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솔로깡 [330158] · 쪽지

2014-05-18 02:39:30
조회수 1,124

수학 문제 해결 법칙 - 1

게시글 주소: https://orbi.kr/0004571189

This is an article about mathematical problem solving. We make three assumption about you, our reader.


1. You may enjoy math.

2. You know high-school math pretty well, and have at least begun the study of math, and logic (NEVER MEANS "KSAT MATH").

3. You want to become better at solving math problems.


First, what is a "Problem"? We distinguish between "Problems" and "Exercises" An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. For example, here is an exercise.


Example 1 : "Compute (1234^3) without a calculator.

Solution : You have no douby about how to proceed. Just multiply. Just do it if you hav your own brain. (by, ORBI Ji Bark Ryung, Jan, 2014)


Example 2 : write (1/1*2)+(1/2*3)+(1/3*4)+...+(1/99*100) as a fraction in lowest terms.

Solution : just calculate it. YOU CAN CONJECTURE THAT FOR ALL POSITIVE INTEGERS n,

(1/1*2)+....+(1/n(n+1))=n/n+1


So, now, we are confronted with a "Problem"; is this conjecture true, and if so, how do we probe that it is true? If we are experienced in such matters, this is still a mere exercise, in the technique of mathematical induction. But if we are not experienced, it is a problem, not an exercise. To solve it, we need to spend some time, trying out different appriaches. The harder the problem, the more time we need. Often the first appriach fails. Sometimes the first dozen approaches fail!


Here is another question, the famous "Census-Taker Problem." A few people might think of this as an exercise, but for most, it is a problem.


Example 3 : A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are.

"I have three daughters, their ages are whole numbers, and the product of the ages is 36," says the mother.

"That's not enough information," responds the census-taker.

"I'd tell you the sum of their ages, but you'd still be stumped."

"I wish you'd tell me something more."

"Okay, my oldest daughter Annie likes dogs."

What are the ages of the three daughters?



At first reading, it seems impossible-there isn't enough information to determine the ages. That's why it is a problem, and a fun one, at that. (If you want a piece of clue, just give me a orbi message)


Example 4 : I invite 10 couples to a party at my house. I ask everyone present, including my wife, how many people they shook hands with. It turns out that everyone shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my wife shake hands with? (I did not ask myself any question-I am genius. HaHa.)


A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it.


These articles(I will write more articles.) will help you to investigate and solve problems. If you are an inexperienced problem solver, you may often give up quickly. This happens for several reasons.


1. You mat just not know how to began

2. You may make some initial progress, but then cannot proceed further.

3. You try a few things, nothing works, so you give up.


An expericed problem solver, in contrast, is rarely at a loss for how to begin investigating a problem. He or she confidently tries a number of approaches to get started. This may not slve the problem, but some progress is made. Then more specific techniques come into play. Eventually, at least some of the time, the problem is resolved. The experienced problem solver operates on three different levels:


- Stragegy : Mathematical and psychological ideas for starting and pursuing problems.

- Tactics : Diverse mathematical methods that work in many different settings.

- Tools : Narrowly focused techniques and "tricks" for specific situations.

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