Index
Every link below was fetched before this site shipped. Contest pages on the Art of Problem Solving wiki sit behind a browser check; they were opened in a browser and the problem pages exist. Paid books are marked. If a famous book has no link, I could not verify a live page, so it is named on the topic page with an ISBN and no URL.
| Topic | Tier | Resource | Cost | Why this one |
|---|---|---|---|---|
| AP Precalculus | Tier 0 | Khan Academy, AP Precalculus | free | Watch once, do not study from this. It is the official unit map, useful for seeing the shape of the course, and a bad place to spend your evenings. |
| AP Precalculus | Tier 1 | Stitz and Zeager, Precalculus, version 4−ε (15 January 2024) | free | Free legal PDF from the authors. This is the book you work. OpenStax Precalculus 2e is the alternate if you want a second explanation of the same material: https://openstax.org/details/books/precalculus-2e |
| AP Precalculus | Tier 2 | Art of Problem Solving, Precalculus | PAID | The competition-flavored textbook. Paid. Use it after Stitz and Zeager, not instead of it. Alcumus at a hard rating is the daily drill: https://artofproblemsolving.com/alcumus |
| AP Precalculus | Tier 3 | Po-Shen Loh, live contest index, and the AMC 12 / AIME archives | free | Real MAA problems, sorted by concept, with difficulty ratings. The AoPS indexes, opened and checked in a browser, are https://artofproblemsolving.com/wiki/index.php/AMC_12_Problems_and_Solutions and https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions . Learn Anything has no precalculus page. |
| Precalculus, competition-hard | Tier 0 | Paul’s Online Math Notes, Algebra | free | Read once, do not study from this. Skim the section list so you can see which algebra moves you still hesitate on. Then close it and go to the book. |
| Precalculus, competition-hard | Tier 1 | Stitz and Zeager, the complex, trig, and conic chapters you rushed | free | Same free book as the previous page. The hardening is in which exercises you choose: every proof-style exercise, every complex-number section, every inequality. |
| Precalculus, competition-hard | Tier 2 | AoPS Precalculus, plus Alcumus on hard | PAID | Paid textbook aimed at contests. Paired with Alcumus: https://artofproblemsolving.com/alcumus . Better than another college precalculus book because the problems are the point. |
| Precalculus, competition-hard | Tier 3 | AIME problems and Sharygin’s plane geometry | free | AIME is the ceiling for precalculus technique. Sharygin, for the coordinate slice only: https://archive.org/details/SharyginProblemsInPlaneGeometryScienceForEveryone . Gelfand’s problem books are the right spirit; legal free downloads were not verified, so they are not linked. |
| Proof and Logic | Tier 0 | Hammack, Book of Proof, the landing page | free | Read once, do not study from this. Read the description and the table of contents so you see that counting, induction, relations, and a chapter of ε–δ are all in one book. Then close the page and open the PDF. |
| Proof and Logic | Tier 1 | Richard Hammack, Book of Proof, 3rd edition, revision 3.4 (5 February 2025), ISBN 978-0-9894721-2-8 | free | Free legal PDF, Creative Commons, AIM open textbook. This replaces Velleman as the working text because you can start tonight. Velleman, How to Prove It, ISBN 978-1-108-43953-4, is the paid book to buy if you want a second, slightly stricter pass. The ISBN was confirmed against the Open Library catalog; a stable publisher page was not. |
| Proof and Logic | Tier 2 | Hammack, every exercise in Chapters 1–14, written | free | The book is the problem set. Do not use the solution section until a writeup exists. Munkres’s Chapter 1 exercises are the harder set-theory drill once Hammack feels short; the book is paid, ISBN 978-0-13-181629-9 for the 2nd edition catalog entry, and the topology page sends you there properly. |
| Proof and Logic | Tier 3 | Kedlaya, The Putnam Archive | free | The public ceiling for timed proof-writing. Enderton, A Mathematical Introduction to Logic, ISBN 978-0-12-238452-3, is the paid logic ceiling; no free copy was verified, so it is named and not linked. |
| Single-Variable Calculus | Tier 0 | 3Blue1Brown, Essence of Calculus | free | Watch once, do not study from this. It shows what a derivative is buying you. It does not make you able to compute one under pressure. |
| Single-Variable Calculus | Tier 1 | APEX Calculus, together with MIT 18.01SC | free | APEX is free and complete; the download page is https://www.apexcalculus.com/downloads . MIT 18.01SC is the lecture, problem set, and exam spine: https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ . Stewart is the usual paid textbook; no edition is pinned here. APEX plus 18.01 is the better working pair because the exams are public. |
| Single-Variable Calculus | Tier 2 | Demidovich, Problems in Mathematical Analysis (Mir, 1970) | free | About three thousand problems, ordered by technique. Apostol, Calculus, Volume 1, ISBN 978-0-471-00005-1, is the paid proof-flavored alternate. Do Demidovich for speed and Apostol when you want the theorem proved before the drill. |
| Single-Variable Calculus | Tier 3 | Spivak, Calculus, ISBN 978-0-914098-91-1, and Putnam analysis problems | free | Spivak is paid; the ISBN is the catalogued one for his Calculus. Its end-of-chapter problems are the English-language ceiling for a first course. The 2024 Putnam paper is one concrete ceiling exam; pick the problems that are limits, inequalities, or series. Keisler’s free infinitesimal book is a side door, not this ceiling: https://people.math.wisc.edu/~hkeisler/calc.html |
| Linear Algebra | Tier 0 | 3Blue1Brown, Essence of Linear Algebra | free | Watch once, do not study from this. It is also the one Learn Anything linear-algebra link that earns the preview slot. Close it before you touch a problem set. |
| Linear Algebra | Tier 1 | MIT 18.06 (Strang) and the Georgia Tech interactive text | free | 18.06 is the course: lectures, problem sets, exams. The free book you can read beside it is Margalit and Rabinoff, Interactive Linear Algebra: https://textbooks.math.gatech.edu/ila/ . Strang, Introduction to Linear Algebra, ISBN 978-1-7331466-7-8, is the paid book the course tracks; the MIT page is https://math.mit.edu/~gs/linearalgebra/ . The free pair is better as a daily diet because the exams are public. |
| Linear Algebra | Tier 2 | Axler, Linear Algebra Done Right, 4th edition, and Treil, Linear Algebra Done Wrong | free | Axler’s PDF is free from the author; ISBN 978-3-031-41025-3. Treil is the determinant-friendly proof book: https://www.math.brown.edu/streil/papers/LADW/LADW.html . Use Axler for the spectral theorem and Treil when you still need to compute a determinant without shame. |
| Linear Algebra | Tier 3 | Borisov’s topic-sorted Putnam list, plus Matoušek’s miniatures | free | The PDF sorts Putnam problems by subject, including linear algebra. Matoušek, Thirty-three Miniatures, is a free PDF of serious applications: https://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf . Prasolov’s problem book was not verified as a stable free file, so it is not linked. Gelfand’s Lectures on Linear Algebra is the right Soviet companion and is paid; no free legal copy was verified. |
| Multivariable and Vector Calculus | Tier 0 | Paul’s Online Math Notes, Calculus III | free | Read once, do not study from this. Use the section list as a map of topics. The notes are a reference you already know how to find, not the course. |
| Multivariable and Vector Calculus | Tier 1 | MIT 18.02SC and OpenStax Calculus Volume 3 | free | 18.02 is the lecture and exam path. OpenStax Volume 3 is the free book: https://openstax.org/details/books/calculus-volume-3 . Marsden and Tromba, ISBN 978-1-4292-1508-4, is the paid text if you want a tighter one. Schey, ISBN 978-0-393-92516-6, is the short paid supplement for the field theorems. |
| Multivariable and Vector Calculus | Tier 2 | 18.02 problem sets and Apostol, Volume 2 | free | Do the 18.02 sets before you shop. Apostol, Calculus, Volume 2, is the paid harder text; the Volume 1 ISBN was verified, the Volume 2 ISBN was not, so it is named without a number. |
| Multivariable and Vector Calculus | Tier 3 | 18.02 exams and Putnam problems Kedlaya files under geometry or analysis | free | The exam page is the same 18.02 site. The Putnam archive is the public ceiling. Borisov’s sorted list helps you find the vector-calculus problems: https://people.math.binghamton.edu/borisov/Putnam/SelectedProblemList.pdf |
| Ordinary Differential Equations | Tier 0 | Paul’s Online Math Notes, Differential Equations | free | Read once, do not study from this. It is a technique index. Knowing the names of the techniques is the preview. Computing them is the course. |
| Ordinary Differential Equations | Tier 1 | MIT 18.03SC | free | Lectures, recitations, problem sets, exams. This is the main text. Boyce and DiPrima is the usual paid book; no edition was verified. Use 18.03 so the problem sets have published solutions. |
| Ordinary Differential Equations | Tier 2 | Tenenbaum and Pollard, Ordinary Differential Equations, ISBN 978-0-486-64940-5 | PAID | The Dover book is the problem mountain and it is paid; the ISBN is catalogued. The link is 18.03’s problem sets, which you can do tonight. Use Tenenbaum when you want more drills of a type 18.03 only assigned twice. |
| Ordinary Differential Equations | Tier 3 | Arnold, Ordinary Differential Equations, ISBN 978-0-262-51018-9 | free | Arnold is paid and is the ceiling: geometry of phase portraits, not another integrating factor. No free legal edition was verified, so the link stays on the 18.03 exams, which are the public ceiling you can download. Do Arnold’s problems if you buy the book. Do the exams either way. |
| Real Analysis | Tier 0 | Tao, books page for Analysis I | free | Read once, do not study from this. It is a prospectus for a construction of the reals. It tells you the subject is about building the line, not about computing more integrals. |
| Real Analysis | Tier 1 | Jiří Lebl, Basic Analysis, and MIT 18.100A | free | Free, current, and written as a course. The HTML home is https://www.jirka.org/ra/ . MIT 18.100A supplies another problem-set culture: https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ . Abbott, ISBN 978-1-4939-2711-1, is the paid book to buy if Lebl’s tone is too dry. |
| Real Analysis | Tier 2 | 18.100B and Pugh’s exercises | free | 18.100B is the harder MIT track. Pugh, Real Mathematical Analysis, ISBN 978-3-319-17770-0, is paid and is the exercise upgrade. Do the 18.100 sets either way. |
| Real Analysis | Tier 3 | Rudin, Principles of Mathematical Analysis, ISBN 978-0-07-054235-8 | free | Baby Rudin is paid and is the qualifying-exam ceiling. No free legal PDF was verified. The link is 18.100B, the public problem set at that altitude. Work Rudin’s Chapter 1–8 exercises if you own the book. Do not read the solutions collections first. |
| Topology | Tier 0 | Quanta, Topology 101 | free | Read once, do not study from this. It is journalism about holes. It is also the only Learn Anything topology link that is a genuine preview rather than a paper or a tool. Then leave it. |
| Topology | Tier 1 | Munkres, Topology, 2nd edition, ISBN 978-0-13-181629-9, read beside Topology Without Tears | free | Munkres is paid and is the main text; the ISBN is the catalogued 2nd edition. Topology Without Tears is the free parallel book when a Munkres paragraph is too compressed. Use both. Do not replace Munkres’s exercises with the free book’s. |
| Topology | Tier 2 | MIT 18.901, Fall 2004, assignments | free | The problem plan, with section and exercise numbers aimed at Munkres. Do these in order. The course page is https://ocw.mit.edu/courses/18-901-introduction-to-topology-fall-2004/ . |
| Topology | Tier 3 | Munkres supplementary exercises, checked only after your writeup | free | The repository is a solution check, not a text. Open it after the 20-minute timer and a complete writeup, one exercise at a time. π-Base is the counterexample catalog when you need a space: https://topology.pi-base.org/ . |
| Abstract Algebra | Tier 0 | Judson, the book’s front page | free | Read once, do not study from this. See that the book runs from groups through Galois theory and that a free PDF exists. Then download the PDF and work there. |
| Abstract Algebra | Tier 1 | Judson, Abstract Algebra: Theory and Applications, 2025/2026 edition PDF | free | Free. The HTML edition is https://judsonbooks.org/aata-files/aata-html/aata.html . Paperback ISBN 978-1-944325-21-3 is the 2026 print edition named on the author’s page. This is the main text because it is complete, exercised, and legal. |
| Abstract Algebra | Tier 2 | Judson’s harder exercises, all of them, plus the Sage checks he includes | free | The exercises are the problem set. Dummit and Foote is the paid source of extra problems; no edition was verified. |
| Abstract Algebra | Tier 3 | Putnam problems on groups, rings, and polynomials | free | Borisov’s sorted list points at the algebra problems: https://people.math.binghamton.edu/borisov/Putnam/SelectedProblemList.pdf . Qualifying exams in algebra are the real ceiling; a public one you can start now is a Putnam algebra problem you cannot start in five minutes. |
| Complex Analysis | Tier 0 | MIT 18.04 course page, the overview only | free | Read once, do not study from this. Read the overview so you see the destination is contour integrals and residues. The same site is the course; the preview is one pass over the description, then you start the problem sets as Tier 1. |
| Complex Analysis | Tier 1 | MIT 18.04 notes and problem sets | free | This is the main text. Ahlfors and Conway are the paid classics; editions were not verified, so they are not cited with fake ISBNs. |
| Complex Analysis | Tier 2 | 18.04 exams and the harder problem sets | free | Do every set. The hardness is justifying arcs and branch cuts, not the residue arithmetic. |
| Complex Analysis | Tier 3 | Putnam problems that are complex analysis in disguise, via Kedlaya | free | Qualifying exams in complex analysis are the real ceiling. Until you have a public quals PDF you have personally verified, the Putnam archive plus a complete 18.04 exam is the honest public ceiling. |
| Measure Theory | Tier 0 | Axler, the measure-theory book site | free | Read once, do not study from this. Confirm the PDF is the free book and see the table of contents. Then work in the PDF. |
| Measure Theory | Tier 1 | Axler, Measure, Integration & Real Analysis | free | Free legal PDF from the author. This is the main text. It goes from measure through functional analysis at a level you can actually finish. |
| Measure Theory | Tier 2 | Axler’s exercises on the convergence theorems and L^p | free | The book is the problem set. Do not skip the “prove that this hypothesis is necessary” exercises. |
| Measure Theory | Tier 3 | A qualifying-exam style reconstruction of dominated convergence | free | Rudin’s Real and Complex Analysis and Folland are the paid ceilings; ISBNs were not verified, so they are not numbered here. The public ceiling is: write the three convergence theorems and a counterexample for each missing hypothesis, from memory. |
| Partial Differential Equations | Tier 0 | MIT 18.152, the course description | free | Read once, do not study from this. See that the course is Laplace, heat, and wave, not a catalogue of tricks. Then use the assignments as the work. |
| Partial Differential Equations | Tier 1 | MIT 18.152 assignments and notes | free | The main text. Strauss is the usual paid book; no edition was verified. |
| Partial Differential Equations | Tier 2 | 18.152 problem sets, every one | free | Separation, maximum principles, and energy estimates are all in the sets. Do them in order. |
| Partial Differential Equations | Tier 3 | Evans is the graduate ceiling; the public ceiling is a full 18.152 exam plus d’Alembert from memory | free | Evans is paid and unnumbered here. Do not start it until the maximum principle proof is yours. |
| Number Theory | Tier 0 | Stein, Elementary Number Theory, the PDF’s table of contents | free | Read once, do not study from this. Skim the contents so you see the path from gcd to reciprocity. Then start at chapter 1 and do the exercises. |
| Number Theory | Tier 1 | William Stein, Elementary Number Theory: Primes, Congruences, and Secrets | free | Free PDF. This is the main text. It includes computation, which keeps the theorems honest. |
| Number Theory | Tier 2 | AIME number theory and Stein’s exercises | free | AIME is the right hard-problem layer. Stein’s later exercises are the other half. |
| Number Theory | Tier 3 | Putnam number theory via Borisov and Kedlaya | free | The sorted list plus the archive https://kskedlaya.org/putnam-archive/ . Ireland and Rosen is the paid graduate-leaning book; no edition was verified. |
| Combinatorics | Tier 0 | Lehman, Leighton, Meyer, the contents page of the PDF | free | Read once, do not study from this. Find the counting and graph chapters. Then work those chapters and skip the rest until you want it. |
| Combinatorics | Tier 1 | Mathematics for Computer Science, Lehman, Leighton, Meyer | free | Free. The counting, induction, and graph chapters are the main text. Better than a paid combinatorics survey you will not finish, because the problems are inline and the book is complete. |
| Combinatorics | Tier 2 | AIME counting problems | free | The hard-problem layer. Filter by combinatorics using the AoPS index or Po-Shen Loh: https://live.poshenloh.com/ |
| Combinatorics | Tier 3 | Putnam combinatorics, Borisov’s list | free | The ceiling at this level. Generatingfunctionology is the right extra book if you find a legal copy you can verify yourself; a stable free URL was not confirmed here. |
| Mechanics | Tier 0 | The Feynman Lectures on Physics, Volume I, opened from the official site | free | Read once, do not study from this. Read the table of contents of Volume I and one lecture if you want the voice. Feynman will not teach you to finish Irodov. Close it. |
| Mechanics | Tier 1 | OpenStax University Physics Volume 1, and MIT 8.01SC | free | Free calculus-based text. The course with exams is https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/ . Halliday is the paid traditional book; no edition was verified. OpenStax plus 8.01 is the better pair because you can start tonight and sit a real exam. |
| Mechanics | Tier 2 | Morin, Introduction to Classical Mechanics, ISBN 978-0-521-87622-3, and Kleppner & Kolenkow | free | Both books are paid. Morin’s ISBN was catalogued. Kleppner’s ISBN was not, so it is unnamed. The link is 8.01’s problem sets, which are the hard layer you can do before you buy either book. |
| Mechanics | Tier 3 | Irodov, Problems in General Physics, and Krotov | free | Irodov is the ceiling. Krotov’s Aptitude Test Problems in Physics is the olympiad companion: https://archive.org/details/ptitudetestproblemsinphysics . Use the mechanics parts. The answers in the back of Irodov are a check after the writeup. |
| Electricity and Magnetism | Tier 0 | Feynman Lectures, Volume II, from the official contents | free | Read once, do not study from this. Open Volume II from that page and read the contents. It is the conceptual preview. It is not where you learn to do Gauss’s law under a clock. |
| Electricity and Magnetism | Tier 1 | OpenStax University Physics Volume 2 and MIT 8.02 | free | Free text plus the course https://ocw.mit.edu/courses/8-02-physics-ii-electricity-and-magnetism-spring-2019/ . Griffiths, ISBN 978-1-108-42041-9, is the paid upgrade when the OpenStax problems feel short on theory. |
| Electricity and Magnetism | Tier 2 | 8.02 problem sets and Griffiths’ problems, if you own the book | free | The public hard layer is 8.02. Griffiths is paid. Do the exam problems before you buy anything else. |
| Electricity and Magnetism | Tier 3 | Irodov, electrodynamics | free | The ceiling. Krotov’s relevant problems sit in the same archive item as the mechanics olympiad set: https://archive.org/details/ptitudetestproblemsinphysics |
| Waves and Optics | Tier 0 | Feynman Lectures, the waves chapters via the official site | free | Read once, do not study from this. Volume I contains the waves and the start of optics. Read the contents and one chapter. Then go to 8.03. |
| Waves and Optics | Tier 1 | MIT 8.03SC and OpenStax Volume 3 | free | 8.03 is the course. OpenStax Volume 3 holds the optics chapters in a standard order: https://openstax.org/details/books/university-physics-volume-3 |
| Waves and Optics | Tier 2 | 8.03 problem sets | free | Normal modes, dispersion, and boundary conditions. This is the hard layer. Lens worksheets are not. |
| Waves and Optics | Tier 3 | Irodov, oscillations and wave optics | free | The ceiling. Path differences, coherence, and waves on strings with boundary conditions. |
| Modern Physics | Tier 0 | Feynman Lectures, Volume III, from the official site | free | Read once, do not study from this. Volume III starts from amplitudes. It is a preview of the taste of the subject. 8.04 is the course. |
| Modern Physics | Tier 1 | OpenStax Volume 3, modern physics chapters, then MIT 8.04 | free | OpenStax is the gentle entrance. 8.04 is the real course and becomes the main text as soon as the infinite well feels easy: https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/ |
| Modern Physics | Tier 2 | 8.04 problem sets | free | This is the hard layer. Griffiths is the paid book to buy when you want a second source; no edition was verified. |
| Modern Physics | Tier 3 | 8.04 exams | free | The ceiling of this page. A graduate quals in quantum is the next ceiling and is not linked because no specific public exam was verified. |