2024-01-12 11:00:42
The Moscow Puzzles, Scribner’s Sons, NY 1972
Let’s look again at the fascinating topic of how we can approach a problem from several different angles.
Previously we have explored how a problem, for example geometric, can be solved in many ways, using the different tools we have available. This gives us a “style” of proceeding: there are those of us who are very careful and methodical even if it takes longer, and there are those who can see a quick and elegant solution from a bird’s eye view.
We have also mentioned how we can approach a problem by trial and error, when we find it difficult to quickly translate it into equations.
And we have even given examples of how to do what we call creative traps: thinking “outside the box” to invent methods that even the person who raised the problem had not considered.
Today we will see another variant: how the same problem can be posed at different levels of mathematical understanding. That is, a child can approach a problem in progressively more sophisticated ways, in the sense of the tools she uses, but assimilates it conceptually from the beginning.
THE PROBLEM
One of the oldest types of problems to practice is the so-called “distribution problem”, which divides a quantity X into several fractions that must be added. I have already shown very ancient examples before, from the Babylonians and the Greeks. The example we will take is very beautiful and is presented in a more modern text, the Moscow riddles (1956), a magnificent compilation by the mathematician Boris Kordemski.
In the best classical tradition, he sets the problem with a funny story:
A goose was flying when it encountered a flock of geese coming in the opposite direction. “Hello, one hundred geese!” he greeted them.
The leader of the flock replied, “We are not a hundred! But if you take twice the number of geese that we are, and you add half of that number, then you add a quarter of that number and then you add yourself, yes we are one hundred.”
The goose was not very good at calculating, so when he saw a crane hunting frogs, he decided to land and ask his opinion, since as we know… cranes are very good at mathematics.
*
This is the end of the approach. Now we are going to solve it in three ways, from the simplest to the most sophisticated: the first will be by approximations (algorithmic solution), the second will be the one shown by Kordemski, which is a graphical solution, and finally the “adult” solution which is algebraically.
ALGORITHMIC SOLUTION
The challenge can be posed to a third grade child without any problem, as long as he or she knows basic arithmetic operations.
To find the solution, we simply create a table to test values until we find the final value.
The idea is very simple: find a number that, if you multiply it by 2, add half, then a quarter and then 1, gives you 100. The thing becomes simpler when you notice that we can only take into account numbers that are can divide by 4.
It doesn’t take many tries to realize, too, that the number must be less than 50, because by simply multiplying it by 2 we have already gone too far. And we can also see that 30 is quite small, so we already have a range that reduces a lot of work. By doing a few experiments we arrive at the value:
Making the table is not complicated, and the key for the child is to realize that by making a few observations (multiples of 4, numbers in a small range), the problem becomes much simpler than it may seem at first.
GRAPHIC SOLUTION
The crane’s solution to the goose is extraordinarily elegant: a graphical solution that clearly shows why:
With its long beak, the crane makes several stripes on the ground: two long equal stripes, one half that size, another a quarter that size, and one very small, almost like a point. We can visualize it like this:
The crane says that the longest line represents the total number of geese in the flock and so all the lines together, plus the dot, represent 100 units. Then he says that knowing that, we must consider everything in quarters. This is:
Once visualized like this, the crane asks the goose, “How many rooms do you see there?” The goose answers that there are 11. The crane says that those 11 rooms are equivalent to 99 geese, so how many geese are there in one of those rooms?
The goose answers that there are 9.
“Very good!” says the crane, “so how many geese is a flock?”
The goose sees the light: “They are 9×4, of course!”
This is a very beautiful way of looking at a problem that is in principle arithmetic, but we can translate it into a clear visual representation and understand it immediately.
Finally:
ALGEBRAIC SOLUTION
When you already know a little more about mathematics, the first thing that comes to mind is to create an equation, which is a more “adult” way of doing it. However, the problem is still accessible enough to pose to a sixth-grade child, who already knows how to use fractions and who has no problem understanding what an unknown is:
x = number of geese in the flock
The equation from the problem is constructed like this:
2x + x/2 + x/4 + 1 = 100
Putting everything with a denominator of 4 and solving:
8x/4 + 2x/4 + x/4 + 1 = 100
11x/4 + 1 = 100
11x/4 = 99
11x = 396
x = 36
Thus we can see that a problem that we normally consider algebraic can be posed early and use simpler tools, gradually progressing until it can be handled in a more abstract way. Including this entire process provides a more intimate understanding.
I was born in Mexico and have lived in China since 2000, where I studied language and history, and then I was a visiting researcher at the Wan Lin Jiang International Center for Economics and Finance, as well as a professor of economics and history for foreigners at Zhejiang University. He currently directs the Mexico-China Center and gives lectures on international science and technological cooperation.
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