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(I underline "distinct" because you cannot repeat powers: otherwise you could write, for example,
How do these examples relate to the Fraudini trick?
1 | 321 | |
2 | ||
4 | ||
8 | ||
16 | ||
32 | ||
64 | ||
128 | Too big! |
Now add up those rows marked with an asterix (*), and you'll get the answer (35952).
Examples: Try these:
Let's look at the simplest example imaginable: divide 32 by 8. We can actually do it by Egyptian multiplication, since 8 divides into 32 evenly:
1 | 8 | |
2 | 16 | |
4 | 32 | * |
So the answer is 4 (how do we get 4?). Rather than building the other half of a product on the left side of the table, we're building the number we're dividing (32 in this case) on the right side of the table. Then we add up the starred entries on the left hand side of the table. It's the opposite of multiplication.
32 was pretty easy to "build", because it appeared on the right hand side directly. It's usually a little harder, as we'll see below.
1 | 8 | * |
2 | 16 | |
4 | 32 | * |
So the answer is 5 (how do we get 5?)
Let's look at an example: divide 35 by 8.
In a way we turn it into a multiplication problem: what times 8 equals 35? So we know the 8, and use it to "double" -- but then to "halve", when 8 won't go evenly into 35:
1 | 8 | |
2 | 16 | |
4 | 32 | * |
1/2 | 4 | |
1/4 | 2 | * |
1/8 | 1 | * |
So the answer is 4+1/4+1/8
(Where have we seen those fractions before? Look to the Eye of Horus!)
But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:
1 | 7 | |
1/2 | 3+1/2 | * |
1/4 | 1+1/2+1/4 | * |
1/7 | 1 | |
1/14 | 1/2 | * |
1/28 | 1/4 | * |
So the answer is 1/2+1/4+1/14+1/28
Dominic Olivastro, "Ancient Puzzles", suggests a third reason why this use of unary fractions is good. Consider the problem Ahmes poses of dividing 3 loaves of bread between 5 people. We would answer "each person gets 3/5-ths of a loaf". If we implemented our solution, we might then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then cut one of the smaller pieces in half, giving the other two people 2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and they will insist that it is not an even division. Some have larger pieces, some have smaller. Ahmes would calculate 3/5 as : 3/5 = ()3 + ()5 + ()15 [ = 1/3 + 1/5 + 1/15 ] Now cut one loaf into fifths, cut two more into thirds, then take one of the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you can now distribute everyone's 3/5-ths share in a way that _looks_ equal, since they will have exactly the same size pieces. (And no, I don't want to argue about the crust.)
(Think about what the answer means, in terms of bread.)