Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 85 minutes, as an unvalidated planning estimate. This collection contains 3 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.
Attempt the unfamiliar constructions before revealing solutions. State every resource and probability assumption.
Support an optimum with a feasible construction and a lower bound. Derive probabilities from the specified sample space.
Record assistance and guidance exposure. Public worked solutions and saved arguments require competent human review; they are not independent proficiency evidence.
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0 of 3 tasks have recorded responses. No overall mastery score is assigned.
A unit-time precedence graph has a starting layer of n−t+1 independent jobs, all preceding a chain of t−1 jobs. For an arbitrary integer deadline D≥t, derive and prove the minimum worker count. Apply it to n=8,t=4 with deadlines4,5,6. Compare a four-job chain plus four isolated jobs at deadline4. Explain why the total-work bound and width alone do not answer the relaxed-deadline question.
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 finite DAG has positive task durations d(v) and nonnegative release times r(v). Unlimited workers are available, but no task may start before its release or before all predecessors finish. Derive and prove a source-removal recurrence for the optimal completion time. State an invariant and a lower-bound certificate for arbitrary DAGs, including the empty graph. Evaluate A:(duration 2, release 4), B:(3,0), C:(4,8), D:(1,0), with A→C, B→C, C→D and B→D. Diagnose both the duration-only claim and the proposed repair that considers release times only at original sources. Explain why a certifying path may start at an internal vertex.
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.
Six labeled one-minute tasks form three disjoint prerequisite pairs A₁→A₂, B₁→B₂ and C₁→C₂. Choose an unordered three-task subset uniformly. Only selected tasks are performed, and a prerequisite edge applies only when both its endpoints are selected; an omitted predecessor imposes no condition in this synthetic model. All selected tasks are available at time zero. With unlimited workers let T be the minimum completion time and X the number of fully selected pairs. Derive the full distribution of T and E[T] using indicators, justify all scheduling claims, and test whether two distinct pair indicators are independent. Then determine the optimal completion time with two workers for every possible selected subset. Explain why substituting the new worker limit into the old recurrence is invalid.
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
A unit-time precedence graph has a starting layer of n−t+1 independent jobs, all preceding a chain of t−1 jobs. For an arbitrary integer deadline D≥t, derive and prove the minimum worker count. Apply it to n=8,t=4 with deadlines4,5,6. Compare a four-job chain plus four isolated jobs at deadline4. Explain why the total-work bound and width alone do not answer the relaxed-deadline question.
A finite DAG has positive task durations d(v) and nonnegative release times r(v). Unlimited workers are available, but no task may start before its release or before all predecessors finish. Derive and prove a source-removal recurrence for the optimal completion time. State an invariant and a lower-bound certificate for arbitrary DAGs, including the empty graph. Evaluate A:(duration 2, release 4), B:(3,0), C:(4,8), D:(1,0), with A→C, B→C, C→D and B→D. Diagnose both the duration-only claim and the proposed repair that considers release times only at original sources. Explain why a certifying path may start at an internal vertex.
Six labeled one-minute tasks form three disjoint prerequisite pairs A₁→A₂, B₁→B₂ and C₁→C₂. Choose an unordered three-task subset uniformly. Only selected tasks are performed, and a prerequisite edge applies only when both its endpoints are selected; an omitted predecessor imposes no condition in this synthetic model. All selected tasks are available at time zero. With unlimited workers let T be the minimum completion time and X the number of fully selected pairs. Derive the full distribution of T and E[T] using indicators, justify all scheduling claims, and test whether two distinct pair indicators are independent. Then determine the optimal completion time with two workers for every possible selected subset. Explain why substituting the new worker limit into the old recurrence is invalid.
