# Putnam Slot Patterns — A1 through B6 > **Data status (2026-05-30).** This file is prose guidance, but the data it > relies on is now real: all 480 archive problems (1985–2024) are topic-tagged > and carry solutions, so the slot+topic filters in the Archive Browser no > longer drop problems. 1965–1984 is not yet loaded (scanned-PDF/OCR gap). Use the Archive Browser slot filter (gamification page → Archive → Putnam slot chips) to grind every problem in a given slot from 1985–2024. This file is the companion: what *kinds* of problems live in each slot, and what the typical first move is. Built from a scan of Kedlaya's archive plus the standard folklore (Bjorn Poonen, Ravi Vakil, the UToronto and UMD notes). Rule of thumb: slot number is a difficulty signal, not a topic signal. A1/B1 are the "fast win" slots — partial credit is rare; you either see it in 10 minutes or you should bail. A6/B6 are open-ended research-flavored. --- ## A1 — Calibrated warmup, ~10 min ceiling **Vibe:** clever observation, one-trick problem, often number theory or clean algebra. The intended solution fits on half a page. **Typical topics (frequency):** algebraic manipulation (~30%), number theory (~25%), combinatorics counting (~20%), elementary calculus (~15%), geometry (~10%). **First-move checklist:** 1. Try small cases (n=1,2,3) — A1 answers are almost always closed form. 2. Look for a clever substitution or symmetry. 3. If you don't see the trick in 5 min, suspect telescoping, induction, or a slick parity/mod argument. 4. If you don't see it in 10 min, skip and come back. **Patterns that recur:** - "Find all integers / polynomials / functions such that …" → start with degree or growth-rate bounds. - Sums with a closed form → telescope or write as a derivative. - "Show that X is rational/irrational/an integer" → modular arithmetic or contradiction via prime factorization. --- ## A2 — Slightly harder warmup, ~15 min **Vibe:** A1 with one extra layer. The trick is the same kind, but you need to combine two observations or do a clean induction. **Typical topics:** sequences/series, combinatorial identities, polynomial roots, basic probability. **First-move checklist:** 1. Same small-case routine as A1. 2. Set up a generating function if the problem is "count the number of …" sequences/tilings/paths. 3. If recursion appears, look for a fixed-point or invariant. --- ## A3 — Real problem, ~25 min **Vibe:** the first slot where you can't bluff. Often a real-analysis or linear-algebra problem requiring a named theorem. **Typical topics:** convergence of series, IVT/MVT, rank/determinant tricks, inequality with a sharp constant. **First-move checklist:** 1. Identify the named theorem in play (MVT, IVT, Cauchy–Schwarz, AM–GM, pigeonhole, dimension count). 2. Try the obvious approach first; A3s often submit to standard tools but require care about edge cases. 3. Write down what you *know* about the object (continuity, finite, integer, etc.) before you start manipulating. --- ## A4 — Mid problem, ~30 min **Vibe:** the inflection point. Either a clever algebra/NT/combo problem that needs a non-obvious construction, or a calculus problem that needs a clever change of variables. **Typical topics:** functional equations, polynomial identities, combinatorial constructions, integration tricks. **First-move checklist:** 1. For "exists" problems: try to construct directly before proving non- existence. 2. For functional equations: plug in 0, 1, x=y, look for fixed points and involutions. 3. For integrals: Feynman trick (differentiate under the integral), symmetry, or contour shifts. --- ## A5 — Hard, ~40 min **Vibe:** A4 plus a synthesis step. Usually requires combining two techniques (e.g., generating functions + a counting bijection, or linearity of expectation + a clever indicator). **Typical topics:** group actions, advanced combinatorics, real analysis with a twist, abstract algebra. **First-move checklist:** 1. Identify the *two* tools needed. A5s rarely yield to a single technique. 2. If a problem says "show that X equals Y" and both look hard to compute directly, look for a bijection or a generating-function identity. 3. Suspect groups acting on sets (Burnside / orbit-counting) when "symmetric" or "fixed by" appears. --- ## A6 — Hardest of session A, ~45+ min **Vibe:** research-flavored. Often number theory or combinatorics with a clever combinatorial-algebraic bridge. Many A6s are unsolved by the median high scorer. **Typical topics:** advanced number theory (p-adic, class numbers in disguise), graph theory (Ramsey-style or extremal), advanced algebra. **Strategy:** if you don't see a foothold in 15 min, **bail**. The opportunity cost is too high. Partial credit on A6 is rare; better to lock in A1–A3 and maybe A4. --- ## B1 — Calibrated warmup, ~10 min Mirror of A1 with different topic emphasis. B-session tends to be more geometry- and probability-heavy. **Typical topics:** geometry (coordinates or synthetic), elementary probability, sequences, NT. **First-move checklist:** same as A1, but with extra checks for: - Coordinate bash on geometry problems. - Conditional probability / linearity of expectation on prob problems. --- ## B2 — Slightly harder warmup, ~15 min Similar to A2. Watch for: - Polynomial root tricks (Vieta, derivative of P). - Combinatorial games (find the invariant). --- ## B3 — Real problem, ~25 min Similar to A3. Topics skew toward: - Linear algebra (often disguised — "consider the linear map …"). - Calculus inequalities (convexity, Jensen). --- ## B4 — Mid problem, ~30 min The other inflection point. Often: - Group theory or abstract algebra (cosets, conjugacy). - Geometry with a hidden algebraic structure. - Polynomial / power-series identities. **First-move pattern:** if the problem mentions a finite group or a finite field, suspect counting orbits, characters, or Lagrange's theorem. --- ## B5 — Hard, ~40 min Same family as A5 — synthesis of two techniques. B5 often hides: - Probabilistic method ("show that there exists …" → count and pigeonhole by averaging). - Reduction to a known inequality (Cauchy–Schwarz, rearrangement, Karamata). --- ## B6 — Hardest, ~45+ min Like A6 — bail if no foothold in 15 min. B6 sometimes hides: - Advanced analysis (operator theory in disguise, Fourier). - Combinatorial constructions with a sharp bound. --- ## How to use this file with the grinder 1. Open Archive Browser → set slot=A1, source=kedlaya. You get ~40 problems, one per year 1985–2024. 2. Time-box: 10 min each. Even just *recognizing the type* is a win. 3. After ~20 problems in one slot, you'll start seeing the patterns above without reading this file. 4. Repeat for A2, B1, B2 (the "must-have" slots for a 30-point score). 5. Save A5/A6/B5/B6 grinding for after the basics are solid. ## Solution checks The browser shows all available published solutions in the reveal-solution pane (Kedlaya = 1 per problem; MathNet shows up to ~5 community solutions per problem). You must write an 18-word attempt before the reveal unlocks — this is intentional (Bjorn Poonen's "write something first" rule). ## Linked reading - Kedlaya's archive index: - Putnam slot-by-slot statistical analysis: see Putnam_Guide.pdf section 1 - Yufei Zhao OTIS-style problem-set classifications (linked in script.js weekly modules for Algebra, Combinatorics, Number Theory tracks).