2024-01-12 09:00:42
The Moscow Puzzles, Scribner’s Sons, NY 1972
Let’s look once more on the fascinating matter of how we are able to method an issue from a number of totally different angles.
Beforehand we now have explored how an issue, for instance geometric, might be solved in some ways, utilizing the totally different instruments we now have out there. This offers us a “fashion” of continuing: there are these of us who’re very cautious and methodical even when it takes longer, and there are those that can see a fast and stylish answer from a fowl’s eye view.
Now we have additionally talked about how we are able to method an issue by trial and error, after we discover it troublesome to shortly translate it into equations.
And we now have even given examples of find out how to do what we name inventive traps: considering “outdoors the field” to invent strategies that even the one that raised the issue had not thought of.
At this time we’ll see one other variant: how the identical drawback might be posed at totally different ranges of mathematical understanding. That’s, a baby can method an issue in progressively extra refined methods, within the sense of the instruments she makes use of, however assimilates it conceptually from the start.
THE PROBLEM
One of many oldest kinds of issues to follow is the so-called “distribution drawback”, which divides a amount X into a number of fractions that have to be added. I’ve already proven very historical examples earlier than, from the Babylonians and the Greeks. The instance we’ll take could be very lovely and is introduced in a extra fashionable textual content, the Moscow riddles (1956), an impressive compilation by the mathematician Boris Kordemski.
In the most effective classical custom, he units the issue with a joke:
A goose was flying when it encountered a flock of geese coming in the wrong way. “Hey, 100 geese!” he greeted them.
The chief of the flock replied, “We’re not 100! However in case you take twice the variety of geese that we’re, and also you add half of that quantity, you then add 1 / 4 of that quantity and you then add your self, sure we’re 100.”
The goose was not superb at calculating, so when he noticed a crane searching frogs, he determined to land and ask his opinion, since as we all know… cranes are superb at arithmetic.
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That is the tip of the method. Now we’re going to clear up it in 3 ways, from the only to probably the most refined: the primary might be by approximations (algorithmic answer), the second would be the one proven by Kordemski, which is a graphical answer, and at last the “grownup” answer which is algebraically.
ALGORITHMIC SOLUTION
The problem might be posed to a 3rd grade little one with none drawback, so long as she or he is aware of fundamental arithmetic operations.
To seek out the answer, we merely create a desk to check values till we discover the ultimate worth.
The concept could be very easy: discover a quantity that, in case you multiply it by 2, add half, then 1 / 4 after which 1, provides you 100. The factor turns into easier whenever you discover that we are able to solely take into consideration numbers which are can divide by 4.
It does not take many tries to appreciate, too, that the quantity have to be lower than 50, as a result of by merely multiplying it by 2 we now have already gone too far. And we are able to additionally see that 30 is sort of small, so we have already got a spread that reduces quite a lot of work. By doing a couple of experiments we arrive on the worth:
Making the desk isn’t sophisticated, and the important thing for the kid is to appreciate that by making a couple of observations (multiples of 4, numbers in a small vary), the issue turns into a lot easier than it might appear at first.
GRAPHIC SOLUTION
The crane’s answer to the goose is awfully elegant: a graphical answer that clearly reveals why:
With its lengthy beak, the crane makes a number of stripes on the bottom: two lengthy equal stripes, one half that dimension, one other 1 / 4 that dimension, and one very small, nearly like some extent. We are able to visualize it like this:
The crane says that the longest line represents the entire variety of geese within the flock and so all of the strains collectively, plus the dot, signify 100 models. Then he says that figuring out that, we should think about every little thing in quarters. That is:
As soon as visualized like this, the crane asks the goose, “What number of rooms do you see there?” The goose solutions that there are 11. The crane says that these 11 rooms are equal to 99 geese, so what number of geese are there in a type of rooms?
The goose solutions that there are 9.
“Superb!” says the crane, “so what number of geese is a flock?”
The goose sees the sunshine: “They’re 9×4, after all!”
This can be a very lovely method of an issue that’s in precept arithmetic, however we are able to translate it into a transparent visible illustration and perceive it instantly.
Lastly:
ALGEBRAIC SOLUTION
If you already know just a little extra about arithmetic, the very first thing that involves thoughts is to create an equation, which is a extra “grownup” method of doing it. Nevertheless, the issue remains to be accessible sufficient to pose to a sixth-grade little one, who already is aware of find out how to use fractions and who has no drawback understanding what an unknown is:
x = variety of geese within the flock
The equation from the issue is constructed like this:
2x + x/2 + x/4 + 1 = 100
Placing every little thing with a denominator of 4 and fixing:
8x/4 + 2x/4 + x/4 + 1 = 100
11x/4 + 1 = 100
11x/4 = 99
11x = 396
x = 36
Thus we are able to see that an issue that we usually think about algebraic might be posed early and use easier instruments, regularly progressing till it may be dealt with in a extra summary method. Together with this complete course of supplies a extra intimate understanding.
I used to be born in Mexico and have lived in China since 2000, the place I studied language and historical past, after which I used to be a visiting researcher on the Wan Lin Jiang Worldwide Middle for Economics and Finance, in addition to a professor of economics and historical past for foreigners at Zhejiang College. He at the moment directs the Mexico-China Middle and offers lectures on worldwide science and technological cooperation.
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