The ancients didn't invent the computer, but they made some interesting observations about "binary numbers."
There is an old African riddle that goes like this: A man must transport a leopard, a goat and a pile of cassava leaves across a river. The boat can carry no more than one at a time, besides the man himself. The goat cannot be left alone with the leopard and the goat will eat the cassava leaves if he is not guarded. How can he take them across the river?
No matter how he managed it, it might be interesting that the number of trips he had to make (assuming he borrowed the boat) equals the number of trigrams of the I Ching. (Later, we will see that there really is a connection!)
As mentioned in an earlier column, the number system consisting of just two values, 0 and 1, is the smallest, complete arithmetic number system. It is the number system on which the theory of the electronic, digital computer is based. And the genesis of "binary expansions" or "binary numbers," where a number is expressed as a string of 0's and 1's or alternately as a sum of powers of 2 is apparently very ancient.
Fu Hsi (originator of the I Ching ) represented 8 numerical values by "trigrams" composed of broken or solid lines (equivalent to all strings of length 3 of 0's and 1's).
In ancient Yoruba, a tortoise shell would be used to represent the value 1 or 0, depending on whether it was turned face up or face down. A necklace of 8 tortoise shell could represent as many as 16 numerical values or all strings of length 8 of 0's and 1's. The Yoruba developed a divination system, called Ifa, similar to the divination system of the I Ching. The necklaces of eight tortoise shells were said to represent 16 "Odu", similar to the 8 "trigrams" of Fu Hsi.
Another ancient African tribe, for whom the binary principle was significant, was the Dogon of the Niger River region in Sudan. To stress the importance of the principle of polarity, the birth of twins was a noteworthy event and central to the the Dogon cosmology and so was the number, 8. The number 8 was, in fact, omnipresent in Dogon life and belief. The Dogon believed that, "in the beginning," the architect of the world order, the "Nummo" spirit, laid out eight covenantstones, in an arrangement which dually indicated the outline of a human soul and also the order of human society. Could this have been the "PreHeaven" diagram of the I Ching, with trigrams characterized by gender, as defined by Fu Hsi (mentioned in a previous column)?
And thousands of years before the birth of Christ, the ancient Egyptians, indirectly, broke numbers down into strings of 0's and 1's when they multiplied two numbers by applying the methods of "mediatio: and "duplatio." This procedure consisted of doubling one factor and halving the other (to within an integral value) as demonstrated in the computation of 27 times 11 in the example below:
* 11 
27 
* 5 
54 
2 
108 
* 1 
216 
where 5 = [11/2] and 2 = [5/2] and 1 = [2/2] in the calculation above, The symbol [.] is the "rounding off" or "greatest integer" symbol.
The sum of those multiples of 27 which correspond to odd numbers (i.e., which are marked by asterisks (*) in the first column) is 297, the desired product.
We could perform an equivalent kind of multiplication (to "mediatio" and "duplatio") by expressing 11 as a binary number. For example, 11 = 2(2)(2) + 0(2)(2) + 2 + 1 or (1011) in binary notation. Therefore, 27 times 11 equals 27{1 + 2 + 2(2)(2)} = 27 + 54 + 216.
The following columns give an example of the ancient Egyptian method of multiplication of 12 by 12. It is taken from the Rhind papyrus, Peet edition, no. 32.
Numbers in parentheses give the transcriptions.
























The first column shows 12 times 1,2,4,8, resp. (the numbers are written "from right to left). The second column shows the extra figuring "written in the margins" of the papyrus, so to speak. The third column shows 12 "halved" by 1,2,4, 8, resp.
The computation is the product of 12x12=48+96=144, where 48 and 96 are the numbers in the third and forth rows indicated by asterisks (*).
Since 1100 is the binary expansion of 12 and this is equivalent to the pattern of dashes in the righthand column headed by (1,2,4,8 resp.), we can imagine that some experience with multiplication by mediatio and duplatio would automatically have given rise to a mental picture of binary expansion of numbers in the mental calculations of early Egyptian arithmeticians.
Let's now return to the African riddle about the tiger, goat and cassava plant. To solve it, we notice that if the man takes the goat across first, then he can take the tiger next without any mishap occurring, because he can then row back with the goat and retrieve the cassava plant, leaving the goat on the other side.
However, if the man takes the tiger over first, then the goat will eat the cassava plant.
This problem turns out to be equivalent to the "Hanoi Tower" puzzle, which involves transferring three rings of graduated sizes, A, B and C (labelled from top to bottom) stacked in a tower or pyramid on one of three pegs, to another peg. A Tower of Hanoi puzzle can be made by cutting eight cardboard squares of graduated sizes and moving them among three spots on a piece of paper. Draw a triangle connecting the spots. The following simple procedure will solve the puzzle: "Transfer the smallest disk [or cardboard square] on every other play, always moving it around the triangle in the same direction. On the remaining plays, make the only transfer possible that does not involve the smallest disk." If we follow this procedure, we solve the puzzle by moving the disks in the order of ABACABA.
The Tower of Hanoi was sold as a toy in 1883 and was invented by Edouard Lucas. It originally bore the name of "Prof. Claus" of the College of "LiSouStian," but these were soon discovered to be anagrams for "Prof. Lucas" of the college of "Saint Louis." This information is given in a book by Martin Gardner. Both D. W. Crowe and the famous Irish mathematician Sir Willam Rowan Hamilton (of 19th century) made interesting observations concerning the "Tower of Hanoi." If we trace a path along the edges of a cube of sides A, B and C, choosing the coordinates in the order ABACABA, the path will form a Hamiltonian circuit, i.e. a complete circuit that passes through every vertex on the cube. Crowe saw that the order of transferring n disks in the Tower of Hanoi puzzle corresponds exactly to the order of coordinates in tracing a Hamiltonian path on a cube of n dimensions.
Returning from this digression back to the African riddle, we notice that transferring the cassava plant, tiger and goat to the other side is like playing the Hanoi Tower game with the following stack:
PLANT
T I G E R
G O A T
The three "pegs" are now the two river banks and the boat.
But now, instead of transferring the small disk first, the big disk,
second and the largest disk, third, we transfer the goat first, the
tiger, second and the cassava plant, third. As we have seen, there
will have to be seven transferals made to complete the task.
According to Martin Gardner, "this sequence of moves is familiar to anyone working with binary computers." To see this, write the binary numbers from 1 to 7 and label the columns A, B and C.
Table of binary numbers
CBA
001 A
010 B
011 A
100 C
101 A
110 B
111 A
Opposite each row in the above table, the letter is written that identifies the "1" that is farthest to the right on each row. The sequence of letters from top to bottom will be the pattern in question. "This pattern is encountered frequently in mathematical puzzles. The most familiar instance of the pattern is the sequence in the sizes of marks on a oneinch segment of an ordinary ruler. The pattern results, of course, from successive binary divisions of the inch into halves, quarters, eights and sixteenths. It is also used in an ancient mechanical puzzle called 'Chinese rings.'"
But the African riddles tells me that the ancient puzzle called 'Chinese rings' may have originated in ancient Africa.
(For more ancient arithmetic, see The Pythagorean Theorem at http://www.fact.fws1.com)
Let me hear from you.
Write ObÈ at RR@Obe.safeshoppermail.com for more information, concerning the proof of the Pythagorean Theorem. Find music at http://ladilamusic.safeshopper.com/
ObÈ's website is http://ladilamusic.safeshopper.com/ and you can email him at RR@Obe.safeshoppermail.com
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