Tips for Improving Your Problem-Solving Skills // Cambridge Student

hey everyone and welcome back to the next video in sort of the problem solving series that i'm making so in the last video i explained why you should be improving your problem solving skills and a few ways to go about doing so and in this video i'm going to be sharing some more general tips on why you should be improving and how you should be improving your problem solving skills so um this is going to be quite a long and rambling video uh there will be time stamps down below to skip to any pots um and i would recommend watching the whole thing and maybe at two times speed or something and i would also recommend bookmarking this video and coming back to it in a few weeks time uh when you are solving more problems and you're better able to appreciate some of the like information i'm going to be showing in this video and revisiting this video a few times during the year is like completely fine and yeah just a fun thing to do so i guess we'll just get started then so i will be

mainly focusing on math olympiads in this video and maths related competitions but like i said you can find competitions for like chemistry physics biology uh computer science astrophysics online and just by searching like astrophysics olympiad or computer science olympiad which is the informatics olympiad or even linguistics and this information sort of applies to all of them but i'm going to be focusing on maths because it's just easier for this video so my first bit of

advice is something to do with choosing which problems to solve now i struggled with this initially because i even chose problems which were way too hard and i was making no progress on or i ended up choosing problems which were way too easy and i just wasn't getting any value out of them now what i would recommend doing is choosing problems which lie just outside your comfort zone so you can sort of imagine your comfort zone as being some like kind of circular blob or something and if uh you do problems which are just on the edge of your comfort zone then doing those problems your comfort zone will gradually increase in size and then a problem which is like a way out over here as your comfort zone increases in size enough then that problem which is way out over there will gradually become within the comfort zone or just on the edge of the comfort zone so doing problems which are just outside your comfort currents comfort zone will improve your problem solving skills cause your comfort zone to increase in size and then make you able to do problems which you thought were no way possible now this requires a little bit of practice because you have to get good at recognizing problems which are just outside your comfort zone and for me um i sort of gained that with a few months or years of experience in that i would look at a problem and just feel like i can sort of do this or see where this is going but i'm not entirely sure and then i would like attempt the problem realize that there is some weird trick at play here or that i wasn't entirely sure and then i sort of have to figure that out but once i become sure and once i solve the problem then my comfort zone is just bigger and bigger in size and problems which i was not able to solve like a year before i am now able to solve so that's sort of the image i have in mind when choosing problems with which problems to solve i'd also recommend

being intentional about what you're trying to improve on and get better at and which specific topics because it is quite easy to just say oh i'm just going to do tons of problems and i'll just somehow get better i would recommend being intentionable and focus intentional and focusing on which problems that you want to be solving and which topics getting better at so let's say you're doing the senior maths challenge or you're doing a few past papers which if we pull up right now so you're on the ukmt website and you go to challenges and you see the senior mathematical challenge and now you can see we have um a few papers here so if we go to the online challenges and download past papers and let's say i start doing a bunch of papers and i bought one of the books to get the older papers as well and i find myself struggling on the later geometry style questions so let's say i'm particularly struggling on these sort of questions down here then i would focus on trying to get better at geometry style geometry questions uh from choosing the last say 10 questions in all the previous years of just geometry questions and then doing some of the solutions and investigations or let's say you were doing bmo1 so if i go to bmo1 then that's on the challenges and then british mathematical olympiad round one which you qualify for after getting a high enough score on the senior math challenge if i go to the papers here uh wherever they are uh which is over here and then it will take me to separate website and i can see all the bm01 past papers so let's say i was looking at the 2016 paper and i find myself really struggling to do some of the number theory problems um and i just was terrible at starting number theory problems then i would try to get better at number theory problems and perhaps get a textbook or some of the uk books and just focus on doing number theory for like a week or two and as you sort of develop your basic skills in number theory uh then you're much better able to like come to some of these problems and know exactly where to be starting uh it won't make the problem tremendously easy for you but i'll just give you like a rough indication of what kind of things you should be trying and doing by attempting some easier problems after learning some of the material in like the books basically what i'm trying to say is that you should consider what objective you have in mind and set yourself small goals and small objectives or what you want to be trying to improve on rather than just like printing out craps and problems and just doing all the problems and seeing what you get better at and wasting your time on like topics which you're already pretty decent at or pretty good at it's helpful to like pay attention to which parts you're struggling on the most and maybe you struggle to really start number theory problems or you struggle to like visualize a lot of the geometry problems or you feel like you just can't i don't know get started with certain ratios or whatever um try to really pinpoint these issues and then focus on what you can be doing so potentially it's trying easier problems on number theory or reading around some of the material so if you went back to the uk mt website and went to say shop then you can see some of the um like a mathematical olympiad companion and mathematical mp primer uh i think one is for one is for bm02 uh but if i went to senior then look an introduction to number theory so let's say it was like particularly bad enough fairy and i tried to do some bmo2 problems on number theory which is the second round for the british mathematical olympiad um then it would be worth reading a book on a theory or just trying a bunch of the problems from these books uh because that can be massively beneficial or even just like searching another theory uh brilliant have some pretty good uh pages related to it on their wiki and they also have some uh particularly good uh free things that you can be practicing um because the courses are paid and i did cover that in my video in brilliant which should be linked above somewhere right now uh but there are a lot of free stuff that you can be practicing on there as well and even on the auto problem solving so also problem solving have some really great books um as well which you can find and it's not a case of having to buy all the books and doing all the books um or something and then feeling like now that you've done all the books you can start like doing some of these olympiad uh past papers just start doing the olympiad pass papers or start doing the senior math challenge papers and then try to pinpoint what you should be focusing on and then like do extra material around that and then you will notice yourself improving so one of the books that many people use is the art of problem solving volume one and uh also volume two now this covers a lot of topics and some people feel like they should be doing the book cover to cover but i would not recommend that i recommend focusing on what you want to be improving and then focusing on actually like getting better at these particular topics now that just lead on

sending points on mixing it up a bit so if you've identified that number theory is like one of your weakest topics rather than doing like 20 30 number theory problems for like the next week or two um like it's much better to be able to mix this up with a few other problems because you're just going to get bored of doing the theory problems um or mix up with some problems that you also find quite difficult um but also enjoyable or like identifying two or three topics that you want to be improving on rather than focusing on like one particular topic for weeks on end because that's just gonna get way too boring now this leads

me on to another point of finding a few good resources and just sticking to them now from the beginning like when you're first introduced to problem solving um and like the fact that you should be actively trying to get better at it and you should be finding more problems that you can be doing it's quite easy to just like spend forever um like just finding new resources that you can potentially be using and kind of like weighing up which resources you should be using and which ones you shouldn't be using and now i was a victim of this as well because uh when i was like oh hey this is something really cool i want to find tons of problems and like find tons of websites which i can be using no it's like much better to just find a few good resources and just stick to that now i would recommend the ukmc websites and the ukmt resources as well um and like i said in my previous video if you can't afford some of the books um then just ask your school as well and they may potentially pay for it for you or your school library or your local library i would just recommend sticking to some of like say the uqmt resources if you're in the uk and even if you're overseas like uk is particularly good so if i go back to uk website and then i go to shop then i can see some of the books like that i could be using and rather than buying all the books i would just like focus on a few books like one or two books and just focus on getting really good at that so let's say i'm in like year 10 year 11. um i think a problem solvers handbook is a pretty good book for that because it's for intermediate olympiads and then i just like focus on getting through this book um rather than like buying a ton of books and um like being overwhelmed by the amount of options or if i really wanted to like make it to the mclaren mathematical olympiad then i would be focusing on getting a high enough score on the intermediate mathematical challenge and once i'm like comfortably hitting the score uh required uh year on year i think online you can find spreadsheets uh with all the previous boundaries as well just by doing a bit of googling um once i'm comfortable that i'm hitting those scores year on year then i can better a then i can move on to like the mclaren mathematical olympiad or if i find that too difficult then i can move on to like the kaley methods column here and just see the problems from there and just so on like i think the uk mt website has some really good progression here uh that you can be working through and some really good books and materials so just sticking to that is like completely fine um and then maybe supplementing it with a few other resources so the auto problem solving forms is particularly good because if you go to the auto problem solving forms you can create an account you can see a lot of like problems that people are posting in math contests and olympiads you can see what kind of discussions they having if you see a problem then you can like start like offer a solution somehow um or you can also post your own problems and yeah just like join the community and get better about everyone else at it so i think if i was restarting and i was doing maths olympiads then i would focus on the uk and t challenges and getting better at them and i would focus uh on say like the aops forms so i'd stick to like ukmt and aops and rather than like going through dozens of websites and just like spreading myself way too thin because that will just lead to me managing more websites and more resources than actually doing the problems themselves so also recommend using focus

and diffused thinking so focus and diffuse thinking or two modes of thinking is thinking which you have in mind so focused thinking is when you're focusing like in intently on one particular problem and like that can be say when you're just i don't know have your head down styling this problem trying to figure out what the hell is going on or like working through a bunch of algebra and not really getting anywhere and that diffuse thinking is where uh your mind is sort of left to wonder a bit more so you may have noticed sometimes when you're stuck in a problem and you get up to like go get a glass of water or you get up to like i don't know just do anything else or during a commute to school you come up with this idea or like sudden realization even though you weren't thinking about that in that thing in particular or let's say you're washing the dishes then you may suddenly come up with a solution to a problem or you may come to some like certain realization that's when you're in your diffused mode of thinking when you mind solve left to wonder because you're not focusing on one particular activity and why this comes in useful is that when you're sitting down and trying to work for a hard problem it can be really tempting to like just stay in focus drinking the whole time and not let your mind wander off into diffused thinking and it can be useful to get up every like 30 minutes or an hour or whenever you feel like you're not making any progress just to get up and then like wander around a bit and like just do some other stuff maybe watch some television maybe speak to someone and then come back to the problem and then maybe you will have like figured something out or something will have clicked in your mind during that time it may not always be the case but often it can be the case and when it is the case it feels like ridiculously good or even moving onto a different problem um which may potentially be easier and then coming back to the original problem you may find that your brain has sort of been working in the background and just figured something out there it's quite a difficult feeling to explain but like when it happens it's really good so do take breaks often and don't just spend three four hours out at your desk trying to like get through one problem even though it's just not working for you so

now my next point is about looking at solutions so this is somewhat related to what i mentioned before about focus and diffuse thinking um and basically don't look at a solution unless you've thought about the problem seriously enough and how long you wait before looking at solution for a problem that you can't do depends on how important you think the problem is to you so i usually try to wait around a day before looking at the solution but this may vary depending on like how important the problem is or how quickly you meant to be able to solve it like for the senior mathematical challenge problems like the first 10 15 problems don't like if you can't do them then look at don't spend like a day or two think about the problem look at the solution say like after an hour or two instead it just depends on how important the problem feels for you and how much it feels like you're getting anywhere like if you keep finding new techniques and new solutions um or new ways of approaching this problem which you feel like is taking you closer to solution then that may be a good sign and you should just keep at it and not look at the solution too soon because remember that once you've looked at someone else's solution then the problem has basically been ruined for you forever and you will no longer gain nearly as much benefits from it as you tried as from trying to do the problem yourself but if you do look at the solution then you will derive vastly more benefit from the solution having attempted the problem yourself and trying to come up with your own solutions which may not necessarily have worked um and then looking at someone else's solutions and realizing that this was a way to go about it or like this was one of the ways to go about it but it's a difficult balance to strike between uh not looking at solutions too soon and spoiling yourself and losing any learning opportunities um but not spending so long on one particular problem or a few particular problems that you just don't look at the solution for many days um because at that point they're just easier problems to try or just different problems to try and get better at um because if you're spending way too much way too long just getting nowhere with a problem it's a good indication that it's a little too off outside your comfort zone and you should potentially try an easy problem or a different problem or get a bit better at the underlying material like say what i mentioned with number theory before and uh learning a few number theory information to tackle easier problems also i think that solution videos are particularly good so if i go back to the bma1 websites then you can find a few solution videos so i think this is the news if i go back to papers then i can find some solution videos i swear they were here somewhere yeah so video solutions for vma1 are available from 2015 to 2016 onwards so if i click on some video solutions then i can see all the video solutions of just people explaining about how they tackle this problem so um like this one i can just watch this like full video about it and that's really beneficial because you can see someone working for the problem themselves uh thinking through the problem if you tackle this problem before yourself then you will have identified uh which parts of your understanding weren't particularly great um or which parts you sort of had gaps in um which is fantastic so i would highly recommend if you've managed to find solution videos for anything then do make the most of the solution video but do watch your pro do attempt the problem yourself and potentially even if you get the problem right then do like skim through the solution video just see if there's any of it interesting with solving the problem and any more tools to add into your toolbox for solving more problems like that also if someone else does present their solution to you then rather than just look at the solution seeing that you follow sort of nodding your head in agreement try to pinpoint any jumps that they're making which you feel like you weren't able to make and sort of ask them how did they make like the jump from say this initial statement to like this first i don't know algebraic line or something or like between these many steps like how do they make these connections in their head focus more on that rather than the actual like content of the solution itself focus on the jumps and connections that they've been making and try to ask some questions around that like how can i get this how can i uh get this solution myself or uh which part of the logic like some i don't know made sense here or made you think of like this other thing probing these sort of questions i think is vastly beneficial when you're learning from someone else's solution which i'm presenting to you but generally solutions i would recommend looking at hints if you're stuck with a problem and try to use those hints because often they're all hints or you can first use the first few lines of a solution as a hint to yourself or like for the video you can use like the first few minutes of the video to sort of uh see how they've set up the problem and then try to solve it from there also i would

recommend discussing any solutions with others so if i go back to the auto problem solving forms and i want to say hey open and clicked on one of the topics that someone was talking about then you can see the solutions that when people are proposed and often you can see some discussion surrounding it um and like just people come commenting on each other's solutions and then you can like even participate um in the description and see any follower problems or just even asking friends at school who are also interested in improving their problem solving skills if you're able to decide in doing a few problems between you and the same problems then discussing solutions with others can be vastly beneficial even just starting a club or getting one of your teachers to start a problem-solving related club at your school can give you a massive opportunity to meet other people in your school who also want to really get better at problem-solving and improve their problem-solving skills now

another tip i have is to like formulate a plan or formulate your own approach for how you tackle unfamiliar problems to begin with so there is a book called how to solve it and the wikipedia page gives a pretty good summary of the book but if you want to then read the book itself and it talks about a few heuristics and if you like questions you can be asking yourself and how to solve difficult problems or potentially finding easier problems to solve instead and this is a fantastic approach that you can probably like use as a baseline and then gradually um when you're tackling like a particularly difficult problem and it's increasingly hard to tell how far this problem is outside your comfort zone then it can give you a way to sort of like figure that out or find easier problems and then come back to this like in great center more challenging problem later so i would recommend sort of having this like plan or approach in mind that you slowly develop through doing many more

problems now i'd also recommend that you be consistent in solving so it's much better to say solve 100 bm01 and bmo2 related problems over the course of six months than it is say two months because you're going to have more time in between doing the problems to actually think about the solutions think about how much progress you're making learn a lot of new material and basically get much better problem solving during that time because you have more time between like say doing a problem every one problem every like two or three days you have much more time to be able to like really tackle the problems and like really try to improve your skills there so being consistent over the long run is going to be massively more beneficial and then like um trying to cram say 10 problems 10 1 problems in a day which is going to be insanely difficult to do unless you're already so used solving one problems in which case it wasn't worth you trying to do 10 problems in a day now how often you do problems depends on how difficult the problems are um because if you're doing like senior maths challenge problems then you can do like say a dozen or so in a day or like a little less and then you can do like even more of the weekends uh whereas if you're solving like bmo2 problems then that can be much more difficult to solve and you may only find yourself being able to solve one two problem like every two or three days it really depends on the problem but just notice that you're pushing yourself hard but not too hard because you don't want

to be burning out but on a more final point about being consistent and sort of considering the long term aim here because your long term aim could be to make it to the training camp or onto uk's team for like the international mass olympiad or the international physics olympiad or whatever um remember the most important thing is like you're having fun along the way and life needs to have a little bit of fun in it and you don't have to be spending your every waking breathing moment trying to be improving your problem-solving skills even many of the people who i met um here at cambridge who've made it onto international olympiad teams like they still had hobbies they still like watched television watched movies hung out with friends played games they basically still had a sort of normal like teenage experience and stuff um but they also like just had a bunch of fun solving a bunch of interesting problems on the weekends and in evenings and just having a lot of fun like learning more maps and discussing with others and like being able to do this during a lesson enjoying school and actually enjoy the lessons much more and discuss it with their teachers or they even made many friends through online communities such as the awesome problem solving forms um or joined discord servers and made friends and like discuss problems with people on there and they formed like group chats with friends and they just like sold problems together or went to math camps and just everything related to that because that is like part of the whole experience so part of the phone as well you don't have to be like stuck yourself thinking it's a solo journey and you don't have to interact with anyone because you're going to be like ruining it for yourself where you're going to help them get way too much better and they're going to like get a high score than your next competition like it's all a bit of fun at the end and you can make a bunch of friends and you can like join these communities and be able to like play online games with them and i know many of my friends they play like i don't know league of legends and among us and other games with a lot of the people who are also insult interested in solving england style problems um so yeah it's all just a bit of fun like you don't have to take this away too seriously it's quite beneficial to sort of take it a bit seriously and um because it means like university or like a levels and just your exams are going to be so much easier in comparison and it's quite a transferable scale so many of the people who i know who are really good at problem solving they happen to do well across the board in their gcse and a levels because it's a very transferable scale so it does help to take it a little seriously but taking it way too seriously is just like um not really too beneficial so basically life isn't about trying to get killer problem solving skills improving your problem solving skills is sort of adds to the fondness of life and having the surrounding community and being able to make friends and sort of bond over like these interesting problems and be able to solve them together and like sort of share that struggle in like i don't know having something click in your heads or trying to explain solutions to other people like all of this is part of the phone and adds to like just having a kind of better teenage life experience overall and not being like a board in class like wondering when i don't know secondary school or sixth one would end because you're distracted and you're enjoying uh tackling these interesting problems and you're enjoying the community surrounding it as well so anyways that's

basically from this video those are most of the tips i had in mind uh do check out the other videos as part of the series there should be a playlist in the description and i will be adding to that playlist over time um and finally i would just recommend like i right now i'm like too old to get involved in some of these competitions and i realized it way too late uh how much the benefits i could have gained growing up by getting involved these competitions so i'd highly recommend getting involved in these competitions and hopefully it will help you enjoy school more you'll meet a bunch of interesting and amazing people along the way and you will just get better problem solving skills which is a massively beneficial throughout like the whole of life so that's it for the video i guess i'll see you next time bye