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   ANSWERS VS. PROBLEM-SOLVING PROCESSES
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   Author  Topic: ANSWERS VS. PROBLEM-SOLVING PROCESSES  (Read 18032 times)
william wu
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ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« on: Jul 27th, 2002, 4:29pm »
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First of all, thanks to everyone for visiting! I'm truly thrilled by the activity of this bulletin board, and there are some very interesting discussions.
 
Now, there have been a few threads recently that are just lists of answers. This doesn't bother me terribly -- actually, I appreciate it to some extent since it lowers the in-flux of e-mail I get from tortured minds. However, I'm not sure how useful it is to tell someone only an answer.
 
As an analogy, this summer my friend Hansen introduced me to an extremely cool card game called Set (www.setgame.com). There's a deck of 81 cards, each card varying across four variables: color, symbol, shading, and number. Given 12 cards, you want to find 3 that share the same patterns. (Kind of like the pattern matching problems on some IQ tests.) Specifically, you want to find a 'Set', which is a group of 3 cards in which each variable is EITHER the same on each card, OR is different on each card. That is to say, any variable in the 'Set' is either common to all three, or is different on each card.
 
 
Examples of valid sets:
 

(All three are red; all ovals; all have two symbols; and all different shadings.)
 

(All different colors; all different symbols; all have different numbers of symbols; and all same shading.)
 
 
and an invalid set:
 

(All different colors; all diamonds; all have one symbol; however, two are open and one is not.)

 
When you play Set with others, you're just trying to find sets faster than everyone else. During my first few games, I remember feeling astounded by my friend's thinking speed. Hansen had written a C++ program that runs the game, and we were staring at the monitor, our hands poised to pounce on the mouse between us.  
 
He'd point to three cards. "See, this is a set." I'd look at them, notice the pattern, and then say "Ah, yes, I see." He'd click the cards, and 3 new ones replaced them.  
 
A few seconds pass. "Now this is a set." Click, click, click, refill. "I see."
 
"This is a set." Click, click, click, refill.  
 
"Set." Click, click, click, refill.  
 
"Set." Click, click, click ...  
 
I'm sitting there with glazed eyes, nodding my head, just verifying that his answers are correct, rarely having ample time to find my own sets before the electric cards refresh.
 
Afterwards, I asked Hansen how he thinks about the game. Do you try to match cards in a certain order -- for instance, first by number, then color, then shape and then shading? Is it just a really fast depth-first search? Or do you run it all in parallel? I had realized it was both trivial and useless to verify the correctness of his sets. What I wanted to know was how he finds sets so quickly in the first place!  
 
Hansen said he would explain his thought processes at the end of summer. In the meantime, the game would be more educational if I developed my own approaches and shortcuts. He gave me some pointers to get started. Now that summer is drawing to a close, I feel comfortable with my playing ability, and it's not very important whether or not he explains his methods, because I have my own.  
 
Conclusively, it's far more meaningful to generate an answer than it is to verify it. That's the difference between checking a set and finding it, between telling a joke and laughing at it, between eating a fish and learning to fish, between checking an NP problem and actually solving it. There will always be new puzzles and problems, and definitely not just during Microsoft interviews. So when I surrender to a riddle and ask for help -- and this does happen often -- I don't just want an answer. I'd like to see a problem-solving thought process. I want to know how a person goes about generating such an answer. What branches of thought were explored to get there. What initial observations and "hooks" did you start with? Did you think about the problem this way? Were you inspired by something peculiar? Maybe you were peeling an orange and realized that you can travel west to get east? You were drinking a glass of water and tilting it toward your mouth? You were traveling in a plane and noticed you had to set your watch backwards? We talk to ourselves when we attack these problems. What was your dialogue like? I don't just want to know your answer. I want you to show me how to think. Because that will get me far more mileage than a hashtable of specific puzzles and their solutions.  
 
These are my opinions about the nature of discussion that I would like to see. Sorry if I was too pedantic. Enjoy the site!
« Last Edit: Jan 12th, 2009, 10:44am by william wu » IP Logged


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raechel bandurchin
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #1 on: Nov 3rd, 2002, 12:22pm »
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hi, ijust read what you had to write about finding answers, we share the same point of view and i was just wondering how to find people like you and i. i go to highschool and i feel like i am surrounded by morrons and geeks. the "cool kids" are the morrons and the smart but immature people are the geeks. i fall in between. i'm smarter than most and have been the cool kid before i just want to know how to find people like me because there don't seem to be any.
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william wu
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #2 on: Nov 18th, 2002, 2:22pm »
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how flattering! i'm glad you agree with my point of view.
 
regarding your problem of finding people like you, i really doubt there's absolutely no one at your high school who is curious and likes to think on his or her own. i think i can relate to where you're coming from though. in junior high i was in this program for supposedly gifted children, but afterwards no high schools in the city had a continuation of that program. so i entered high school very aloof, earning flawless grades while being one to two years younger than everyone else, closing myself and silently believing that everyone around me was an idiot. but eventually i realized that there were quite a few very interesting people in my class who were just as smart or smarter than myself. i just needed to get to know them to discover this. unfortunately, by the time i fully realized this simple fact, i had already established a  reputation for being a real pretentious bastard Tongue in retrospect, i regret having been so close minded, so don't make the same mistake. furthermore, it turned out that the "gifted" students academically didn't do any better than the rest. so i would be careful about making such sweeping negative generalizations as the ones you have made. even statements like "i'm smarter than most" are dangerous ... i've met lots of smart people whose intelligence isn't immediately obvious in a school environment, because they tend to direct their efforts toward independent endeavors outside the classroom. to make a long story short, if you're having trouble finding kindred intellects, relax your convictions for a while and just try looking a little harder  Smiley
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alice wong
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #3 on: Dec 11th, 2002, 12:14am »
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"The aim of education should be to teach us rather how to think, than what to think--rather to improve our minds, so as to enable us to think for ourselves, than to load the memory with thoughts of other men."
-- Bill Beattie
 
This quote totally speaks through your message.  I agree with you and also believe that the way to intellectual growth is to learn the "how", the methods and strategies of thinking, because methods can be applied to all fields. For instance, the way you identify a set in card game can also be applied to the identification of underlying meaning/ideas in a text.  It is a play of patterns that our mind familiarizes itself with, so that eventually, they become second nature.  You might also want to try the card game which is based on 6 cards that we show at a time and try to add/substract/divide/multiply to get 24 as fast as possible. Kings=13, Queen=12, Jack=11, and the rest... Have fun!
 
Alice
p.s.: thanks for compiling the riddle database, though i'm not very good at solving them...=P
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #4 on: Apr 15th, 2003, 3:38pm »
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When WE get into college?  I think the significant majority of US either are in or have graduated from college.
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #5 on: Apr 15th, 2003, 7:27pm »
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Quote:
When WE get into college?  I think the significant majority of US either are in or have graduated from college.

 
Hey Aero_guy, i was refering to me, raechel and anyone else who has not yet graduated from highschool. Just clearing it up.
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #6 on: Mar 6th, 2007, 10:09am »
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Guys I am a teacher in the Netherlands. Now you wonder why I am on this forum.  
I like to give my thoughts a push. just seeing if I can solve a riddle or a problem.  
 
Reason I post in here:
I agree we should guide the way of thinking. I am trying to bring them logic instead of things they really need to know, because if they get the logic behind the lessons. they can find the rest out themselves.
 
Logic is not easy given. Thinking and reading are sometimes worse then you think. But then again you can train it. The only real thing thats against you is the motivation of the students. you cant always teach em logic because if they dont see it.. they wont go with you and disrupt the lessons.
 
so technically: good thinking and lets get it done
practical: not always working
 
regards,
Chekov
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #7 on: May 7th, 2007, 6:54am »
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Hi, I have simillar experience in general and particulary with the game sets, too.
 
For the starting period plaing with woman which are known to have better "peripheral thinking" I had no chance to win.
 
But after a while I have invented algorithm with high success chances and the situation reversed...
 
In each dimension calculate sizes of projections. Take a subset with maximal such size (among all dimensions). Problem is now divided into two parts ...  
A) work inside the subset  
B) one point in the set and two points in the corresponding small sets.
 
B) As the reamining sets are usually small, the number of "lines" they generate is small so you can check quickly if it goes through the set.
 
We have reduced one dimension in the A) case, so it is easier here, too.

 
Of course this is not big fun to simulate an algorithm pretending you are playng game, but it usually works best for me. (Actually I don't compute the dimension sizes, I just guess which is the biggest Wink )
« Last Edit: May 10th, 2007, 6:43am by Hippo » IP Logged
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #8 on: May 8th, 2007, 5:18am »
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on May 7th, 2007, 6:54am, Hippo wrote:
Hi, I have simillar experience in general and particulary with the game sets, too.
 
For the starting period plaing with woman which are known to have better "peripheral thinking" I had no chance to win.
 
Those who like playing sets ... please stop reading now ... Wink
 
But after a while I have invented algorithm with high success chances and the situation reversed...
 
In each dimension calculate sizes of projections. Take a subset with maximal such size (among all dimensions). Problem is now divided into two parts ...  
A) work inside the subset  
B) one point in the set and two points in the corresponding small sets.
 
B) As the reamining sets are usually small, the number of "lines" they generate is small so you can check quickly if it goes through the set.
 
We have reduced one dimension in the A) case, so it is easier here, too.
 
Of course this is not big fun to simulate an algorithm pretending you are playng game, but it usually works best for me. (Actually I don't compute the dimension sizes, I just guess which is the biggest Wink )

You mean you don't just maintain a running plot in a 3*3*3*3 hypercube and look for lines?
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #9 on: May 8th, 2007, 1:54pm »
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on May 8th, 2007, 5:18am, rmsgrey wrote:

You mean you don't just maintain a running plot in a 3*3*3*3 hypercube and look for lines?

Smiley no looking for (n choose 2) lines in 4D and checking that some of them is there trice is not the best method for mine brain ... especialy when n>11. Wink
It works best for you? Unbelievable Wink
 
BTW: The same algorithm can be described in oposite way ... not looking to maximal "dimension" but to minimal dimension ... than you start with the 3D lines from it and if unsuccessfull, take just the maximal corresponding dimension and the problem is reduced from 4D to 3D ...
« Last Edit: May 8th, 2007, 2:09pm by Hippo » IP Logged
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #10 on: May 9th, 2007, 12:40am »
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on May 8th, 2007, 5:18am, rmsgrey wrote:
You mean you don't just maintain a running plot in a 3*3*3*3 hypercube and look for lines?

That won't cover all cases anyway.  There are 216 possible "all different" sets, but only 8 grand diagonals in a hypercube.  Sad
 
By the way, the "all different" sets (where none of the characteristic is repeated) are those I have the most trouble finding.  And I don't see how your method of "maximum dimension" helps for that.
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #11 on: May 9th, 2007, 1:50am »
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This is the B) case ...  
The algorithm does not work well when the projections are almost equaly sized. But if some projection is big and therefore the remaining are small, the number of lines generated by remaining pairs is small enough to be checked fast. ... So finding all different precedes finding same in the coordinate with bigest projection in my algorithm.
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #12 on: May 9th, 2007, 2:10am »
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I see.  Using brute-force, with 12 cards, you would check 12*11/2 pairs and see whether the 3rd card is there.
 
But if you see that for instance there are 5 greens, you check (A) 3 cards among the 5 greens and (B) pairs among the 7 non-green cards, of different color, (max 3*4), and see if the 3rd matching card is among the greens.
 
I have to try that.
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #13 on: May 9th, 2007, 6:22am »
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on May 9th, 2007, 12:40am, Grimbal wrote:

That won't cover all cases anyway.  There are 216 possible "all different" sets, but only 8 grand diagonals in a hypercube.  Sad

Yeah, you're right, you need to either try multiple hypercubes, or tesselate the hypercubes to fill hyperspace (or treat the hypercube as the hypersurface of a hyper-torus - or whatever the appropriate 5-dimensional object is called...)
 
Just as well I don't consciously use that method Tongue
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #14 on: May 9th, 2007, 6:36am »
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Reduction of the line count to test is very important ... in the case of 5 green 4,3 of other colors I would check other dimensions ... if there are just 2 diamond suits, the number of cross lines to be checked is at least 10...
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #15 on: May 9th, 2007, 7:15am »
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on May 9th, 2007, 6:22am, rmsgrey wrote:

Yeah, you're right, you need to either try multiple hypercubes, or tesselate the hypercubes to fill hyperspace (or treat the hypercube as the hypersurface of a hyper-torus - or whatever the appropriate 5-dimensional object is called...)
 
Just as well I don't consciously use that method Tongue

That would be (Z/3)4 or Z34, i.e. vectors with 4 integer components mod 3.
Makes me wonder if we could do a Fourier transform on such a thing?
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #16 on: May 9th, 2007, 1:28pm »
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Oh yes, now I understand what are you talking about. Cheesy Of course the lines I was talking about are in Z34.  
 
I don't thing so Fourier transform would be usefull here Roll Eyes ... why to apply harmonic analysis here?
 
William: I am sorry to spam out the thread about solving process by talking about SETS ... Embarassed
« Last Edit: May 9th, 2007, 1:29pm by Hippo » IP Logged
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Re: ANSWERS VS. PROBLEM-SOLVING PROCESSES  
« Reply #17 on: May 10th, 2007, 8:33am »
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on May 9th, 2007, 1:28pm, Hippo wrote:
William: I am sorry to spam out the thread about solving process by talking about SETS ... Embarassed

You could always look at it as an illustration of the problem-solving process in action... Cheesy
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