This post was inspired by a somewhat similar one at:
Did great Mathematicians of the Past Leave Unsolved Problems Behind?
Can You Solve This Problem?
An ancient Egyptian scribe recorded this problem in the Rhind Papyrus.
**“Go down I times 3, 1/3 of me, 1/5 of me is addded to me; return I, filled am I.
What is the quantity saying it?”
**From “The World of Mathematics,” Vol. I, Pg. 177, by James R. Newman, 1956.
Hint: Solve by the Egyptian method.
Let’s try to translate that into English:
Go down I times 3, 1/3 of me
Take 3 1/3 times I from an unknown X
1/5 of me is addded to me
Then add X/5 to the result
return I
Add I to that result
filled am I.
And you get X.
So:
X - 10*I/3 + X/5 = X
X/5 = 10*I/3
3X = 50I
X = 16 2/3 times I
You can only find the unknown X if you know what value I has. (Of course, my understanding of the problem could be wrong).
I just looked in my copy of The World of Mathematics to see if the problem made more sense in context. It doesn’t. I can’t even figure out what the problem is here, let alone how to solve it. Before someone claims that they do know what it means, I suggest that they look at the statement of the problem in the book, where there’s a two-page layout showing the original Egyptian hieroglyphs in which the problem is supposedly solved. I can’t even figure what number in the messy layout is supposed to be the solution.
It seems the problem is solved by the “Egyptian Method” of multiplication. (surprising that the Egyptians were not that mathematically advanced). Here’s an explanation of the “method”:
http://mathforum.org/library/drmath/view/57542.html
It also seems that there are many references to be found about the “Rhind Papyrus”. This is as far as I’ll go toward solving the problem. I’m not going to learn some obscure procedure that I’ll never use again.
The “Egyptian method of multiplication” wouldn’t be the issue here: that’s just an algorithm for multiplying positive integers, which would give the same result as any other valid algorithm.
What may be in issue is Egyptian fractions. They only allowed fractions with numerator 1 (1/2, 1/3, 1/4, …, 1/n), and the specific fraction 2/3 (though that isn’t strictly necessary – see MathWorld on Egyptian fractions
In my solution, 1/3, 1/5 and 2/3 are involved, which are all Egyptian fractions. This may be a point in faviour of my solution.
The problem is that your solution isn’t really a solution. It just expresses one variable in terms of another. It doesn’t give a value for either variable. In so far as I understand the problem as given in the book, there is only one variable and it is given a value.
I’d say the “I” in the OP isn’t a variable, but a pronoun. The problem is phrased as a number (“quantity”) speaking about itself. Let’s call that number “x”.
Secondly, and I am not completrely sure about this, “Go down I times 3” could mean “x-3” rather than “-3x”, just because only the first interpretation makes sense:
“x - 3 +x/3 + x/5 = x” makes sense: x = 5 5/8
“-3x + x/3 + x/5 = x” is a contradiction, unless x = 0 (which I doubt, Egyptians had no number zero, as far as I know).
As I said, it’s just an interpretation and I am not completely sure about this, so any comments are welcome.
When the question says "“Go down I times”, the “I” can’t be the first person pronoun: if it were, you would have “me”, not “I”. So the only interpretation that I could give was that “I” is the name of a variable. I don’t think the ancient Egyptians used variable names like that, but someone translating into English might.
If that’s not what it means, I have no idea what "“Go down I times” could possibly mean.
Just a question: is there a translation issue here? Considering the kinds of disputes that arise in just translating one modern language to another, can anyone be sure of what the original hieroglyphs even say, let alone mean?
The notion of a variable arose far later than this problem was stated. “I” is a pronoun, referring to the same value as “me”.
As far as they were concerned, zero was not a quantity. Look at some annotated translations of Euclid VII for parallels (not the geometric kind).
Not true: “Go down I” is just an archaic way of phrasing “I go down”, as in “Able was I…”. So IMO “Go down I times 3” simply means “I go down 3 times”.
I’d buy that if “I” were directly after the verb, i.e. “Go I down …”. But, even if that’s the word order in Egyptian, it’s a lousy translation into English: you don’t preserve word order in translations, you preserve meaning. If I’m translating from German into English, I don’t put verbs at the ends of clauses like they do in German.
(“Down I go …” or “Down go I …” would be possible word orders in English too.)
MartinL from what I’ve seen this is a fairly simple proof of the existence of zero even though the Egyptians had no symbol for zero. Their math was used in practical applications and rarely did any theoretical math ever show its face in Egyptian life. That doesn’t mean that Egyptian mathematicians weren’t aware of the fact that zero or the concept of zero did exist. Merely that there was no specific symbol representing nothing.
The math used then involved a lot of subtraction which was represented by “go down” and fractions were of course divisions. I think you probably got the translation damned close. It’s pretty much the same thing I came up with.
I kept dodging the zero concept myself but I don’t see why an equation instead of a symbol couldn’t be representative of the idea. Perhaps this problem was their proof of zero.
another translation, that I like better… [(x-3)/3] + [x/5] = x
where x therefore =2 1/7 and lest I forget the last line >return 1 to me.
Which must surely mean ADD one which then gives us the value pi’
just a WAG?
This would be a useful thing to know.
Again, I would really urge people to get a copy of The World of Mathematics and try to understand the problem from the entire layout there, rather than trying to understand it from the problem as given in the OP. There’s a lot more in the article in the book, including supposedly a solution (although I don’t understand the solution). The article in The World of Mathematics, Volume 1, pages 170-178, is by James R. Newman himself. It originally appeared in a 1952 issue of Scientific American, if someone would rather look there, but you shouldn’t have any problem finding a copy of The World of Mathematics in a large library.
So are you saying the answer isn’t Pi?
No, the answer is 42.