is similar to the one used today for eliminating one unknown. Another of his problem which attributed to Euclid, is as follows:
Three merchants saw [ dropped ] on the way a purse [ containing money]. One [ of them ] said [ to the others], " If I secure this purse, I shall become twice as rich as both of you with your moneys on hand." Then the second [ of them ] said, " I shall become three times as rich." Then the other [ the third ] said, ' I shall become five times as rich." What is the value of the money in the purse, as also the money on hand [ with each of the three merchants]?

Intermediate Problems

In the Greek Anthology( 500 AD), there are two problem as follows:
1. " The three Graces were carrying baskets of apples and in each was the same number. The nine Muses met them and asked them for apples, and they gave the same number to each Muse, and the nine and three had each of them the same number. Tell me how many they gave and how they all had the same number."

2. The second ( xiv, 144) is a dialogue between two statues:
A. How heavy is the base on which I stand, together with myself!
B. My base together with myself weighs the same number of talents.
A. I alone weigh twice as much as your base.
B. I alone weigh three times the weight of yours."

4) Magic Squares

The eight triagram
It was probably Won- wang (1182-1135 B.C) who wrote the I-king, and which included the eight triagram ( magic square ). The magic square was shown:


The original in the book described the numbers of the Magic square as the faces of the dice. The author included that the Magic square was written upon the back of a tortoise which appeared to Emperor Yu ( 2200 B.C) when he was embarking on the Yellow River.

Magic squares and Magic circle in Japan.

Show only a half of a Magic square as given in Hoshino Sanenobu's Ko-ko-gen Sho,1673. The magic square numbers were written in Japanese language, and included the Magic circle of 129 numbers which from Muramatsu kudayu mosei's jinko- ki, 1665. The numbers in each radius add to 524, or with the center1. India in Magic square, It first appears in a jaina inscription in the ancient town of Khajuraho, India (unknown author, 870 - 1200 ).

The sum =34

This Magic square display a somewhat advanced knowledge of the subject, for it has each of the four minor squares has a relation to the other:

19 19

19 19
The wheel of life ( Thibetan )

The three triads ( which locates around the center of the wheel of life) made up of the nine digits:


The first had the following relation:
1 = gold = the sun: symbol circle ( O ) with a dot at the center
2 = silver = the moon symbol looks like a lune .
3 = tin = Jupiter symbol looks like the hand grasping the thunderbolt

The second triad was as follows:
4 = gold again = the sun symbol as the number 1.
5 = mercury = Mercury Symbol looks like the key with a little circle on its top, and a half circle stays on the top of the circle.
6 = copper = Venus: Symbol looks like the key with a little circle on its top

The third triad was as follows:
7 = silver again = the moon Symbol looks like the number 2
8 = lead = Saturn: Symbol looks like a letter h
9 = iron = Mars: Symbol ( a circle with a arrowhead point 60 degree of East-South).

The outer of the Triad, which is still on the circle, show the wheel of life by representing twelve types of animals to indicates the time in day and the year cycle. One year cycle is twelve years.

AT the bottom of number Eight ( West ) starts with :
mouse, and next animal is buffalo, tiger, cat, dragon, snake, house, goat, monkey, chicken, dog, and pig. Each animal represent the hours of a day. Horse is at noon. Next horse is Goat and it shows 1 PM, monkey is 2 PM, chicken is, dog is 4pm, pig is 5pm, mouse is 6pm, buffalo is 7pm, tiger is 8pm, cat is 9pm, dragon is 10pm, snake is 11pm, horse is midnight, goat is 1am, monkey 2am, chicken is 3am, dog is 4am, pig is 5am, mouse is 6am, buffalo is 7am, tiger is 8am, cat is 9am, dragon is 10am, snake is 11am, and return to horse is noon. 12 years would be a circle, and each type of animal would represent each year in one circle.

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