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By Andy Newman, Hal Just
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Extra resources for What Every Uncle
Sample text
Once again, we have both the hieratic script and the hieroglyphics, with modern numbers printed under the hieroglyphics. Look at the numbers in the left-hand column first; they are a little surprising. The numbers are 7, 49, 343, 2301, 16807, and 19607. The first five numbers are so nearly the first five powers of seven that we may assume Ahmes made a slight clerical error in his transcription and meant 2401 for the fourth number. r htp- t'wI :,,. n tt m; 9999&1 1^ 108, 94 3 1999 4211 99 1 4 0t,11 9 II o.
An arithmetical progression is a sequence in which every number differs by a constant from the one before it. " For example, 5, 9, 13, 17 is an arithmetical progression in which the "difference of share" is 4. Before reading Ahmes' solution, the reader should try it alone. It is easily solved by modern methods, but to get a better feel for the world of the ancient Egyptians, try solving it using only the method of false position. Ahmes' solution itself is typically cryptic, and gives only a hint: Do it thus: Make the difference of the shares amounts that the five men receive will be 23, 5 /2.
Now we have 1/2 + 1/3 + 1/7 = 41/42 Since the sum is of the form dl(d+ 1), we have a proper puzzle in which a merchant divides his 41 camels among his sons in the proportions 1/2, 1/3, and 1/7, If you play at this long enough you will sense that the number of fractions that work is actually limited. Fortunately, we find right away that a cannot equal anything other than 2. Assume the opposite, that a equals 3. Then the smallest value of (d+ 1) we can get is 12, which arises when a = 3, b = 4, c = 6.