32 · Triage at Dawn

The exam triage rule. During each 90-minute session: scan all available questions for eight to ten minutes; identify the most promising entry point; attack one problem for thirty-five to forty-five minutes; abandon dead ends early; use the remaining time to secure partial progress or complete a second solution; reserve ten minutes for proof cleanup. The Putnam is not a contest of emotional loyalty to your first idea.

The operational definition of "prepared perfectly": a finite guarantee does not exist, so use a measurable readiness standard. You are fellow-ready when your last twelve protected blind papers satisfy all of the following — median score 90-plus; at least six of twelve papers scoring 100-plus; at least three scoring 110-plus; complete solutions on A1, A2, B1, B2 nearly automatic; complete solutions on A3 or B3 frequent; occasional but real completions on A4 through A6 or B4 through B6; near-zero avoidable proof errors; and time wasted on dead approaches declining consistently. You are "120 on half of mocks" ready only when you literally score 120 on at least half of a protected batch of unseen papers. Do not redefine the target after missing it — humans have invented enough accounting tricks already.

The daily training block, for a serious 20–25 hour week. Monday to Friday, three hours: ninety minutes solving two hard problems cold; thirty minutes writing one clean proof; thirty minutes studying one official solution; twenty minutes updating the error log; ten minutes reviewing previous triggers. Saturday, six hours: four timed 90-minute Putnam sessions under strict exam conditions. Sunday, three to four hours: grade ruthlessly, rewrite proofs, categorise missed triggers, schedule spaced reviews. That is sustainable brutality — not a heroic fourteen-hour binge followed by five days of intellectual decomposition.

The priority order for scoring: you do not begin by worshipping B6. The 2025 score distribution shows why — among the top 507 participants, full-credit solutions were far more common on A1, A2, B1, B2 and B3 than on the hardest questions; 432 of the top 507 earned full credit on B1 while only 12 earned full credit on B6. Train in this order: guarantee A1 and B1; make A2 and B2 highly reliable; become dangerous on A3 and B3; develop occasional kills on A4 and B4; only then invest heavily in A5–A6 and B5–B6. A stable 60 beats a chaotic obsession with impossible final problems. Once 60 is routine, push toward 90. Once 90 is routine, pursue 110 to 120.

The first fourteen days. Week one: sit one full diagnostic, start an error log, read the proof-methods section of Putnam and Beyond, solve twelve problems on induction, contradiction, invariants, extremal arguments and pigeonhole, write six clean proofs. Week two: complete MIT's hidden-independence set, solve twelve algebra and number theory problems, sit one 90-minute mixed set, re-solve every failed week-one problem without notes, rewrite the three ugliest proofs. At the end of fourteen days calculate: solve rate by topic, average time to first useful observation, percentage of proofs completed cleanly, most common missed triggers, and the diagnostic score. Then train the weaknesses — not the topics that make you feel clever.

Bottom line — the sufficient practical system: core theory, then 800 to 1,200 deep attempts, 500-plus written proofs, the MIT seminar sets, 25 to 40 protected full mocks, a ruthless failure log, spaced re-solving, repeated 90-to-110-plus blind scores. A 120 cannot be promised. But this is the closest thing to a complete preparation machine: narrow resources, enormous deliberate volume, protected mocks, proof-quality discipline, and no tolerance for fake progress.