Federico Kereki
2005-07-29 13:30:52 |
To compute, or not to compute
What do you think about using computers to solve puzzles?
Personally, I think that sometimes itīs "overkill" -- some problems can be worked out in a few minutes, and writing/testing/running a program can take quite longer.
Obviously, you should fit the tool to the problem. If you cannot work the problem out otherwise, use the computer! But sometimes, I feel it's more like laziness... I don't believe there are many (if any) Perplexus users who CANNOT solver problems!!
Why the "rage" for/against computers?
Why include notes for/against their usage? |
Old Original Oskar!
2005-07-29 18:46:02 |
Re: To compute, or not to compute
Saying "NO COMPUTERS" is like saying "NO ANALYTICAL GEOMETRY" for a geometrical problem, or "NO DICTIONARY" for a word problem... or like a cook saying "Eat my food as I prepared it; do not add salt to it."
However, I do agree that some people rush in to program before thinking... |
owl
2005-07-30 16:31:22 |
Re: To compute, or not to compute
I spend a lot of time thinking about the use of technology in math education. This community has definitely affected my thoughts in this area, especially Charlie's solutions to some problems.
It seems in pencil vs. computer discussions there is a tendency to assume that computers are an easy escape. This completely misses the deep and careful thinking that must go into creating a CORRECT computer solution. Consider a combinatorics problem; a computer does allow one to bypass the slickest counting or summation arguments through sheer speed, but one must still make a correct counting argument AND code it correctly AND do both of these so that the program doesn't take forever. This can be just as high order of thinking as the slick counting argument.
Another example is the probability problems. Correct probabilistic thinking does not, in general, come naturally for even the most mathematically talented. Why this is true is a great separate topic. But given this, the education world is moving to the idea of simulation being the first approach to probability problems. Why? Because the students must understand the question to correctly simulate the situation, and the empirical results help give the students a "feel" for how there theoretical conclusions should come out. And this approach gives practice in the use of technology that is unavoidable in the next step of statistics. This may be overkill in basic probability problems, but with conditional probabilities it is an almost unbeatable approach. And as we have seen in this community, there are other deep issues that get raised with such approaches, like the idea of the (pseudo)random number generator and how many legitimate trials can be produced or even how many trials does one need to have a certain confidence in the results.
And let us not forget visualizations. The ability to create different USEFUL representations is a powerful tool that does not come easily. And rarely do these representations prove something as much as guide one to a superior understanding. For instance, I would claim that a true student who is a skillful user of Geometer's Sketchpad will almost always have a huge advantage in theoretical geometric understandings of planar problems.
But ... we know that when one becomes enamored with a hammer, the world looks like a bed of nails. And thus, sometimes technological approaches totally miss the sweetest solutions (as Erdos would say, the solution in God's book). And the reverse is also true, a lot of ugly mental gymnastics can be invested in a problem that is a slam-dunk with a few lines of code. This problem has been around before computer tools were available. I know complex analysts that view every geometry problem as a vector/complex number exercise. And I would point out that similarly to coding, those who do not have a lot of experience with vector approaches in geometry tend not to appreciate them and even claim their inferiority to constructive approaches. What is the "best solution", however, has always been a communal decision and truly dependent on the eyes of the beholders. |
Sam
2005-07-30 18:19:45 |
Re: To compute, or not to compute
I basically agree with owl. I think what people forget, when they howl that one should solve the puzzle ONE way or the OTHER is that people are solving the puzzles for their own enjoyment. No one makes anyone solve a puzzle, and if it were not enjoyable they wouldn't. If it's more enjoyable, or intellectually stimulating for someone to solve a puzzle through a computer solution, then who's to stop them from doing so?
For example, I'm a student in AI. Though I never got around to trying to solve Conversing Club 3, I thought it would be very interesting to see what kind of method of solving it I could come up with. I think I could have written an interesting Genetic Algorithm to do the job, and it would have been fun and intellectually stimulating. It wouldn't have detracted from the puzzle at all.
I do, to be clear, understand the point of someone stating "No computers or calculators!" in their puzzles. However, I believe that the reason for such a restriction is that the puzzle author knows a smart, interesting way of solving the puzzle by hand, and hopes that others will see this interesting solution. It's a slightly less restrictive version of the cook example above: my cook is saying "While I know you like to put salt in your food, I think you'll find that this is actually really good without. Just give it a try before you put your salt on."
In that case, the author has a valid reason for suggesting that people at least try to do it by hand first, with the promise of a fun and interesting solution at the end if they succeed, but this still allows those who wish to solve it by computer to do so. The author must shrug and say "Their loss, it would have been fun if only they realized," though of course the specific user may have had more fun doing it the computer way.
We can't seek to restrict other people's enjoyment of a puzzle, unless it encroaches on the enjoyment of others. so we don't put solutions in the headings, but we allow any method of finding a solution that the user wants. |
owl
2005-07-31 20:29:21 |
Re: To compute, or not to compute
A couple of other thoughts come to mind on this,
I suppose the habit of annotating computer assisted solutions in the title is an okay idea, though there are grey areas and it accepts that the use of computers (or calculators?) is bothersome.
What I would request is that solutions that use "programs" are described as well as other kinds of proofs/solutions. Giving the code is possibly enlightening to those that code in a similar language, but calling this terse is kind. I am suggesting that when someone cracks something with coding, they explain carefully how the code works. I am not talking about psuedo-code or flow-charts; I am talking about an explaination of the approach that is understandable to non-coders. In this way we can all see and appreciate the cleverness (or lack thereof) of the approach. I will personally try doing this when using Mathematica.
Does this make sense? |
