How to Solve It

your brain solves math problems all day long without you noticing. that quick mental math when you pick the shortest grocery line or eyeball the best spot for your couch? the same kind of thinking helped sea traders calculate their routes and astronomers predict eclipses. every time you portion out a dinner bill or pack a car trunk efficiently, you're using skills that mathematicians have studied and refined for centuries. 

this chapter explores how to solve it – a 1945 masterwork that remains vital because its insights are timeless. you'll learn practical ways to make better decisions and cleaner plans. by the end, you'll spot solutions that most people miss and break down tricky situations into simple, solvable pieces. the same strategies that mathematicians use to crack complex equations can help you sort out everything from career choices to daily puzzles – and you'll get to see exactly how that works.

ready? then let’s get started.

a key trick in solving problems – whether you’re working with numbers or real-life situations – is one that many people skip right past. before you do anything else, stop and look carefully at what’s in front of you.

think about a student looking at a geometry problem about finding the length of a rectangular room’s floor. sure, they could grab their calculator right away. but people who are good at solving problems will pause and ask three basic questions: what are they trying to find? what facts do they already know? how do these pieces fit together? just by asking these questions, what seemed hard becomes much clearer.

looking closely at a problem works like putting together a jigsaw puzzle. you wouldn’t just grab random pieces and try to force them together. you’d look at the picture first, find the edges, and sort out pieces with similar colors. this same idea works for any problem you’re trying to solve. look at where you want to end up, see what you’re starting with, and figure out what rules you need to follow.

when people are teaching this to students, many want to jump in and give answers right away when someone’s stuck. but asking good questions often works better: “can you tell me what the problem's asking for?” “what do you notice about these numbers?” “does this remind you of anything you’ve solved before?” these aren’t just good teaching questions – they’re exactly what you should ask yourself when solving problems.

here’s a simple example that shows why this works so well. say you need to draw a square inside a triangle. the bottom corners of the square have to touch the bottom of the triangle, and the top corners need to touch the triangle’s sides. instead of drawing lines randomly, someone who’s good at solving problems will first picture how it should look. they’ll think about what makes a square special (sides that are the same length, corners that are perfect right angles), and how this fits with the triangle’s shape. once you understand these pieces, finding the answer becomes much simpler.

taking time to understand requires stopping yourself from jumping in too fast. yes, you might want to start working right away – but that's like starting a trip without checking where you're going. when you spend time understanding first, solving the rest of the problem becomes much easier.

make it a habit to stop, ask questions, and really look at what you’re dealing with before trying to solve anything. this turns problem-solving from guesswork into a clear path forward.

after you’ve seen what the problem’s all about, you need a smart plan to solve it. this is what sets great problem-solvers apart from people who just try things at random. good planning makes the difference between getting stuck and finding your way through smoothly.

this planning process often starts with looking at your past experiences. picture sitting down for a chess game against someone who always beats you. before making your first move, you might think back to your past games – what moves got you in trouble, and what strategies almost worked? this way of thinking helps with any problem. your brain builds up a collection of patterns and solutions you can use when facing new challenges.

these patterns become especially powerful as you gain experience. people who solve problems well do this naturally. say they need to find the middle point of a pyramid shape. they might notice it’s similar to finding the center of a triangle, which they already know how to do. by spotting this connection, they turn a tricky 3d problem into something more familiar. the trick isn’t just about remembering similar problems – it’s about seeing how problems connect to each other, and building bridges between what you know and what you’re trying to learn.

of course, this pattern-matching is just the beginning. making a plan doesn’t mean figuring out every single step before you start. think of it more like planning a climb up a mountain. you pick out some major points along the way, but stay ready to change your route if you hit a tough spot. sometimes you can’t see the best path until you’ve started moving.

this flexibility leads us to another powerful tool in problem-solving: finding similarities to other situations. when math experts first tried working out the size of 3d shapes, they used what they knew about 2d shapes. they saw how ideas from flat shapes could help them understand solid ones. you can use this same trick – look for problems that work the same way as yours, even if they seem different on the surface.

balance becomes crucial at this stage. planning takes a careful touch. plan too much, and you might miss good opportunities. plan too little, and you’ll waste time going nowhere. try to make a plan that keeps you on track but lets you try new ideas. it’s like having a good map for a trip – detailed enough to stop you from getting lost, but with room to explore interesting places along the way.

all of this preparation pays off in the end. don’t rush into solving the problem right away. take time to look at different ways to solve it, and trust that good planning will help you find a better answer. the plan you make now will guide you through all the tricky parts ahead, making your path to the solution clearer and more certain.

now that you have your plan, it’s time to put it into action. this next phase is all about careful execution – turning your strategy into precise steps while checking your work along the way. taking notes as you go helps turn your ideas into real steps forward, making sure each piece works on its own and fits with everything else.

to put this approach into action, start by breaking big problems into smaller pieces to check your work as you go. if you’re working on a geometry proof, you’d check each connection before moving to the next one. with algebra, you’d make sure each step makes sense before continuing. it’s like building something complicated – you need to know each part is solid before adding more on top.

this careful checking gets even better when you look at your work in different ways. when solving a big equation, try checking each step using different math tricks. if you solved something with geometry, see if you can prove it with algebra, too. if you got a number answer, draw it out to see if it looks right. this doesn’t just catch mistakes – it often shows you new connections you didn’t see before.

beyond these individual checks, your solution needs to make sense when you look at all the pieces together. in geometry, your answer should work no matter how you turn or resize the shapes. in algebra, plug your answer back into the original problem to make sure it works. think of it like a detective checking a story – the truth comes out when you ask questions from many different angles.

all of this checking needs good record-keeping, which helps in two ways. first, it keeps you on track while you work. second, it creates a map showing which paths worked and which didn’t. later, when you face similar problems, you can look back at your notes to see what worked best. sometimes, just writing down your steps shows you patterns you wouldn’t have noticed otherwise.

following this organized approach leads to faster solutions over time. as you spot patterns, you find shortcuts and cleaner ways to solve things. you might notice that several similar problems all work the same way, showing you a faster method to solve them all. what seems like extra work in writing things down actually saves time in the long run.

to bring it all together, the last step is stepping back to look at your whole solution. this big-picture view shows how all the pieces work together, whether there might be an easier way, and how you could use similar ideas to solve other problems. like a mathematician looking over a finished proof, this final check turns one solution into lessons that help with future challenges.

moving beyond just checking our work, let’s look at some powerful techniques that can help solve really tough problems. these methods – breaking problems into pieces, finding connections to similar problems, and looking at broader patterns – can turn what seems impossible into something you can handle step by step.

here’s a common situation: students get stuck on a hard geometry problem until they learn to break it into smaller parts. instead of trying to solve the whole thing at once, they work on simpler related problems first. take finding the diagonal of a complex 3d shape – you might start by working out the diagonals of its flat faces. each smaller problem becomes a step toward solving the bigger one.

this connects to another useful trick: seeing how different math ideas mirror each other. the way a circle has a radius from its center to its edge works just like how a sphere has a radius from its center to its surface. a cylinder’s volume relates perfectly to the flat area of its base. seeing these connections lets you use what you already know to tackle new problems.

sometimes, oddly enough, making a problem bigger makes it easier to solve. when you’re stuck on a specific problem about triangles, try thinking about how it would work for any shape with straight sides. often you’ll spot patterns you couldn’t see before – like stepping back from a painting to see the whole picture instead of just one small part.

but you can also learn a lot by going the other way – looking at specific examples. if you’re working on an idea that’s supposed to work for all triangles, try it first with the simple ones: triangles with equal sides, or triangles with right angles. these simple cases often show you important clues about how the bigger idea works.

these different approaches work best when you use them together. making a problem more general might help you see connections to other problems, while specific examples might suggest ways to break the problem into pieces. each method helps the others work better, giving you more tools to solve problems.

once you get good at using these approaches, you’ll start seeing math problems differently. each problem becomes a chance to find new patterns and connections. these techniques don’t just help you find answers – they help you see how different parts of math fit together, showing you patterns that connect ideas that seemed completely different before.

we’ve looked at different strategies for solving problems, but there’s one more crucial element: what happens in your mind while you’re working. even if you know all the right techniques, the way you handle the mental side of problem-solving can make the difference between giving up and finding the answer.

when we think about solving tough problems, we often imagine sudden flashes of insight. but real progress rarely comes in one big “aha” moment. most of the time, you build understanding bit by bit, through attempts that don’t quite work. a mathematician might spend days trying different approaches that fail, but each attempt teaches something new. it’s like a sculptor slowly chipping away at stone – the pieces you remove help show you the final shape.

this kind of patient progress requires a special skill: handling frustration. your brain has natural cycles of being sharp and being tired, and knowing these patterns helps you know when to take breaks. often, a problem that seems impossible becomes clear after you step away for a while. these breaks aren’t just rest – they give your brain time to sort through ideas and make new connections.

speaking of brain patterns, your mathematical gut feelings develop in an interesting way, too. at first, you get hunches based on patterns you’ve seen before, but you need to check these hunches carefully. as you get better at math, these gut feelings become more trustworthy. but even expert mathematicians stay careful – they use these feelings as guides, not guarantees.

your body and mind will tell you when it’s time to pause: your work starts feeling mechanical, you keep thinking the same thoughts, or you feel frustrated without new ideas. many mathematicians set specific time limits, knowing that pushing too long usually doesn’t help. often, new ideas come after you’ve done something completely different for a while.

this brings us to a fascinating aspect of problem-solving: your unconscious mind does a lot of the work. people often get breakthroughs at unexpected times – while walking, showering, or just waking up. this happens because your brain needs time to reorganize information and see new connections. many famous mathematicians solved big problems during breaks, after they’d already spent lots of time working hard on them.

all of these experiences contribute to getting better at spotting patterns. what matters isn’t how many problems you solve, but how well you think about each try. good problem-solvers keep notes about what they tried, even when it didn’t work. these notes help them see patterns in what works and what doesn’t.

this whole process leads to building real confidence – the kind that comes from understanding that math discoveries have their own rhythm. sometimes you might feel stuck for a while before making a breakthrough. understanding this helps you keep going through tough spots, even when you can’t see progress right away. getting really good at math means accepting this process, knowing that every struggle helps you understand things more deeply.

in this chapter to how to solve it by george pólya, you’ve learned that great problem-solvers combine methodical thinking, pattern recognition, and mental resilience to crack any challenge.

the journey begins with deep observation before acting, then moves through careful planning and systematic execution. by breaking problems into manageable pieces and connecting them to familiar patterns, complex challenges become solvable. understanding your mental process – including how to handle frustration and when to step back for clarity – transforms the way you tackle every problem. these proven approaches turn challenges of any size into clear, conquerable steps.

okay, that’s it for this chapter. we hope you enjoyed it. if you can, please take the time to leave us a rating – we always appreciate your feedback. see you in the next chapter.