Solving new problems (ones that have not yet been solved)

[Fascheue]

let’s say I’m trying to solve a mathematical problem that has not yet been solved or that I have not encountered yet. One where I don’t yet have the mathematical toolset to solve the problem.

Assuming I make no mistakes, should I end up doing things that don’t work? Or in other words, should there sometimes be instances where you can’t know weather or not you can solve a problem a certain way until you try it? Kind of like going through a maze, you have to try things knowing that they might not work and you’ll have to try something else?

When math is being taught, the teachers usually say something like, to solve this problem you do this, then this, then this, then this.

When I’m solving particularly complex problems that I don’t have much experience it seems like there is a lot of fiddeling involved. Should that be the case, or is that indicative that I’m making mistakes while solving these problems?

 

[kuruman]

I am not sure who said it first, but here it goes, "If we knew what we are doing it wouldn't be called research." Welcome to the club.

 

[scottdave]

Do you have any specific examples of what you have done, or is this just general?

I think I understand what you are asking. If you don't know how to solve something, then you can do some trial and error of different methods. You should keep track of what works, and what does not. You may come up with something new (or at least new to you). You will most likely remember the method which works the best, than if somebody had just told you "do it this way".

 

[jedishrfu]

You could compare this the explorers problem centuries ago looking for the northwest passage from th Atlantic to the Pacific Ocean. Many routes had to be tried and failed before one was found. The explorers believed it existed but couldn’t know for sure until it was discovered.

The end result was a passage that gets blocked by sea ice much of the time.

https://en.m.wikipedia.org/wiki/Northwest_Passage

 

[Fascheue]

Do you have any specific examples of what you have done, or is this just general?

I think I understand what you are asking. If you don't know how to solve something, then you can do some trial and error of different methods. You should keep track of what works, and what does not. You may come up with something new (or at least new to you). You will most likely remember the method which works the best, than if somebody had just told you "do it this way".

Just in general really.

For the sake of simplicity I’ll choose something simple.

a/b = c

If I want to solve for b I can multiply each side by b to get

a = bc

Let’s call this step 1

Then divide both sides by c to get the answer

b = a/c

Let’s call that step 2

But how did I know to do step 1? Only because from my experience I know that if I have something like a = bc I can just divide by what I’m not looking for, or c in this case to find what I am looking for, b in this case, alone.

If I didn’t know this, I’m not sure how I could know how step 1 got me any closer to the answer without doing it and then looking at the new equation to find that you are now able to solve for b in one step.

If I didn’t know that, it seems like I would have to fiddle around with the problem a bit, and when it’s not such an easy problem, there’s seems to be a lot more fiddling.

 

[fresh_42]

But how did I know to do step 1? Only because from my experience I know that if I have something like a = bc I can just divide by what I’m not looking for, or c in this case to find what I am looking for, b in this case, alone.

If I didn’t know this, I’m not sure how I could know how step 1 got me any closer to the answer without doing it and then looking at the new equation to find that you are now able to solve for b in one step.

You know by experience, or for beginners by the principle to eliminate step by step what is disturbing. Here $\frac{1}{b}$ is disturbing and the knowledge about the definition of a multiplicative inverse gives you the tool to use $b \cdot \frac{1}{b} = 1$ to get rid of it. The next disturbing thing is then the factor $c$ at the $b$.

If I didn’t know that, it seems like I would have to fiddle around with the problem a bit, and when it’s not such an easy problem, there’s seems to be a lot more fiddling.

And here you start to get sloppy in my opinion. As far as you don't know, where $a,b,c$ are taken from, as far you are not allowed to divide by $c$ or better, multiply by $c^{-1}$ assuming existence, use the associative law, and the definition of the neutral element $1$ again. All these are hidden assumptions. It might look very petty in this example, but it demonstrates the pitfalls when doing something on your own: the risk of hidden assumptions which might not always be fulfilled. It is easy to trick yourself into errors.

So the main advice is: tackle what's disturbing step by step and always make sure, that you're indeed allowed to do what you want to do.
There are more complex examples, which are by no means as obvious as the one above, e.g. the Sierpiński curve or the Weierstraß function.

 

[Fascheue]

You know by experience, or for beginners by the principle to eliminate step by step what is disturbing. Here $\frac{1}{b}$ is disturbing and the knowledge about the definition of a multiplicative inverse gives you the tool to use $b \cdot \frac{1}{b} = 1$ to get rid of it. The next disturbing thing is then the factor $c$ at the $b$.

And here you start to get sloppy in my opinion. As far as you don't know, where $a,b,c$ are taken from, as far you are not allowed to divide by $c$ or better, multiply by $c^{-1}$ assuming existence, use the associative law, and the definition of the neutral element $1$ again. All these are hidden assumptions. It might look very petty in this example, but it demonstrates the pitfalls when doing something on your own: the risk of hidden assumptions which might not always be fulfilled. It is easy to trick yourself into errors.

So the main advice is: tackle what's disturbing step by step and always make sure, that you're indeed allowed to do what you want to do.
There are more complex examples, which are by no means as obvious as the one above, e.g. the Sierpiński curve or the Weierstraß function.

What exactly does disturbing mean in this context? I’m not sure how I would know that the 1/b is disturbing without the experience that it has to be gotten rid of from solving similar problems.

 

[mfb]

Mathematics only gets interesting if you don't know which steps you have to follow.
Often the first approach won't directly lead to a solution, but it will help to understand the problem better, and choose better approaches afterwards. Ideally this leads to a solution in a limited time.

There are problems where hundreds of mathematicians worked on them, sometimes for centuries, and many of these problems are still unsolved. This involves many problems that even children can easily understand. One example
Take a number, if it is even divide it by two, if it is odd multiply by three and add one. Repeat with the new number. As an example: 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> ...Will you always reach 1 eventually? This is the Collatz conjecture, and we don't know the answer.

 

[fresh_42]

What exactly does disturbing mean in this context? I’m not sure how I would know that the 1/b is disturbing without the experience that it has to be gotten rid of from solving similar problems.

Well, you know that the goal is $b = \ldots$ , so the $b$ in the denominator has to vanish. In addition, a clear problem statement is the key to proper work. We have $\frac{a}{b}=a \cdot \frac{1}{b}$ which uses a certain notation, namely $b^{-1}=\frac{1}{b}$. So it is the power $-1$ which is disturbing. Thus we have to think about what this means. And it means $b \cdot b^{-1}=1$. That's all we have. So to get rid of the power $-1$, we will have to multiply by $b$ and see where it leads us to.

 

[scottdave]

So this reminds me of when the Rubik's Cube first came out (early 1980's). I was in high school - several of us got Rubik's cubes and started playing around with them. We started to recognize certain sequences of moves could be used to generate a result, which gets you closer to a solution (like getting all of the corners in position), so we would memorize that sequence. We collaborated with friends, sharing different sets of moves and what they would achieve.

I discovered that there is more than one way to "solve" a Rubik's cube - a screwdriver or knife can pry out one of the cubes, then you can take it apart and reassemble as solved. Is that cheating or just an alternative method? That was my method for my first several "solved cubes". Eventually though, I did learn the sequences to solve a cube on my own.

There was a lot of trial and error - recognizing and remembering what works. But also learning from what others had done. I hope this gives you some insight.

 

[jedishrfu]

I discovered that there is more than one way to "solve" a Rubik's cube - a screwdriver or knife can pry out one of the cubes, then you can take it apart and reassemble as solved. Is that cheating or just an alternative method? That was my method for my first several "solved cubes". Eventually though, I did learn the sequences to solve a cube on my own.

There was a lot of trial and error - recognizing and remembering what works. But also learning from what others had done. I hope this gives you some insight.

You were here just following in the footsteps of Alexander the Great when he cut the Gordian Knot.

https://en.m.wikipedia.org/wiki/Gordian_Knot

The original use of the Rubiks Cube before it became the cube was to teach engineering design to students. They had to design the mechanism that the cube used.

Rubik's invention

In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest.[14] Although it is widely reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart. He did not realize that he had created a puzzle until the first time he scrambled his new Cube and then tried to restore it.[15] Rubik obtained Hungarian patent HU170062 for his "Magic Cube" in 1975. Rubik's Cube was first called the Magic Cube (Bűvös kocka) in Hungary. Ideal wanted at least a recognisable name to trademark; of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980.

https://en.m.wikipedia.org/wiki/Rubik's_Cube