Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 95 minutes, as an unvalidated planning estimate. This collection contains 4 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
State domains, initial conditions, enabled transitions and the exact operation being counted before proving a claim.
Attempt the changed cases before opening solutions. Explain safety, termination and cost separately; graphs or successful examples alone do not establish them.
Public solutions and rubrics support revision with a competent reader. Recorded responses are evidence of work, not automatic proof certification or an independently invigilated examination.
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Two accounts have capacity7, total10, initial state(3,7), and permitted transfer sizes4 and6 in either direction. Characterize all reachable states and decide whether(5,5) is reachable, using a residue argument and a complete closed-set certificate. Then add size2 and construct a route to(5,5). Separately use capacity4, total4, initial(2,2) and sizes3,5 to refute the claim that gcd one always suffices. State why a finite queue search stopped before exhausting successors cannot certify impossibility.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A pipeline must produce and process N jobs, where N is a fixed nonnegative integer. State (p,q,d) records produced, queued and processed jobs; initially all are zero. A Produce transition, enabled when p<N, adds one to p and q. A Process transition, enabled when q>0, subtracts one from q and adds one to d. Either enabled transition may be chosen; when none is enabled, execution stops. Discover and prove a sufficient invariant, characterize every reachable terminal state, and prove termination for every choice sequence using R=2(N−p)+q. Derive the exact number of transitions. What fails if Process forgets to increment d? What changes if an always-enabled no-op is added?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Change the counter machine: from natural-number state (a,b), if b>0 subtract one from b; if b=0 and a>0 replace it by (a−1,3a+1), using the old a. Stop at (0,0). Prove termination without a bound on the initial counters. Derive an exact remaining-transition rank as a finite sum, simplify it, and prove one-unit decrease at both kinds of transition. Compute the total work from (3,2). Refute the claim that a+b decreases at every step, and explain why measured browser time is a different quantity.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A procedure on n records does nothing for n=0 or 1. For n≥2 it splits into floor(n/2) and ceil(n/2), processes both recursively, then performs three charged operations per original record and two additional charged operations. Derive its exact cost E(n). Let C(n) count one charged operation per record at each nonleaf call in the same split tree, with C(0)=C(1)=0. Prove E(n)=3C(n)+2(n−1) for n≥1, and use C(n)=n(k+1)−2^k, k=ceil(log₂n), to calculate E(7) and state an all-integer Θ(n log n) bound with explicit constants and threshold. Explain what would change if the split were n−1 and 1. Do not treat n=0 as an input to log₂.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Two accounts have capacity7, total10, initial state(3,7), and permitted transfer sizes4 and6 in either direction. Characterize all reachable states and decide whether(5,5) is reachable, using a residue argument and a complete closed-set certificate. Then add size2 and construct a route to(5,5). Separately use capacity4, total4, initial(2,2) and sizes3,5 to refute the claim that gcd one always suffices. State why a finite queue search stopped before exhausting successors cannot certify impossibility.
A pipeline must produce and process N jobs, where N is a fixed nonnegative integer. State (p,q,d) records produced, queued and processed jobs; initially all are zero. A Produce transition, enabled when p<N, adds one to p and q. A Process transition, enabled when q>0, subtracts one from q and adds one to d. Either enabled transition may be chosen; when none is enabled, execution stops. Discover and prove a sufficient invariant, characterize every reachable terminal state, and prove termination for every choice sequence using R=2(N−p)+q. Derive the exact number of transitions. What fails if Process forgets to increment d? What changes if an always-enabled no-op is added?
Change the counter machine: from natural-number state (a,b), if b>0 subtract one from b; if b=0 and a>0 replace it by (a−1,3a+1), using the old a. Stop at (0,0). Prove termination without a bound on the initial counters. Derive an exact remaining-transition rank as a finite sum, simplify it, and prove one-unit decrease at both kinds of transition. Compute the total work from (3,2). Refute the claim that a+b decreases at every step, and explain why measured browser time is a different quantity.
A procedure on n records does nothing for n=0 or 1. For n≥2 it splits into floor(n/2) and ceil(n/2), processes both recursively, then performs three charged operations per original record and two additional charged operations. Derive its exact cost E(n). Let C(n) count one charged operation per record at each nonleaf call in the same split tree, with C(0)=C(1)=0. Prove E(n)=3C(n)+2(n−1) for n≥1, and use C(n)=n(k+1)−2^k, k=ceil(log₂n), to calculate E(7) and state an all-integer Θ(n log n) bound with explicit constants and threshold. Explain what would change if the split were n−1 and 1. Do not treat n=0 as an input to log₂.
