Learn about types of problems and common approaches to solving them. Created by Carole Yue.
Want to join the conversation?
- Would I be wrong if I went and cut all the matches in half and made those halves into equilateral triangles?(20 votes)
- For the match problem... couldn't you do a square with an x in it for 4 equilateral triangles?(6 votes)
- you can't, the diagonal of the square is not the same as the side, so the matches in the x can never touch both corners.(10 votes)
- Why wouldn't means end analysis be considered a type of algorithm. You are essentially taking a logical step by step approach by breaking it down into little pieces and solving them aren't you?(4 votes)
- Because the means-end analysis is a mental shortcut that helps with breaking down larger problems into more manageable ones. The algorithm method is just a logical step-wise process to get an answer. For means-end analysis, the process might not be in the most logical format, but will get you to your goal.(1 vote)
- At1:46, "With trial and error you're not necessarily keeping track of what you have already done"?
Is she sure about that? I went to the Wikipedia page on this and although there's no specific mention of having to keep track, all examples and manifestations do keep track of the errors, and it's what I've always thought about trial and errors-that you could at least exclude previous errors on subsequent trials.
Is it by definition that with trial and error you could actually err in the same way for an unlimited number of trials?(2 votes)
- In a practical sense, trial and error is a way to find the most successful solution, and the best way to do that would be to not repeat the same error. So, she wasn't very clear about that. Unless she means somebody is putting in random passwords without thinking about what they're typing, then no, we wouldn't constantly make the same error since trial and error is about moving towards the best solution. We tend to learn from our mistakes and repeating the same error endlessly would go against this concept.(4 votes)
- our brain is just a big chunk of tissues. how does it do so much?(1 vote)
- Nobody knows exactly how the brain works, even though a lot of people have spent a lot of time trying to figure it out. We do know some things though. For example, we know that the brain works by cells called neurons, and that these do (a bit simplified) one thing: fire off an electrical impulse, or not fire off an electrical impulse. The neurons are connected together, and if a neuron receives a lot of impulses, then it will fire off an impulse of its own. Some neurons connect to our senses, and receive impulses from for example light hitting the eyes, or something touching a part of the skin. Other neurons connect to the muscles, and when these neurons fire off an impulse, the muscle it connects to contracts. The way the brain does so much lies in how the connections between the neurons cause for example an impulse from neurons connected to the eyes to trigger a cascade of firings that end up causing the neurons connected to the muscles in the arm to contract, causing you to pull up your hand to catch a ball.
There is a whole lot more to the brain than this, but it captures the idea of how the individual neurons are pretty simple, but the whole, connected mass of neurons and other cells and chemicals called the brain is massively complex. Again using a simplification that can't really be taken too far, we can think of the brain as a computer. The individual components that make up a computer are very simple. However, when you connect them together in the right way, we can send an electrical impulse from our keyboard that will trigger a cascade of electrical impulses in the computer that end up causing a letter to appear on the screen. Since we built it, we know exactly how the computer does this, while we don't know exactly how the brain does all the things that it does, but the principle is the same.(6 votes)
- What will video games do to a child mind? Will it cause their development to slow down?(0 votes)
- Quite the opposite actually. Studies have shown that video gamers are better problem solvers and perform well in academics.(7 votes)
- how does problem solving relate to cognition?(2 votes)
- Cognition is another word for thoughts, or the way we process information. So, problem solving is a large part of how we process information. For instance, I couldn't just write down random words to answer this question, I had to process your question and then with cognitions come up with a logical reply.(2 votes)
- Didn't quite understand working backwards ?
If we work our way back to the maze, as the given example, that would take take ages... ?(2 votes)
- What are the problem solving strategies in math!(1 vote)
- There are a lot of different tricks for MCAT math. Sometimes it is just memorizing a different form of the equation that will get you straight to your answer (such as, for a weak acid, [H+}=sqrt(Ka*[HA]) where you assume HA is just your original molarity of the weak acid), but what seems to help me the most is finding different patterns to the math problem that make it easier to chunk everything together. For example, knowing the sine and cosine of different angles (0, 30, 45, 60, and 90) are very important for the MCAT so:
sin(0) = 0________cos(0) = 1
sin(30)=0.5_______cos(30) = ~0.9
sin(45)=~0.7______cos(45) = ~0.7
sin(60)=~0.9______cos(60) = 0.5
sin(90)=1_________cos(90) = 0
Notice that you are just counting upwards for the angles that you should know (0, 30, 45, 60, 90) and you are almost counting up by odd numbers (5, 7, 9) for the answer.
For cosine (since they are complementary) you are just going the opposite way.
*I am using the ( ~ ) sign to indicate rounding.
These numbers should get you close to the correct answer. Let me know if you have a more specific "problem solving strategy in math" question.(3 votes)
- [Narrator] What's the best route to your new job? How do you decide who to marry? How do you satisfy your growling stomach? Each of these questions represents a problem that we sometimes face. You may not even realize it, but you are an excellent problem solver. Just a few seconds ago, you figured out how to start this video, and that may not feel like a huge accomplishment compared to, say, coming up with the theory of relativity, but every time you engage in an action or thought pattern to move from your current state toward a goal state, you're solving a problem. Problems can be generally broken down into two categories; well-defined problems and ill-defined problems. Well-defined problems have a clear starting and ending point, such as how to make it bright in a room that's currently dark. You know exactly what you're starting with and exactly how you wanna end up. Ill-defined problems, on the other hand, have a more ambiguous starting and or ending point, such as how to live a happy life. It's something you can still try to solve, but you may not know exactly what the outcome will look like. Because you face such a variety of problems each day from the simple to the complex, your brain has several different ways of solving problems. One method of solving problems is trial and error. This means that you just take random guesses until something finally works. For example, imagine you're trying to log in to an email account you haven't accessed in a while. You know that the password is eight numbers long, but you don't remember what it is. If you're using the trial and error method of problem-solving, then you would just try any random combination of letters and numbers until something works. As you can probably tell, that approach would take forever. And with trial and error, you're not necessarily keeping track of what you've already done, so you could get lucky and hit on the right password early, or it could take a very, very long time. A more methodical approach would be to use the algorithm strategy. An algorithm is a logical, step-by-step procedure of trying solutions until you hit on the right one. So if we're still trying to get into our email account, we might start with one, two, three, four, five, six, seven, eight, and if that doesn't work, we'll change one number at a time. Two, two, three, four, five, six, seven, eight, then three, two, three, four, five, six, seven, eight, and so on. This could still take a really long time, but you are guaranteed to find the right solution eventually. Usually though, we don't have time to try every possible solution, so a more common method of solving problems is to use some sort of heuristic. A heuristic is a mental shortcut that allows us to find a solution more quickly than the other two methods we've talked about so far. For example, you probably made a password out of familiar number combinations, so you might try something that includes your birthday or something else that stands out in your mind. This drastically reduces the number of solutions we need to try because it eliminates any unlikely ones right off the bat. Heuristics don't guarantee a correct solution, but they do simplify complex problems and reduce the total number of solutions that we'll try in order to get to a more manageable number. Since heuristics are so common, let's talk about a few different ones that work for different types of problems. One type of heuristic is means-end analysis. This heuristic means that we analyze the main problem and break it down into smaller problems. Then we attack the biggest subproblem in order to reduce the most difference between our current state and the goal state. If you're planning a trip to another country then, the biggest problem might be the distance between you and that country, so your first step would be to book a plane ticket. And that creates another group of subproblems, which you solve one at a time, usually starting with the biggest one. A second heuristic is working backwards. Now, with means-end analysis, we were trying to work from our current state toward our goal state. With working backwards however, you start with your goal state and use it to suggest connections back to your current state. This strategy is commonly used in mathematical proofs. Another example of it is if you've ever done a maze and started at the end and worked your way backwards toward the beginning. Let's try to solve another problem. What if I gave you these six matches and asked you to use them to draw four equilateral triangles? Go ahead, give it a try. If you had trouble solving that problem, you're not alone. Most people get stuck on thinking about this problem in a two-dimensional way. This act of getting stuck is called fixation. The answer, though, requires you to think about the problem in three dimensions. You need to create a triangle pyramid with the six matches in order to form four equilateral triangles. If you did solve that problem, try to think about how the solution came to you. You probably didn't do a series of step-by-step arrangements of matches, and the heuristics we've talked about don't quite work. What probably happened is something called insight, which is that sudden aha moment when the solution just pops into your head. Insight is tricky, it's hard to predict, and harder to encourage, particularly when you're fixated on seeing a problem from the same ineffective perspective. If you do get stuck on a problem, you can let it incubate, or just sit in your mind while you're not really thinking about it. Often, insight comes after a period of incubation. It's like when you're trying to think of the name of that actor in a movie you saw, but it only comes to you later that night after you thought you stopped thinking about it.