The Art Of Problem Solving Vol 1 The Basics

Working Through The Basics Without Losing Your Mind

The Art Of Problem Solving Vol 1 The Basics is honestly one of the few math competition prep books that doesn't waste your time, though it's still going to feel like work. I picked it up around 2008 when I was trying to get someone ready for AMC 10, and I kept coming back to it myself because the problem sets actually teach you how to think about unfamiliar problems rather than just drilling procedures. The book covers algebra, counting, geometry, and number theory at a level that sits somewhere between standard high school curriculum and full-on olympiad training. It's split into two major sections. The first part walks through fundamentals of algebra and counting with increasingly difficult example problems. The second part dives deeper into geometry and number theory. Each topic starts with a conceptual explanation, moves into worked examples, and then gives you problem sets that range from medium to genuinely hard. The problems are where the real value lives. Reading the explanations alone won't do much for you unless you're already comfortable with the material. Here's something most people miss about this book. The authors intentionally sequence problems so that the early ones in each set teach you a specific technique, but the later ones require you to combine techniques from completely different chapters. I remember working through a number theory problem around page 280 that looked like it belonged in the algebra section because it demanded you set up a Diophantine equation using methods from Chapter 3 before applying modular arithmetic from Chapter 6. That cross-chapter thinking is exactly what AMC and AIME questions test, and the book forces you to practice it repeatedly without making it feel like a separate skill.

How To Actually Use This Book Instead of Just Reading It

Most people buy this and go through it like a textbook, which defeats the purpose. The book was written for people who already have basic math skills and need to develop problem-solving stamina. If you're struggling with the algebra from a standard curriculum, start somewhere else first. The book assumes you're comfortable with equations, factoring, and basic functions. Here's my actual process when working through a chapter. I read the concept section once without stopping. Then I attempt every example problem on my own before looking at the solutions. The examples aren't graded problems, but they're still harder than textbook examples, and skipping them means you'll hit the exercise sets and realize you don't understand what just happened. After the examples, I do the problem sets in order. When I get stuck on a problem, I leave it for at least an hour before checking the solution. Not immediately. The struggle is where learning happens, and if you pivot to the answer key too fast, you train yourself to give up rather than develop persistence. I had a specific issue with the geometry section where I kept making the same angle-chase mistake involving inscribed angles and central angles. I'd draw the diagram, identify what I needed, and somehow mix up which arc each angle subtended. I realized I was memorizing theorems without actually visualizing the relationships. The workaround was to stop solving problems for two days and instead redraw every diagram in the chapter from scratch, labeling every arc and central angle pair explicitly. Once I could see the connections spatially instead of just symbolically, my accuracy improved dramatically. That's not a general study tip, that's specific to how this book structures its geometry problems.

The Downsides Nobody Talks About

The book has real limitations. The difficulty curve is steep and sometimes inconsistent. You'll finish a chapter feeling confident, then immediately encounter a problem set where maybe two or three questions feel accessible and the rest seem designed to humiliate you. That's intentional, but it can be demoralizing if you're not prepared for it. The solutions section at the back is sometimes brief in ways that leave gaps, particularly in the counting section where the authors skip steps that seem obvious to them but aren't obvious to anyone encountering the technique for the first time. Another problem is that this book doesn't cover everything on modern competitions. The AMC 10 now includes some trigonometry and coordinate geometry topics that get light treatment or are omitted entirely. If you're specifically preparing for the current AMC 10/12 or AIME, you'll need supplementary material. I used AoPS forums and older competition archives alongside this book to fill those gaps. For pure foundational problem-solving skills though, it's still hard to beat.

What To Expect Time-Wise

Working through the entire book properly takes maybe 80 to 120 hours if you're doing the problems seriously rather than skimming. That's not a casual read. If you're tackling it alongside regular schoolwork, plan on three to six months depending on your schedule. The counting and combinatorics chapters alone can eat two or three weeks each because those problem sets are dense and each question can take 20 to 45 minutes depending on difficulty. If you just want the download link and are looking to print it out, the book is officially published by AoPS and available through their store. There's a printed edition and a Kindle version. I'd recommend the printed version because you'll be writing in it, drawing diagrams, and flipping back and forth between pages constantly. The Kindle version works fine but the experience is noticeably worse for a book this interactive. One more thing worth noting about the number theory coverage. The book introduces modular arithmetic and Diophantine equations well, but it doesn't go deeply into advanced topics like lifting the exponent lemma or quadratic reciprocity. Those show up in later AoPS volumes and in actual olympiad problems. If you finish this book and feel hungry for more number theory, that's normal. It's designed as a foundation, not a complete treatment. That's by design, and it works if you accept that limitation upfront instead of expecting it to solve every problem you'll ever see on a competition.