Cold re-solves and memory locking: making it permanent enough for one Saturday
Twenty-six weeks of technique now gets locked for one specific Saturday. The mechanism is the cold re-solve: problems you solved weeks ago, attempted again from a blank page. Re-solving is not review — review recognizes, re-solving RETRIEVES, and retrieval under mild pressure is precisely the operation December demands. The week's second thread is the confidence set: a deliberate diet of clean wins, because the machine that shows up to the exam includes its own morale.
The exam tests retrieval, so the last full training week practices retrieval — and a solver who has recently succeeded sixty times walks in different from one who has recently drowned in nightmares.
Key ideas — the week on one card
Cold re-solves exercise RETRIEVAL — the exam's operation; review only exercises recognition. Blank page means blank page.
Blanks are the yield: classify trigger-vs-pathway, run the matching repair, re-solve cold in two days — effortful recovery locks hardest.
The deck locks statements; re-solves lock pathways — they decay independently, so both run.
The taper: volume down, per-problem intensity flat, sleep protected, nightmares retired — the banked tier gets every remaining hour.
1The cold re-solve protocol
Selection: pull from your own solved history — battle log wins, mock solves, the technique cards' worked instances — biased toward problems that were HARD-WON the first time (they encode the most technique per problem) and toward your banked slots (A1-B2 tier, where December points actually live). Twenty to thirty problems across the week, timed at exam pace, from a genuinely blank page: statement only, no notes, no memory-jogging peek at your old solution first.
Scoring is binary-plus: full re-solve (the pathway is locked), partial re-solve (reached the key idea, fumbled execution — a write-up rep re-locks it), or blank (the solution did not survive the weeks — and THAT is the finding). Blanks are not failures of the week; they are the week working: each blank marks a pathway that would have silently failed on exam day, and its repair — re-study, then re-solve again two days later — is the highest-value hour available this close to the exam.
The spaced-repetition deck runs its normal schedule underneath: the deck locks STATEMENTS (theorems, formulas, triggers); the re-solves lock PATHWAYS (whole solution arcs). Both kinds of memory decay independently, which is why both get their own maintenance — a locked statement with a decayed pathway produces the exam-day experience of 'I know the tool but cannot make it work', which the re-solve protocol exists to prevent.
Worked example
A blank, worked: you re-attempt a Week-16 Catalan-path problem you solved cleanly a month ago, and produce nothing in 25 minutes. Run the repair.
Nudge 1
Before re-studying, decide: did the TRIGGER fail or the PATHWAY?
Reveal step 1
Classify the blank before re-studying: did the TRIGGER fail (never thought 'reflection') or the PATHWAY (thought 'reflection', could not execute the first-touch bijection)? The distinction takes one minute of honest introspection and picks the repair.
Nudge 2
Match the repair to the failure type — cue re-reads for triggers, closed-book re-derivation for pathways.
Reveal step 2
Trigger blank: re-read the technique card's cue line ('never-below-zero / non-crossing → Catalan, check 1-2-5'), then classify five path-flavored statements without solving. Pathway blank: re-derive the reflection argument once from the card, closed-book, writing the bijection AND its inverse.
Nudge 3
Schedule the same problem cold again in two days; recovered items lock hardest.
Reveal step 3
Schedule the re-lock: the same problem returns in two days, cold again. A blank that re-solves after repair is locked better than one that never blanked — the retrieval literature is unambiguous that effortful recovery beats smooth recall for retention, which is the entire reason this week exists.
Answer
Classify trigger-vs-pathway, run the matching one-hour repair, re-solve cold in two days — the blank becomes the best-locked item in the set.
Pitfall. Peeking at the old solution 'to warm up'. The peek converts retrieval into recognition and voids the measurement — the entire diagnostic value of a re-solve lives in the blank page.
2The confidence set, and tapering the volume
The confidence set is deliberately engineered: problems at or slightly below your banked tier, in your strongest topics, done at full exam ceremony — read, triage, solve, finish-clean, verify. The goal is not learning; it is accumulating recent, honest wins so that the exam-morning self-model is 'someone who solves these', built from last week's evidence rather than last month's. This is not a psychological indulgence — calibrated confidence changes triage behavior on the day: under-confident solvers park solvable problems, and that error costs more points at your level than any technique gap.
Volume tapers this week: total problem hours DOWN relative to any training week, intensity per problem UNCHANGED. The analogy to athletic tapers is exact — reduce load, preserve intensity, protect sleep. What fills the reclaimed time: the light warmups gate (keep the hands moving), the deck (cheap, high-value), and honestly, rest. Fatigue-driven errors and pressure-driven errors look identical on paper, and the taper is how you remove the first kind from December entirely.
Nightmare problems are DONE after this week — Week 22-24 built that capacity, and it does not decay in a fortnight. What decays in a fortnight is speed and sharpness on the banked tier, so that is where every remaining problem-hour points. The discipline of NOT doing hard problems is genuinely difficult for people wired like Putnam competitors; do it anyway. The competitor who spends the last week losing to A6s arrives having practiced losing.
Worked example
Design today's confidence session: 90 minutes, your strong topics are analysis and number theory.
Nudge 1
Choose winnable-but-real problems from your strongest topics.
Reveal step 1
Pick three: one analysis A1-tier (a limit or integral with a visible trick), one NT B1-tier (a congruence/order argument), one A2-tier stretch in either — the third slot keeps the session honest without threatening it.
Nudge 2
Run the FULL ceremony anyway — the ritual is what transfers to Saturday.
Reveal step 2
Full ceremony: five-minute triage read with written tags (yes, even knowing the problems are chosen to be winnable — the RITUAL is what transfers), checkpoints set, finish-clean write-ups, five-minute verification pass on each.
Nudge 3
Log the wins; the recent log IS the exam-morning confidence artifact.
Reveal step 3
Log wins in the battle log like any session. The log's recent pages ARE the confidence artifact: on exam morning, 'my last sixty logged attempts ran 80% banked' is a fact you can stand on, which is worth more than any pep talk precisely because it is a fact.
Answer
Three winnable-but-real problems, full exam ceremony, logged — manufacturing evidence, not comfort.
Pitfall. Reading the taper as slacking and sneaking in a nightmare set 'to stay sharp'. Sharpness on the banked tier is what December pays; hard-problem capacity is already banked and keeps; fatigue does not taper itself.
Before you open the gates
Re-solve 20-30 own-history problems cold, biased hard-won and banked-tier; blank page means blank page.
Blanks are the yield: classify trigger-vs-pathway, repair accordingly, re-solve in two days.
Deck locks statements; re-solves lock pathways — run both, they decay independently.
Confidence set at full ceremony, logged: recent honest wins are exam-morning evidence, not indulgence.
Taper: volume down, intensity flat, sleep protected, nightmares retired — the banked tier gets every remaining hour.
Check yourself
1. A cold re-solve differs from review because it exercises:
Recognition ('I know this') and retrieval ('I can produce this now, unprompted') are different memory operations, and only the second scores points.Repair: revisit §1 The cold re-solve protocol ↑
Filter $(1+x)^n$ modulo $3$ at $\omega = e^{2\pi i/3}$, and use $1 + \omega = e^{i\pi/3}$.
Worked resolution
With $F(x) = (1+x)^n$ and $\omega = e^{2\pi i/3}$, the mod-$3$ filter gives $\sum_{k}\binom{n}{3k} = \frac13\bigl[F(1) + F(\omega) + F(\omega^2)\bigr]$. Here $F(1) = 2^n$, and $1 + \omega = \tfrac12 + i\tfrac{\sqrt3}{2} = e^{i\pi/3}$, so $F(\omega) = e^{in\pi/3}$ and $F(\omega^2) = \overline{F(\omega)} = e^{-in\pi/3}$; hence $F(\omega) + F(\omega^2) = 2\cos\frac{n\pi}{3}$. Therefore the sum equals $\frac13\left(2^n + 2\cos\frac{n\pi}{3}\right)$. (Check $n = 6$: $1 + 20 + 1 = 22 = \frac13(64 + 2)$.) [Source: this week's endgame keep-warm — the closed-book roots-of-unity-filter recall.]
✓ rung 2 done
Rung 3 (cold re-solve, B1 pigeonhole). Prove that among any $n+1$ integers chosen from $\{1, 2, \dots, 2n\}$, there exist two such that one divides the other.
One hint
Write each chosen number as $2^a m$ with $m$ odd. How many odd values $m$ are available in $\{1, \dots, 2n\}$?
Worked resolution
Write each chosen integer as $2^{a}m$ with $m$ odd. Every such odd part $m$ lies in $\{1, 3, 5, \dots, 2n-1\}$, a set of exactly $n$ values. With $n+1$ chosen integers, two must share the same odd part by pigeonhole: say $2^{a}m$ and $2^{b}m$ with $a < b$. Then $2^{a}m \mid 2^{b}m$, so one divides the other. [Source: this week's cold re-solve set — a B1-tier pigeonhole rep (Mixed Putnam).]
✓ rung 3 done
Prove it — constructed response
A closed-book roots-of-unity-filter recall, promoted to a proof. State the filter identity and prove it: for a polynomial $F(x) = \sum_j a_j x^j$, an integer modulus $n \ge 1$, and $\omega = e^{2\pi i/n}$, show $\sum_{j \equiv r \,(n)} a_j = \frac{1}{n}\sum_{t=0}^{n-1}\omega^{-rt}F(\omega^t)$. Then apply it to prove that $\sum_{k}\binom{N}{2k} = 2^{N-1}$ for every integer $N \ge 1$.
Self-grade against the rubric — completion requires the judgment, not the text
Model proof (compare AFTER grading yourself)
Let $\omega = e^{2\pi i/n}$ (a primitive $n$-th root of unity) and $F(x) = \sum_j a_j x^j$. Claim: $\sum_{j \equiv r \,(n)} a_j = \frac{1}{n}\sum_{t=0}^{n-1}\omega^{-rt}F(\omega^t)$. Proof: the right side is $\frac1n\sum_{t=0}^{n-1}\omega^{-rt}\sum_j a_j\omega^{tj} = \frac1n\sum_j a_j\sum_{t=0}^{n-1}\omega^{t(j-r)}$. If $j \equiv r \pmod n$ then $\omega^{j-r} = 1$ and the inner sum is $n$; otherwise $\omega^{j-r} \ne 1$ and $\sum_{t=0}^{n-1}(\omega^{j-r})^t = \frac{(\omega^{j-r})^n - 1}{\omega^{j-r} - 1} = 0$, since $(\omega^{j-r})^n = (\omega^n)^{j-r} = 1$. Thus only the terms with $j \equiv r$ survive, each with weight $\frac1n\cdot n = 1$, giving $\sum_{j \equiv r}a_j$. Application: take $F(x) = (1+x)^N$, so $a_j = \binom{N}{j}$, and filter with modulus $2$, $r = 0$: $\sum_k\binom{N}{2k} = \frac12[F(1) + F(-1)] = \frac12[2^N + (1-1)^N] = \frac12\cdot 2^N = 2^{N-1}$ for $N \ge 1$. $\blacksquare$
The gates
Cold Re-Solves
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- Do: Default cold set: Putnam 2010 A1, 2010 B1, 2011 A1, 2011 B1, 2012 A1, 2012 B1, 1985 A2, 2013 B2, 2014 A1, 2014 B1, 2015 A2, 2015 B2 - zero notes. Replace only with solved-history equivalents in the same slots.
- Track: retained vs. relearned; note which topics need one more pass
SourceMichael Penn — Real Analysis (playlist) — Strategy companion for “Cold Re-Solves” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
SourceSupplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Writeup companion for “Cold Re-Solves” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Problems12 — Default cold set: Putnam 2010 A1, 2010 B1, 2011 A1, 2011 B1, 2012 A1, 2012 B1, 1985 A2, 2013 B2, 2014 A1, 2014 B1, 2015 A2, 2015 B2 — zero notes. Replace only with solved-history equivalents in the same slots.Putnam
Light Warmups
Light Warmups
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- Archive: 2 A1/B1 per day · 10-min cap · confidence-building only
SourceMichael Penn — Real Analysis (playlist) — Strategy companion for “Light Warmups” — 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 — Real Analysis (playlist) — Writeup companion for “Confidence Set” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
Problems3 most reliable topic areas — 3 most reliable topic areasPutnam
Spiral Reinforcement
Spiral Reinforcement
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- Archive: 3 Putnam A1/B1 · each of your top 2 confidence topics
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SourceMichael Penn — Real Analysis (playlist) — Strategy companion for “Spiral Reinforcement” — a live problem-solving session: watch how the solver chooses a first move under uncertainty, then apply the same selection discipline to this gate.
SourceSupplemental — Lupu TTU MATH 4000, Lecture 4 (direct) — Writeup companion for “Spiral Reinforcement” — attend to how each claim is justified aloud; steal one justification phrase for this gate's written artifact.
ProblemsArchive Browser — 3 Putnam A1/B1 · each of your top 2 confidence topicsPutnam
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.
Mandatory Reflect gate from AGENTS.md for every week 3–28.
Ritual: Pick the cold re-solve that felt least automatic — the one your hand hesitated on. Close all notes and the solution. Rewrite the full solution from memory at speed. Then lock the trigger: When I see [pattern], my first move is [opening], no hesitation.
SourceMichael Penn — Newton's Sums — 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.
SourceMichael Penn — Newton's Sums — 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.
Endgame keep-warm: linear algebra · number theory · geometry · probability · complex numbers (30 min)
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Retention keep-warm (2026-07-02 audit): the mock endgame left linear algebra, number theory, geometry, probability and complex numbers untouched for 3-4 weeks before the exam. 25-30 minutes: one conversion-speed rep per topic; log any miss into repair.
