Taper and exam: the week where doing less is the discipline
Exam week. The mathematics is done — twenty-seven weeks of it, none of which changes in the next six days. What remains is delivery: a short taper protocol, the finalized runbook, and the four sessions themselves. This lesson is deliberately the shortest of the twenty-eight, because the correct amount of new input this week is almost none, and the site practices what it preaches.
You do not rise to the occasion; you fall to the level of your systems — and after twenty-seven weeks, your systems are good. Let them run.
Key ideas — the week on one card
Days 1-3 light, days 4-5 near-zero, day 6 off: sleep outperforms any final drill, and nothing new consolidates in time.
Exam-week shakiness is usually arousal misread as ignorance: check the evidence, then win ceremonially or run the pre-written containment plan.
In session, the trained ritual runs verbatim: triage read, bank first, minute-30 checkpoints, verification pass, partial artifacts.
The reset sentence between sessions: this score is fixed, the next one is not.
You are repeating a measured performance, not attempting a miracle.
1The final six days
Days 1-3: light warmups only — two or three banked-tier problems daily at full ceremony, deck reviews, and one read-through of your technique cards per day (reading, not studying: the cards are yours, written in your own compressed voice, and a full pass takes under an hour). Nothing new. A gap discovered now costs sleep and buys nothing; the not-fixing list from Week 26 exists precisely so that late discoveries have pre-decided answers.
Days 4-5: taper to near-zero — one warmup problem daily to keep the hands moving, runbook read-through, logistics locked (location confirmed, materials staged: pens you like, watch, water, food for the between-session breaks). Sleep is the actual assignment: two consecutive full nights before the exam outperform any final drill by a margin the sleep literature finds embarrassing to the drills. Day 6, the day before: a twenty-minute morning warmup at most, then genuinely off. Anxiety will suggest studying; the runbook's answer is written down and it is no.
Exam morning: the runbook runs — wake time, food, arrival with margin, and the identity fact from Week 27's log: your recent evidence says you solve these. Between sessions: the trained reset — food, walk, no post-mortems, next session fresh. The December rule from Mock #1, now load-bearing.
Worked example
It is Wednesday of exam week and you suddenly feel shaky on generating functions. What does the runbook say?
Nudge 1
Pull the logs before believing the feeling — what does the evidence say?
Reveal step 1
Check the feeling against evidence: pull the battle log and the Week 27 re-solve results for generating functions. Almost always the log shows banked-tier competence — the shakiness is arousal misread as ignorance, which is the standard exam-week illusion and worth naming to yourself in exactly those words.
Nudge 2
Solid log: one ceremonial win and close the topic. Soft spot: run the containment plan.
Reveal step 2
If the log agrees you are solid: one confidence problem from the topic at ceremony, log the win, close the topic. If the log shows a genuine soft spot: it is on the Week 26 routing list with a containment plan already written — read the plan, run its rehearsal once if it never got one, done.
Nudge 3
Either way, bound the episode at forty minutes and end it deliberately.
Reveal step 3
Either way the session ends the same: the topic is HANDLED — by evidence or by plan — and handling it took forty minutes, not a panicked evening. That conversion of anxiety into a bounded procedure is the runbook's entire job, this week and Saturday.
Answer
Consult evidence, not the feeling; solid → one ceremonial win and close; soft → the pre-written containment plan. Forty minutes, bounded, done.
Pitfall. Emergency studying. Every hour of exam-week cramming trades sleep and calibration for material that will not consolidate in time — negative on all three axes. The list of things you are not fixing was written two weeks ago by a calmer, better-informed version of you; trust that version.
2The four sessions, and after
In-session, everything is trained and nothing is new: the five-minute triage read with written tags, bank-first ordering, minute-30 checkpoints, parking notes, the verification pass on every banked problem, partial-credit artifacts on the parked ones. Ninety minutes, three problems, four times. The single most valuable in-exam sentence, for when something goes sideways: 'this session's score is fixed; the next session is not' — the reset rule, which two mocks have already made yours.
Expectations, calibrated once: the median score on this exam is historically a single-digit number OF 120. Problems will defeat you Saturday — they defeat everyone; the top-200 target is roughly two banked problems per session plus honest partials, which is exactly what your last two mocks measured you doing. You are not hoping to perform Saturday; you are repeating a measured performance a third time.
Afterward: log it, rest for real, and then the fork in the road this site was quietly built for — the archive, the technique cards, the FSRS deck, and the training system itself carry directly into Putnam 2027, research mathematics, and everything downstream. Week 28 ends the program; nothing about it ends the mathematician. See you in the archive.
Worked example
Session 2 opens with three problems that all look bad after the five-minute read. Run the protocol.
Nudge 1
Rank the three by least-bad — 'no full arc' and 'no entry' are different verdicts.
Reveal step 1
Trust the tags anyway: rank the three by least-bad — the read produced a candidate first move for at least one (it essentially always does; 'no full arc visible' and 'no entry visible' are different verdicts, and the tags record which is which).
Nudge 2
Open with fifteen minutes of small-case data; costumes come off under computation.
Reveal step 2
Open the least-bad at fight posture: phase-one data gathering (small cases, concrete instances) for fifteen minutes. On a bad-looking set, the Week 22 investigation protocol IS the triage — data converts 'looks bad' into 'is actually an A2 wearing a costume' more often than not, because looking bad is a costume choice exam writers love.
Nudge 3
Harvest partials on all three under checkpoint discipline and use the reset sentence.
Reveal step 3
If the session stays hard: partial-credit artifacts on all three beats zero-zero-ten in expectation, and the minute-30 checkpoints enforce the allocation automatically. Two sessions remain after this one; the reset sentence is already in your pocket. Bad sessions are survivable precisely because you decided how to survive them in October.
Answer
Rank the least-bad, investigate with data, harvest partials under checkpoint discipline, reset. The protocol does not require the problems to cooperate.
Pitfall. Abandoning the ritual because the paper looks unusual. The year the problems look weird is the year the ritual pays most — everyone else is improvising.
Before you open the gates
Days 1-3 light, days 4-5 near-zero, day 6 off; sleep is the assignment, the runbook is the authority.
Exam-week anxiety → evidence check → win ceremonially or run the containment plan; never emergency-study.
In session: the trained ritual verbatim — triage, bank first, checkpoints, verification, partial artifacts.
The reset sentence between sessions: this score is fixed, the next one is not.
You are repeating a measured performance, not attempting a miracle — and the system you built outlives Saturday.
Check yourself
1. The correct response to discovering an apparent gap on Wednesday of exam week is:
Exam-week shakiness is usually arousal, not ignorance; the log and the routing list convert it into a bounded forty-minute procedure either way.Repair: revisit §1 The final six days ↑
2. The last two days before the exam, the priority ordering is:
Two full nights of sleep outperform any final drill; logistics remove morning variance; the warmup only keeps hands moving. Nothing new consolidates in time.Repair: revisit §1 The final six days ↑
3. When an entire session looks bad at the read, the protocol says:
Data gathering unmasks costumed problems, and disciplined partials across three beat a heroic zero — the expectation arithmetic was settled weeks ago.Repair: revisit §2 The four sessions, and after ↑
Since $n^2 + 1 = (n+1)(n-1) + 2$, we have $(n+1) \mid (n^2 + 1)$ if and only if $(n+1) \mid 2$. For a positive integer $n$, $n + 1 \ge 2$, so the only possibility is $n + 1 = 2$, i.e. $n = 1$ (and indeed $2 \mid 2$). Thus $n = 1$ is the unique solution. [Source: this week's familiar warm-up re-solves — an A1/B1 number-theory opener.]
✓ rung 2 done
Rung 3 (taper warm-up, pigeonhole). Prove that among any five lattice points in the plane, some pair has a midpoint that is also a lattice point.
One hint
Classify each point by the parities of its coordinates. How many parity classes are there?
Worked resolution
Classify each lattice point $(x, y)$ by $(x \bmod 2,\, y \bmod 2) \in \{0,1\}^2$ — four classes. Among five points, two share a class by pigeonhole, say $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \equiv x_2$ and $y_1 \equiv y_2 \pmod 2$. Then $x_1 + x_2$ and $y_1 + y_2$ are both even, so the midpoint $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ has integer coordinates — a lattice point. [Source: this week's A1/B1 taper warm-ups — a pigeonhole confidence re-solve.]
✓ rung 3 done
Prove it — constructed response
A confidence-set warm-up re-solve, written exam-clean. Prove that for every positive integer $n$, $1^3 + 2^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$. Use induction, and show the algebra of the inductive step in full.
Self-grade against the rubric — completion requires the judgment, not the text
Model proof (compare AFTER grading yourself)
By induction on $n$. Base $n = 1$: the left side is $1^3 = 1$ and the right side is $\left(\frac{1\cdot 2}{2}\right)^2 = 1$, so they agree. Inductive step: assume $\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2$. Then $\sum_{k=1}^{n+1}k^3 = \left(\frac{n(n+1)}{2}\right)^2 + (n+1)^3 = \frac{(n+1)^2}{4}\left[n^2 + 4(n+1)\right] = \frac{(n+1)^2}{4}(n+2)^2 = \left(\frac{(n+1)(n+2)}{2}\right)^2$, which is exactly the claimed formula with $n$ replaced by $n+1$. By the principle of induction, the identity holds for every positive integer $n$. $\blacksquare$
The gates
Core assignment
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### Week 28 — Taper + Exam
*Due: Dec 5 (EXAM DAY)*
- Mon–Wed: 2 A1/B1 warmups/day · 10-min cap only · no new material
- Thursday: STOP solving. Re-read all notes once.
- Friday: rest. Re-read exam-day protocol.
- **Saturday Dec 5: EXAM.**
Ritual: ### Week 28 — Taper + Exam
*Due: Dec 5 (EXAM DAY)*
- Mon–Wed: 2 A1/B1 warmups/day · 10-min cap only · no new material
- Thursday: STOP solving. Re-read all notes once.
- Friday: rest. Re-read exam-day protocol.
- **Saturday Dec 5: EXAM.**
Source3Blue1Brown — Essence of Linear Algebra — Strategy companion for “Core assignment” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
SourceMichael Penn — Number Theory v2 (playlist) — Writeup companion for “Core assignment” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Problem-Solving Reps — weekly homework
Problem-Solving Reps — weekly homework
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Standing weekly homework — the problem-solving book stack, every week. Solve ALL listed; write ONE full clean solution (the rest may stay scratch). Up to 2 due re-solves from your review queue surface first.
sources & assignments (2)
ProblemsPnB — Final skim of own weak-area notes (reference)Putnam
ProblemsMAA archive — 4 familiar warm-up re-solves (no new material) + the Putnam examPutnam
Reflect
Reflect: reconstruct and log the pattern
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Mandatory Reflect gate from AGENTS.md for every week 3–28.
Ritual: Pick one problem you solved this week. Close all notes and the solution. Rewrite the full solution from memory. Then compare and write: When I see [pattern], do [first move].
SourceSupplemental — Lupu TTU MATH 4000, Lecture 6 (direct) — Lecture companion for “Reflect: reconstruct and log the pattern” — full treatment; pause at each theorem statement and predict the proof's first move before it plays.
SourceMicrosoft Research — The Wonders of the Probabilistic Method — Intuition companion for “Reflect: reconstruct and log the pattern” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Verify: fatal-slip audit before closing
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unlocks after: Reflect: reconstruct and log the pattern
Mandatory Verify gate from AGENTS.md for every week 3–28.
SourceSupplemental — Lupu TTU MATH 4000, Lecture 6 (direct) — Lecture companion for “Verify: fatal-slip audit before closing” — full treatment; pause at each theorem statement and predict the proof's first move before it plays.
SourceMicrosoft Research — The Wonders of the Probabilistic Method — Intuition companion for “Verify: fatal-slip audit before closing” — the picture behind the machinery; afterwards write one sentence connecting the visual to this gate's exercises.
Exit contract — Week 28
Verification remaining
reading progress…
Carry-forward repairs
reading queue…
Next week opens with
Nothing — this is the final week. The exam is next.
