After today's class when we were walking back to our department, Vinod and me were talking about some puzzles he used to post for PhD students at ISyE. he then asked me "what is a hard puzzle?" or better to say "is a problem hard because the solution is complicated or maybe something else is involved?"
I answered Vinod's question by bringing this example that we all know:
suppose we are training a child to learn the mathematical sum operation. To help him understand the "relationship" better we present him a story: suppose there are 3 birds sitting on a tree and 2 birds just come and sit next to them. how many birds are now on the tree? the child hopefully will answer 5. we repeat this problem with different numbers until he learns the whole concept of sum. now we switch the gear and try to teach him "subtraction" we ask him what will happen if we have 3 birds on the tree and 1 of them flies away. he will say 2 birds will remain on the tree. we repeat this experiment over and over until he learns this operation as well. now we test him on a "slightly" different problem. we tell him to suppose that there are 3 birds on the tree, we have a gun to shoot them, so we shoot and hit one of them. how many birds are now on the tree?
the first time I encountered this problem I immediately answered 2. I used the fact that since 2 problems are slightly different (actually they are exactly the same but some features are different. it's like we use 2 different kind of birds in 2 problems), the answer should be the same. but as we all know the answer is 0 since if you shoot one bird, other birds will fly away before you find a chance to shoot them.
I would call this problem "hard" and why so? before giving my reasons I would like to hear what you think.
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When the answer to that kind of problem is revealed to me, I'm always disappointed in myself for not thinking "outside the box" and coming up with the same answer. That phrase (thinking outside the box), I feel, is a very common way of classifying what you are calling "hard" problems. My most immediate thought is that for me to have answered "0" in the first place, I needed to consider information beyond what was freely provided before I rushed to give an answer. Ultimately, I think there could be a number of solutions to that problem. 1) There ARE still two birds because they are just born and can not fly away and are stuck in their nest. 2) There is only 1 bird left because that one didn't notice the third bird being shot and didn't realize what was happening while the 2nd bird did and got out of there. 3) 0 birds are left as originally offered. So my take on the proposal of what defines a "hard" problem is that a problem is "hard" if more information than what is initially given must be inferred or assumed in order to arrive at a solution. We definitely see this in engineering where an exam problem will require us to make assumptions about the information given in the problem and must be explicitly stated to prove we had that knowledge while solving the problem. I think the inability or failure to make these presumptions, in combination with a lack of previous knowledge (or lack of having similar problem-solution sets) makes a problem "hard".
ReplyDeleteThis spirals into the discussion of the notion that when I am given a "hard" problem, am I told that it is a hard problem and how does my problem solving process differ from when I think it is an "easy" problem or when there is no indication of difficulty given about it? How does the initial set of knowledge that I pull from memory (either LTM into WM or just MEMORY) used to solve a problem change based on what I understand about a problem in terms of its difficulty? I realize I just dug a little deeper without providing a way out. It is "hard" to tell if I'm actually addressing your post, but hopefully I've provided a helpful perspective.
Another way to look at what constitutes a 'hard' problem is using semantic networks.
ReplyDeleteTo take a specific (class of) example, consider mathematical olympiad problems. Most people would consider that they are really hard despite the fact you only need to use the stuff you learnt in high school mathematics.
It turns out that the solution to such problems involves two things - creatively combining concepts and non-obvious substitutions (equivalent to longer paths in the semantic net from the hypothesis node to the solution node).
So I guess a hard problem is one that necessitates lot of creativity and non-routine thinking.