How To Figure Out Math Proofs On Your Own

In this video I provide several strategies that you can use in order to figure out proofs. Note that this is a response to an email I received from a subscriber. Do you have any advice for people? If so, please leave a comment below.

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[joef]

Best advice I have so far (still a student myself): As you read or study, if you don’t know why something is true, make it a lemma and prove it, anytime a question pops into your head, make it a lemma and prove it, any time the author says something is clear and it isn’t, make it a lemma and prove it. When taking proof based math classes, you should have a notebook that ends up getting cover to cover front and back covevered in lemmas, examples and counter examples, separate from hw and exams study. That’s why, in my opinion, grad school is so hard for people. There were a lot of things you probably took as true, but you never actually figured out why, you just believed it. Believing without seeing was a very hard habit to break. One line at a time friends, and thank you for your videos sir!

[amydebuitleir]

Another tip: when you get stuck, before you look at the answer, ask yourself if there's some assumption or intermediate result that, if true, would enable you to solve the problem. That will help you improve your analytical skills. Also, when you look at the answer, you may discover that you were almost there; you were just missing the one piece. That will boost your confidence.

[Cambo]

There's a lot of great advice in this video and this comment section. One thing that I have struggled with and still do when faced with hard problems is a feeling that I'm stupid or other people in the class are understanding the material so much easier than I am. This often leads to wanting to give up. I think that it's really important to realize that this negative self talk is not based on reality and is mostly made up by your imagination.

[dhickey]

Totally agree with MS on using the material help solve the problem. Also, rather than spend endless hours on one problem, come back to it after a break of several hours or even a day. I often filter out key information or make a wrong assumption at first glance. Many times I can return with a fresh perspective and solve the same problem easily. Best of luck

[magvargas]

As someone who's repeating their first year in their math major (and someone who's slowly getting the hang of coming up with proofs) I would say that most of the tips I would've given to myself two years ago are things that you have already said: understand the definitions (this was something important to me, since I thought I understood them at first glance when it wasn't the case), look up previous things that have been mentioned before, such as theorems, propositions and definitions and apply them to your problem and reflect on the solution that are already given (or the one you just came up with.)

Another thing I would have told myself (similar as looking up the solutions to proofs and summarizing them in your own words) is to look up and also summarize the proofs given in the books you're reading. This will give you a feeling on how proofs should be written, when can certain proof techniques be used and also keep your thoughts organized when it comes to a specific problem.

Finally, sometimes the issue I had in not being able to solve a problem is that I wouldn't even understand what the problem was (as in, I didn't understand what exactly it was that I wanted to prove.) What I had to do in order to get over this hurdle was to relearn the habit of writing down what I already knew about the topic of the problem and the goal I wanted to achieve solving this problem (regarding definitions, propositions, etc. Sometimes I write these using the logical quantifiers and symbols to get an overview of the problem.)
This habit now helps me get hints on how to solve a specific problem and it's also something my high school physics teacher was very adamant on so we could solve physics problems easily (these would mostly involve equations and numbers instead of definitions and concepts, but I have to say that in the end the results are similar in regards to developing problem-solving abilities.)

I hope these tips are helpful!!

[KMMOS]

There is a type of math course for majors to show how to do proofs. Often it is taught in college sophomore year after calculus and before abstract algebra. The textbooks for those kinds of classes often have a variety of proof ideas and techniques. Search for proof methods, mathematical logic, and discrete math to get a variety of answers. Look for well-reviewed and highly rated books, then look for used copies of these texts. Most of these ideas are not brand new, so cheaper, older books may be OK for logic and proof methods.

[Adam-cnib]

Having a strong background is always vital. Without decent algebra skills you're gonna suffer in calculus. That applies to number theory too. Also, make sure you don't spend 3 hours on one problem. Take a 5-10 minute break whenever you get stuck for too long (30-60 minutes of struggle). Ponder on the problem on a high level. Let your mind free to find the out-of-the-box solutions for you.

[rupen]

The part about thinking how you could come to the answer on your own is really important. It applies to other things too: I often use it when solving chess puzzles or watching someone solve sudoku/pencil puzzles.

It's also important to realize the path for solving is often different from the path of understanding the problem and finding the proof. A lot of times, it's completely backwards. You may start with "ok, this is what I want to prove. What would have to be true for my conclusion to follow? And what would have to be true for _that_? What are some _consequences_ of this conclusion? Maybe that consequence is an "if and only if", so it's also a prerequisite and j could just prove that the consequence is true". Completely backwards or even wrong from a logic standpoint, but very useful in practice.

[shahirabdullah]

I am so glad I found this video. Also grateful to that person who asked this wonderful question. I am going through almost a similar kind of problem. I am doing a lot of problem solving but I feel like I haven't reached that level where answers or hints would naturally come to your head. Solving lots of problems or seeing the answers won't necessarily allow you to reach that level cause it's not about solving similar problems, it's something else. I couldn't figure out what that "something else" is till now.
Thank you so much. Both of you.

[ranjits]

Teach someone, teach, teach and more! I can't tell you how much trying to teach someone has helped me figure out things. Concepts that I thought I knew, really gave me interesting twists when I would teach, and slowly but steadily the CONCEPTS would solidify. Just giving free lessons to folks has helped me immensely in the learning journey!

[henriquel]

Understanding how it works is more important than memorizing it

[samuelp.]

One thing you can always do while studying (not only doing exercises and problems but also when reading the theory) is to try to play spontaneously with the theory before attacking the problem set. This freedom will allow you to discover small but interesting facts, which might be key to the solution of some subsequent problems.

[andrewdias]

My obvious but not obvious advice would be to be methodical. I always start with a literal written list of things I know (givens, assumptions, definitions) and then write a mathematical expression of what I'm trying to prove (eg: proving a number n is even means showing that n = 2k for some integer k). From there, I often let the definitions lead me, constantly referring to the "end goal" as a lodestone. Also, working backwards can help (starting with your conclusion and then see if you can create a backward path from that statement) - I often use the analogy that mazes are often easier to solve backwards.

[MrCliverlong]

I suggest in addition to everything the Math Sorcerer has said (particularly understanding the provided solution - it may leave out some key points) - is after writing out the solution from memory is (i) make a note of the part of the solution that you didn't find yourself - if you have really tried the problem the key step will leap off the page and (ii) go back to the problem a few days later - certainly no longer than one week - and try the problem again and (iii) once you have mastered the original problem find a very similar but different problem then try that. The part you got stuck on will eventually become clearer in your mind and become a key technique when you see a similar problem in the future. Such an approach takes time and effort.

[abdullah]

Had a professor recommend learning a theorem prover for proofs, using it now for solving a good chunk of proofs as a PMATH major, it will definitely be worth it.

[petrabassey]

thank you for this video! i’m in geometry and often get hung up and even skip proofs; i know i’ll have to face them eventually so this motivates me! continue making videos!! God bless you and have a wonderful day ! 😄😄

[pichirisu]

Math proof advice: ignore math proof rules and just use philosophy proof rules. Same destination, different road.

[brrrrrrruh]

You grow not at the moment you solve the problem, but during the hours you are trying to solve it. It's completely okay to be trying to solve a hard problem for a week sometimes.

Upd: and never read the entire solution. Read it until the "ahh, that's the next step I haven't tried", and then try to finish it yourself.

[CGExp]

Not the big brother we deserved but the big brother we needed.

[GarryBurgess]

The camera isn't quite focused enough for me to read the titles of the wonderful books that you have in the background, although I know from other videos what to expect since you went through them. Still, it's fun to see them.