(Preface: I wasn’t sure where to put this, so I put it in GD because it is not so factual).
I’d like to know how a person could improve their logic and reasoning skills. I’ve been exposed to a lot of logic problems recently, at math competitions and also in studying for the SATs. Some are more math oriented than others. For example, some might be similar to this:
(I made that one up but I’ve seen very similar ones). This one didn’t involve any math.
A lot also involve math problems dealing with integers.
Example:
I’m guessing the best way is just to practice, practice, practice. Are there any other ways? Are there any great web sites or books on logic problems?
Logic (and by extension, math) is one of those things that is a lot easier to pick up if you want to. I think this is contrary to history or biology which, at lower levels, can be almost synonymous with memorization. But even liking it isn’t a perfect fix. I know lots of people who seem to exhibit great reasoning skills but can’t make the move to the symbolism.
I’ve seen plenty of books that attempt to introduce people to logic, but my advice (as someone who struggles now and again) is to see if you can find someone that knows it and can help. That allows them to shape their presentation to your terms specifically.
Well, it involved logic. We could symbolize it by trying to find the case where
A & B & ~C
is true (two truth tellers and one liar). As an example. (In this case, none of the three could have done it if there is only one liar; if there are two then it could be either of the two who proclaimed innocence but there is no way to be sure which.)
My strongest grasp of logic came from dealing with digital circuits, something that used Boolean algebra and something that I was interested in. When I try and tackle regular propositional logic or other kinds I’m not so strong and tend to make mistakes. Doesn’t stop me from trying, though.
Well, you need to practice, yes, but not just practice. Practice alone can actually tend to make one frustrated and give up. Finding someone who can explain it to you will make it go much smoother. Either a friend, or a personal tutor is, IMO, the best bet. They should be able to tailor the presentation to your needs much better than a class could. After you’ve become more comfortable, using books is easier because you can read them without too much effort. Being able to read a math book is not very easy, but getting there is helpful.
read the strategies in the SAT preparation book. i mean like for a multiple choice you might want to just plug each answer into the question and see if the result is consistent.
all questions on SAT math are easy, none of them require truly out-of-the-box thinking or complex computations. you just have to know the basic rules/formulas and there are only so many topics covered. an SAT prep book should have that info. like there is a lot of geometry, lots of questions on prime numbers or whatever.
practice is certainly a good idea, but its not about how many sample problems you attempt, its about understanding EXACTLY WHY you failed to solve a certain problem every time you fail. for every time you fail - you need to answer the question why. i wasn’t lucky enough is not an answer.
but questions repeat so much its amazing, i mean it may not be exactly the same but you are almost guaranteed to have some question about a right triangle that is a multiple of a 3,4,5 triangle or whatever. you have to go through maybe 3 - 4 sample SAT tests and you will get the feeling for which questions you will be asked. obviously its better to use the more recent SATs as samples.
I found that the first time I read through all the Sherlock Holmes stories, I was much more attentive to details and to connecting the dots between details.
I should offer a couple of qualifications to this. First, the conclusions that Holmes draws, while always accurate, are NOT always logical: he always makes the correct assumption even when the clues suggest other equally valid conclusions. But still. And, as far as a recommendation from me, who is sometimes UNreasoning in this forum, of course these skills are only good when you use them, and not went you don’t. (How’s THAT for reasoning?)
I find that an area of math that isn’t taught in school (that I am aware of) that will train your reasoning is discrete math. Discrete math is cool because you can use it for a lot of real world problems. I especially learned to reason by messing around with programming. Not saying that you have to program to reason, but it will help a lot, because every program you write makes you think logically about a problem. Try solving the problems found on Project Euler. Other ways to train your reasoning skills are to play abstract games such as chess, or better, model them with math, proving and reviewing proofs for math theorems, or studying physics or engineering. Like the guy above me who talked about boolean algebra said, it’s not good to only practice, but also contemplate the things you learn.
Even if the hypothetical stands as given, it can be true if there are two culprits: the party who admitted guilt and one of the parties pleading innocence. Then, one liar, two culprits. We know only that “a crime [has been] committed.” Perhaps it was a conspiracy!
Way back when (the rainbow was a new idea at the time) my parents got me the “Wff ‘n’ Proof” game. Now, screw the game; the damn thing is utterly worthless as a game.
But as a logic system? Very nice! I sat down and worked all the examples in the book, and thereby picked up a healthy working knowledge of conventional symbolic logic. (Also of “Polish notation,” which was also very useful when Hewlett Packard calculators were the hot new “in thing.”)
(Hey, I’m old. I used to punch IBM cards…by hand!)
There’s logic and then there’s reasoning. I would submit that the latter far exceeds the bounds of the former. So the question then becomes, which is really the one that is important to you?
To improve logic skills, one can study and practice using logic. There are many ways to do this, one “flavor” of logic study is called Boolean Logic, and Boolean Algebra.
This will be useful in understanding the basic logical operators, and how they work in combination (that’s the algebra part.)
I would suggest crawling into the great logical minds. Knowing the steps, memorizing the formulas, etc is all well and good but understanding the philosophy of logic, if you will, goes a lot farther in cementing your mode of thought in my opinion.
Read some Plato, Aristotle, Galileo, Descartes, Einstein and the other great minds.
I was tempted to reply, and then I noticed that this is a zombie thread from 2004. Oh well, I’ll reply anyway, just in case the OP is still around or in case anyone else is interested. Raymond Smullyan has written several good books of logic puzzles which are possibly the best of their kind.
There are several books out there about mathematical problem solving. The classic in the genre is George Polya’s book How To Solve It.
There’s a good series of lectures on mathematical problem solving available from The Teaching Company. (See if you can get it from your library, or, if you’re tempted to buy it from The Teaching Company, be aware that they regularly put their courses on sale for far less than the “regular” price.)
See the MAA Minute Math blog for a daily practice problem, with solution. (The problems there are in geometry, number theory, probability, and other areas of mathematics.)
And, a technique: Learn to break a question down into the possible cases, and consider each case in turn to eliminate those that don’t work out or lead to a logical contradiction. This is the approach I would take with both of the problems given as examples in the OP.
