How does superintelligence work when an AI is asked which task must finish on time for a project to meet its deadline? A school magazine needs a plan, article drafts, images, fact-checking, layout and final publishing. Some tasks can run at the same time, while others cannot start until a predecessor finishes. A delay in one task may change the entire finish date, while another delay can be absorbed without changing it.
The mechanism is critical-path scheduling. A project is represented as tasks with durations and dependency links. By calculating the earliest start and finish for each task, a planner identifies the longest dependency chains that determine the earliest possible project finish. The difference between a task’s allowable delay and a deadline-changing delay is called slack or float under a specified scheduling convention. A reliable AI must check every predecessor and distinguish a hypothetical schedule from actual completed work.
This easy guide uses six completely fictional school-magazine tasks. We calculate a nine-day earliest finish, identify two equal-length critical paths, and test what happens when image preparation or drafting is delayed. The dates and durations are invented for teaching, not actual publishing work or a benchmark of any AI system. “How Superintelligence Works” is an educational series title and does not claim contemporary AI has achieved artificial superintelligence.
The project at a glance
- A — Outline: 2 days, no predecessor.
- B — Draft articles: 4 days, after A.
- C — Prepare images: 3 days, after A.
- D — Fact-check: 2 days, after B.
- E — Layout: 2 days, after BOTH B and C.
- F — Final publication step: 1 day, after BOTH D and E.
- Earliest finish: end of Day 9 from a Day 0 start.
- Two critical paths: A→B→D→F and A→B→E→F, each 9 days.
- C’s image work has 1 day of slack under the initial plan.
1. A project is more than a task list
A list names what must be done. A schedule also says how long each activity takes and which activities cannot begin until another has finished. These relationships determine which work can happen in parallel.
An AI that simply adds every duration would assume all tasks run one after another, overstating the time required when some can proceed simultaneously. An AI that assumes everything can overlap would ignore the dependencies and may promise an impossible date.
2. What does a dependency mean?
A finish-to-start dependency says one task must finish before another begins. In the fictional magazine project, drafting must follow the outline, and fact-checking must follow drafting.
The words before and after are not decorative; they encode a scheduling constraint. The assistant should build the dependency graph from those facts instead of using the order in which the user mentioned tasks.
3. A directed acyclic graph
We can represent each task as a node and each dependency as a directed arrow. Acyclic means the arrows do not create a loop that leads back to a task through its successors.
A cycle such as A waiting for B while B waits for A would prevent ordinary finish-to-start scheduling without some additional change. The fictional six-task plan is acyclic and therefore can be ordered for calculation.
4. Define the start of the timeline
We use abstract elapsed working days, with Day 0 as the beginning of the project. A two-day task starting on Day 0 finishes at the end of Day 2 in this simplified arithmetic.
This avoids ambiguity about weekend calendars, public holidays, half days and time zones. The source does not give a real date, so an AI must not convert Day 9 into a specific November calendar date by guesswork.
5. The fictional Cedar Magazine Project
Imagine the school magazine team planning a single publication. Its work is divided into outline, writing, image preparation, fact-checking, layout and final publication.
Every duration in the fixture is fixed for the purpose of the mathematical exercise. We also assume enough people and resources are available for tasks with no direct dependency to proceed in parallel. Real schedules need additional staffing and approval checks.
6. The six-task dependency ledger
- A — Outline; duration 2 days; predecessors: none.
- B — Draft articles; duration 4 days; predecessor: A.
- C — Prepare images; duration 3 days; predecessor: A.
- D — Fact-check; duration 2 days; predecessor: B.
- E — Layout; duration 2 days; predecessors: B and C.
- F — Final publication stage; duration 1 day; predecessors: D and E.
- All constraints are finish-to-start with zero transition time.
- Day 0 is the origin; there are no real calendar dates or completed operations.
7. Calculate the first task
A has no predecessor, so it can start at Day 0. With a two-day duration, its earliest finish is Day 2.
This is a planned early finish, not confirmation that an outline was actually written. The calculation determines what could happen if the stipulated durations and dependencies hold.
8. Find the earliest draft start
B must wait for A. Since A’s earliest finish is Day 2, B’s earliest start is Day 2. Its four days of work lead to an earliest finish of Day 6.
An assistant that starts drafting at Day 0 has violated the source dependency. The result might look faster but would not satisfy the specified project.
9. Start image work in parallel
C also depends only on A, so it can start at Day 2, alongside B. Its three-day duration gives an earliest finish of Day 5.
This is a legitimate overlap: drafting and image preparation run in parallel because the fixture has no dependency between them. The ability to overlap work reduces the total project time compared with a fully sequential list.
10. Fact-check follows the draft
D depends on B, which finishes at Day 6. With a two-day fact-checking duration, D starts at Day 6 and finishes at Day 8.
Fact-checking cannot be counted as completed merely because a draft is scheduled. A real workflow requires the actual content and verification work before declaring an article fact-checked.
11. Layout waits for two predecessors
E requires both drafting B and images C. B finishes at Day 6; C finishes at Day 5. Therefore E cannot start before the later of those two finishes, which is Day 6.
E takes two days and finishes at Day 8. The correct start is max(6,5)=6 rather than the average 5.5 or the earlier Day 5. Every required predecessor must be finished.
12. The final step waits for two branches
F depends on both D and E. Each has an earliest finish of Day 8. Therefore F’s earliest start is Day 8.
F requires one additional day and has an earliest finish at Day 9. This is the mathematical project completion time under the frozen rules, not a claim that anything was actually published.
13. The complete earliest schedule
- A: earliest start 0; finish 2.
- B: earliest start 2; finish 6.
- C: earliest start 2; finish 5.
- D: earliest start 6; finish 8.
- E: earliest start 6; finish 8.
- F: earliest start 8; finish 9.
- All durations use abstract days.
- The earliest project finish is Day 9.
- There are no real-world completion receipts in the teaching case.
14. Why the longest dependency chain matters
A dependency chain gives the minimum time required along its tasks, because every successor must wait for the predecessor. When several chains feed one final outcome, the longest chain sets a lower bound on how early the project can finish.
The method does not simply favour the largest individual task. A short task on a decisive chain can determine the end date, while a longer task on another branch may have some spare time.
15. First critical path: through fact-checking
The chain A→B→D→F has durations 2 + 4 + 2 + 1 = 9 days.
Because it ends at the project finish of Day 9, it is critical under the fixed-duration, resource-unconstrained schedule. Delaying a task on that chain, while holding all else fixed, can move the earliest project finish.
16. Second critical path: through layout
The chain A→B→E→F has durations 2 + 4 + 2 + 1 = 9 days. E depends on C too, but the B branch is the later predecessor under the baseline.
This is a second critical path, equal in total duration to the fact-checking route. A project can have more than one critical chain; reporting only one may hide another source of deadline risk.
17. The image branch is shorter
The chain A→C→E→F contains durations 2 + 3 + 2 + 1 = 8 days. E also needs B, so this eight-day chain does not by itself make layout begin earlier than Day 6.
Relative to the nine-day project finish, the image branch has one day of scheduling flexibility under the simplified model. This explains why a limited image delay need not postpone publication.
18. What slack means here
Slack or float is the amount an activity can be delayed without changing a specified project finish, under the relevant scheduling rules. In this example, C has one day of float: it could finish at Day 6 rather than Day 5 and E could still begin at Day 6.
Slack is not permission to ignore a task or guarantee that a real team can move it freely. It is a property of the hypothetical network schedule and may change if durations or resource constraints change.
19. Delay C by one day
Suppose image preparation takes four days instead of three, while beginning on Day 2. C would finish on Day 6. B also finishes on Day 6.
E still begins on Day 6 and finishes on Day 8. D is unchanged and finishes Day 8, so F runs Day 8–9. The extra image day uses up C’s slack without changing the earliest final date.
20. Delay C by two days
Now suppose C takes five days instead of three. Starting at Day 2, it finishes at Day 7. B still finishes Day 6, so layout E must wait until Day 7.
E then finishes at Day 9. Fact-checking D still finishes at Day 8. F must wait for both, so it runs Day 9–10. The project finishes one day later than the original plan.
21. Why two days of delay cause only one day of project slip
C originally had one day of slack. A two-day increase first consumes that one free day, then pushes E’s start one day later.
The resulting project shift is one day rather than two. The calculation helps planners distinguish a task’s own delay from the resulting deadline delay.
22. Delay B by one day
If drafting B takes five days rather than four, starting on Day 2, it finishes at Day 7. D then runs Day 7–9; layout E also must wait for B and runs Day 7–9.
F begins after both at Day 9 and finishes Day 10. Because B lies on both initial critical paths, a one-day draft delay moves the project finish by one day under the specified model.
23. Delay D by one day
If fact-checking D takes three days rather than two, it runs Day 6–9. Layout E still finishes Day 8.
F must wait for D’s Day 9 finish, and so it ends Day 10. The delayed fact-check branch becomes the later constraint. A single critical activity can move the deadline even while another branch stays on schedule.
24. Delay E by one day
If layout takes three days instead of two, it runs Day 6–9. D still finishes at Day 8, so F starts on Day 9 and completes on Day 10.
This is symmetrical to the fact-check delay in its effect on the final date, but the underlying work is different. The two critical paths meet before F.
25. Accelerating a non-bottleneck task may not help
Suppose C finishes a day faster, at Day 4 instead of Day 5. Layout still waits for B until Day 6, so the final publication stage still ends on Day 9.
Making image preparation faster has not improved the project completion time because the draft branch remains the limiting predecessor. An AI optimiser should identify the actual bottleneck before promising a shorter schedule.
26. Accelerating B alone may still leave two branches
Suppose B takes three days rather than four and finishes on Day 5. C also finishes at Day 5. D and E then both run Day 5–7, and F can finish at Day 8.
Here reducing B by one day helps because B appears on both initial critical paths and the C branch catches up at Day 5. Any further compression would require examining the new critical structure rather than assuming unlimited improvement.
27. Do not confuse duration with effort
A two-day task may require one worker for two full days or several workers for a shorter amount of time. The mathematical durations here are elapsed time needed under the stipulated resource plan.
An AI should not convert every duration directly to staff-hours without knowing the allocation. Critical-path scheduling primarily handles dependencies and elapsed task durations; real resource levelling adds another layer.
28. A task may have several predecessors
E cannot begin until both B and C are complete. This AND condition matters. If an assistant starts E when either B or C completes, it has changed the dependency from both to either.
The correct earliest start is the maximum of the predecessor finish times. This simple rule prevents the schedule from showing an impossible layout start before all required materials are ready.
29. A task may have several successors
B is needed by both D and E. A delay in B therefore propagates into two later activities rather than just one.
Such shared upstream tasks can be especially consequential in complex projects. The network shows which branches inherit a delay, but its actual impact still depends on other predecessors and available slack.
30. What is a topological ordering?
A topological ordering lists the tasks so every predecessor appears before its dependent successor. For this graph, A, B, C, D, E, F is one valid ordering, and A, C, B, E, D, F is another.
An ordering is not a claim that the tasks actually execute one at a time. It is a way to compute dependency-constrained schedules without visiting a successor before its necessary inputs.
31. The critical-path and delay scorecard
- Baseline project finish: Day 9.
- Critical path 1: A→B→D→F = 9 days.
- Critical path 2: A→B→E→F = 9 days.
- Image branch A→C→E→F = 8 days.
- Baseline C slack: 1 day.
- C duration 4 instead of 3: finish remains Day 9.
- C duration 5 instead of 3: project finish Day 10.
- B duration 5 instead of 4: project finish Day 10.
- D duration 3 instead of 2: project finish Day 10.
- E duration 3 instead of 2: project finish Day 10.
- C duration 2 instead of 3: finish remains Day 9.
- B duration 3 instead of 4: project finish Day 8.
32. Twenty-four claims for a careful AI
Use SUPPORTED for claims established by the network and duration rules, CONTRADICTED for conflicts with that model, and NOT ESTABLISHED for actual completion and authorisation claims absent from the source.
Planning calculations can prove that a timetable would be feasible under assumptions. They do not prove that the tasks were performed, the magazine was reviewed or publication succeeded.
- 01. “Task A lasts two days” — SUPPORTED.
- 02. “Task B depends on A” — SUPPORTED.
- 03. “Task C depends on A” — SUPPORTED.
- 04. “Task D depends on B” — SUPPORTED.
- 05. “Task E depends on both B and C” — SUPPORTED.
- 06. “Task F depends on both D and E” — SUPPORTED.
- 07. “The earliest A finish is Day 2” — SUPPORTED.
- 08. “The earliest B finish is Day 6” — SUPPORTED.
- 09. “The earliest C finish is Day 5” — SUPPORTED.
- 10. “The earliest D finish is Day 8” — SUPPORTED.
- 11. “The earliest E finish is Day 8” — SUPPORTED.
- 12. “The earliest F finish is Day 9” — SUPPORTED.
- 13. “The only critical path is A→C→E→F” — CONTRADICTED.
- 14. “Both A→B→D→F and A→B→E→F take nine days” — SUPPORTED.
- 15. “Baseline C has one day of slack” — SUPPORTED.
- 16. “Adding one day to C necessarily delays final completion” — CONTRADICTED.
- 17. “Adding two days to C moves the earliest finish to Day 10” — SUPPORTED.
- 18. “Increasing B by one day moves the earliest finish to Day 10” — SUPPORTED.
- 19. “Increasing D by one day moves the earliest finish to Day 10” — SUPPORTED.
- 20. “A shorter C duration alone guarantees completion by Day 8” — CONTRADICTED.
- 21. “Shortening B to three days can produce Day 8 completion” — SUPPORTED.
- 22. “The project was actually published on Day 9” — NOT ESTABLISHED.
- 23. “Every real editor approved the proposed schedule” — NOT ESTABLISHED.
- 24. “E may begin before both B and C finish under the given constraints” — CONTRADICTED.
33. Reconcile the answer key
The twenty-four claims comprise 18 supported, 4 contradicted and 2 not-established statements. They total 24. Each deadline and dependency fact follows the invented six-task network.
The unknown claims distinguish calculated project possibilities from actual published artefacts and human approvals. No article or magazine has been created by the fictional planning example.
34. Why fastest individual task is not always the key
F lasts one day, shorter than any of B, C, D or E. But it lies on both critical paths, so a delay in the final step could still delay completion.
Conversely, C lasts three days yet has slack. A model that ranks task importance only by duration can overlook where the task sits in the dependency graph.
35. A real schedule has uncertainty
Task durations may change because of revision requests, missing inputs, staff capacity or other events. A critical path calculated from estimates can move as actual information becomes available.
A trustworthy AI should report the baseline assumptions, show which activities are critical today and update the analysis when a genuine change is verified. It should not pretend that fixed toy durations predict real completion times with certainty.
36. Buffers and calendars need explicit rules
Real work may exclude weekends, need review appointments or require extra time between tasks. Our timeline uses abstract days with zero transition delay.
To produce an actual dated schedule, an assistant would need the work calendar, holidays, team capacity and approval requirements. Adding a date without those facts would turn a transparent duration model into an unsupported commitment.
37. Critical path is not always the only resource problem
Two tasks may have no dependency but still need the same person or computer. In that case they cannot necessarily run in parallel, even if the graph has no arrow between them.
Our fixture assumes sufficient resources for B and C to overlap after A. A more complete resource-constrained project schedule could have a different earliest finish.
38. A practical school transfer exercise
Imagine a school video requiring a script, voice recording, visuals, editing and approval. Some visual work can begin as soon as the plan is agreed, while editing might require both audio and visuals.
The same max-of-predecessor-finish rule determines when editing can start. An AI can help map dependencies, but must not report a video rendered or uploaded until real systems confirm those actions.
39. A classroom calculation activity
Give learners six task cards labelled A through F with durations and arrows. Have them write earliest start and finish numbers, then colour each nine-day path.
Next increase image duration by one and two days separately. Ask pupils to explain why the first extra day is absorbed but the second changes completion. This makes float visible without complicated software.
40. How AI builders can test project reasoning
Create networks with multiple critical paths, shared predecessors, non-critical tasks, changing durations and simultaneous constraints. Check whether the model computes the maximum predecessor finish rather than the minimum or average.
Also test whether the assistant invents calendar dates, external publication receipts or approval. Reliable scheduling requires both sound mathematics and accurate reporting of operational status.
41. The central lesson
A project’s finish date is controlled by its longest dependency chains, not simply by the sum of every task or the duration of the largest individual task. Some work has slack, while delays on critical paths can move the whole deadline.
A dependable AI planner keeps all predecessor rules visible, recalculates after changes and distinguishes a feasible schedule from completed actions. That turns a plausible timetable into a checkable model.
Frequently asked questions
- What is a critical path? — A longest dependency chain that controls the earliest project completion under the schedule assumptions.
- Can a project have two critical paths? — Yes, the Cedar example has two nine-day chains.
- What is slack? — Time a task can be delayed without changing a specified finish in the model.
- Why does a one-day image delay not affect the project? — C has one day of baseline slack.
- Why does a two-day image delay matter? — It exceeds that slack by one day.
- Does a calculated Day 9 finish prove publication occurred? — No. It is a plan, not an execution record.
- What if the same worker is needed by parallel tasks? — Resource constraints must be included in a more complete schedule.
Connected lessons and sources
- How Super Intelligence Works — main learning curriculum.
- Order and Dependencies — basic predecessor logic.
- Impossible Plans — resource and time constraints.
- Time Intervals and Overlapping Schedules.
- Critical Path Method — formal background on task networks and float.
- Google OR-Tools: Job Shop Scheduling — a distinct resource-constrained scheduling example.