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Module Quiz

Complete this quiz after finishing all concept and practice pages.

Current Module Questions​

Question 1: Understanding Phase Output​

What is the required output of Polya's understanding phase, and why is that output required rather than optional?

Answer: A written artifact: a restatement, an input/output specification, a constraint list, an assumption list, and a worked tiny example. It is required because "I understood the problem" is unverifiable in your head, while the artifact is checkable and can be handed to a peer.

Question 2: Plan vs. Solution​

What is the difference between a plan and a solution?

Answer: A plan is a short strategy sketch -- related problem, heuristic, subgoal, success criterion -- that can be written in a paragraph. A solution is the complete execution. The plan is a falsifiable hypothesis; the solution is the experiment.

Question 3: Per-Step Verification​

What does a margin ledger buy you that simply writing a proof does not?

Answer: A justification for every step, written at the time. It catches silent arithmetic errors, unstated assumptions, and misapplied lemmas while they are cheap to fix.

Question 4: Looking Back​

List the four questions that structure the looking-back phase.

Answer: Can I verify the result? Can I verify the argument? Can I derive the result differently? Can I use the method elsewhere?

Question 5: Working Backwards​

When is working backwards more effective than forward search?

Answer: When the goal is concrete and the space of forward moves is large. Backward search narrows the funnel by asking "what would immediately produce the goal?"

Question 6: Special Cases​

Why is n=1 often a bad special case to start with?

Answer: n=1 is frequently degenerate (empty products, trivial truths) and may satisfy the claim for reasons that do not generalize. The smallest interesting case is typically n=3 or n=4.

Question 7: Pattern vs. Analogy​

What is the difference between surface similarity and structural analogy?

Answer: Surface similarity is shared vocabulary or domain (both problems mention integers). Structural analogy is shared relationships: same recurrence, same invariant, same decomposition. Only structural analogy supports transfer.

Question 8: Contradiction​

State the skeleton of a proof by contradiction.

Answer: (1) Assume, for contradiction, the negation of the claim. (2) Derive consequences by valid logical steps. (3) Reach a contradiction with a premise, a theorem, or itself. (4) Conclude the original claim.

Question 9: Invariants​

What must you verify when using an invariant argument to prove unreachability?

Answer: That the invariant is preserved by every legal move type, not most; and that the starting state and the goal state have different invariant values. If either fails, the argument does not apply.

Question 10: Construction Requirements​

What three properties must a construction have to count as an algorithm?

Answer: Finiteness (terminates on every valid input), definiteness (each step is unambiguous), and effectiveness (each step is executable in finite work).

Question 11: DP vs. Divide-and-Conquer​

What structural feature distinguishes DP from divide-and-conquer?

Answer: Overlapping subproblems. DP applies when the recursion re-encounters the same subproblem on different branches; divide-and-conquer applies when subproblems are independent.

Question 12: Debugging Categories​

What are the three categories of reasoning bug, and why do they require different responses?

Answer: Model bug (wrong problem), plan bug (wrong strategy), execution bug (wrong step). A model bug requires re-reading the prompt; a plan bug requires redesigning the attack; an execution bug requires fixing the step. Treating them the same wastes time.

Question 13: Generalization​

What is the difference between generalization and "replacing numbers with variables"?

Answer: Generalization identifies why the specific case worked and preserves that reason under variation. Replacing numbers with variables is syntactic and may not preserve the underlying argument.

Question 14: Transfer Notes​

Why are transfer notes considered a required output of a solved problem?

Answer: Without a transfer note, a solved problem teaches only itself. The note turns a one-shot solution into a reusable technique by identifying the problem class it applies to.

Question 15: Heuristic Selection​

Given a problem, name a one-sentence test for when to reach for invariants rather than construction.

Answer: If the question asks whether something is possible or always holds, and you suspect it does not, reach for invariants or contradiction. If the question asks you to produce, find, or compute an object, reach for construction.

Question 16: Spec vs. Prompt​

What is the minimum content a formal specification must carry that an informal prompt typically lacks?

Answer: Input and output types with ranges and null behavior, pre- and post-conditions, invariants, determinism and tie-breaking rules, failure modes for out-of-range or missing input, and at least one explicit performance constraint. An informal prompt almost always under-specifies tie-breaking and failure modes; leaving them implicit is how two engineers produce behaviorally different "correct" solutions.

Question 17: Reading Code​

You are handed an unfamiliar function and asked whether it has a bug. What are the two artifacts you should extract from any non-trivial loop or recursion before answering?

Answer: (1) The invariant: a sentence that is true each time the loop body is entered. (2) The termination measure: a quantity that is strictly monotonic toward a bound. Together they tell you what the loop is trying to preserve and why it ends; most real bugs are a violation of one or the other. Reading without extracting these two is guessing.

Question 18: Triage Ordering​

Under a deadline, a definitive sub-problem takes 20 minutes and a fast-but-shallow sub-problem takes 5 minutes. You have 30 minutes. How do you use them, and why?

Answer: Start the definitive sub-problem running first (in parallel or in the background if possible, since it needs the full 20 minutes anyway). Use the foreground time on the shallow sub-problem and any fast leads, then integrate the definitive result near the end. Reserve the last 10-15% of the budget for a written summary stating the decision, the evidence, what was not checked, and what would change the conclusion. This keeps you from blowing the budget on one path with no fallback.

Interleaved Review Questions​

Prior Module Question 1​

What is the difference between proof by cases and proof by contradiction?

Answer: Proof by cases exhaustively handles all legal scenarios and concludes the claim in each. Proof by contradiction assumes the negation and derives an impossibility. Cases is constructive; contradiction is not.

Prior Module Question 2​

Why is a probability without a sample space usually meaningless?

Answer: Because the probability rule P is defined on a sample space of outcomes. Without a specified sample space, the events are not well-defined and the numbers do not refer to anything.

Prior Module Question 3​

What is a loop invariant?

Answer: A property of program state that holds before the loop, is preserved by each iteration, and, together with the exit condition, implies the desired postcondition.

Prior Module Question 4​

Why is linearity of expectation a "method" rather than just a property?

Answer: Because it lets you compute expectations of sums by summing individual expectations, even when the summands are dependent. It converts many "hard to enumerate" expectation problems into trivial ones.

Prior Module Question 5​

What does it mean for a matrix to be singular?

Answer: Its determinant is zero, equivalently its columns (or rows) are linearly dependent, equivalently it has a non-trivial kernel. It is non-invertible.

Self-Assessment and Remediation​

Mastery Level (90-100% correct):

  • Ready to advance. Focus on consolidating transfer notes and making the Polya protocol your default.

Proficient Level (75-89% correct):

  • Review only the missed concept pages and redo two problems of each missed type. Rewrite any weak answers with full written-form reasoning.

Developing Level (60-74% correct):

  • Rework Practice 1 (Polya Framework) and Practice 3 (Structural Reasoning). Commit to a written plan block on every problem for the next two weeks.

Insufficient Level (<60% correct):

  • Return to Cluster 1 and Cluster 3 of the concept sequence. You are skipping the written phase artifacts; re-anchor the habit before advancing.