Capacity flow routes
Respect edge capacities and conserve quantities at intermediate nodes.
Adult planning suggestion: about9–12 · Logic & rule checking
- Players
- 2–4 per group
- Time
- 20–25 minutes
- Materials
- S/A/B/T node map, five arrow-capacity labels and flow table; no water.
Prepare
Draw directed arrows S→A capacity 3, S→B capacity 2, A→B capacity 1, A→T capacity 2 and B→T capacity 3. S supplies and T receives; A and B cannot store anything between rounds. Capacity is a maximum per shared round, not a cost or distance.
Play
Assign nonnegative whole-number flow to every arrow, no larger than its label. At A and B total inflow must equal total outflow. Add the flow entering T and try to make it as large as possible. Erase and revise together; no real water or physical racing is involved.
Worked example
Send 3 on S→A,2 on S→B,1 on A→B,2 on A→T and 3 on B→T. A receives 3 and sends 1+2. B receives 2+1 and sends 3. T receives 5. Sending 3 on A→T would violate its capacity 2 even if supply were available.
Finish and check
Show why five is maximal: only 3+2 can leave S. A feasible flow reaching that upper bound proves this example’s maximum. Drawing a promising route alone does not prove a maximum. Finish after a partner checks every capacity and both balance equations.
Adaptation and limits
Thirty learners can form ten trios with allocator, accountant and checker. Supply a table for arithmetic and keep arrows prominent. The technical reference concerns network flow; our small paper network and upper-primary grouping are a separate teaching adaptation.
Purpose, observation and provenance
Within this activity: Respect edge capacities and conserve quantities at intermediate nodes.
Check whether a proposed improvement violates conservation or merely shifts the bottleneck.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
CP Algorithms — Maximum flow — Technical capacity and conservation definitions; our four-node network and classroom task are original.
Age bands are planning suggestions. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Player materials · Capacity flow routes
S→A cap 3 / S→B cap 2 / A→B cap 1 / A→T cap 2 / B→T cap 3.
For each arrow: flow ___ ≤ capacity ___; A balance ___=___; B balance ___=___ .
| Arrow | Capacity | Flow |
|---|---|---|
| S→A | 3 | ___ |
| S→B | 2 | ___ |
| A→B | 1 | ___ |
| A→T | 2 | ___ |
| B→T | 3 | ___ |
Player record · keep the host key separate
| Arrow | Capacity | Chosen flow |
|---|---|---|
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
| ________ | ________ | ________ |
Host key · separate from players
Flow (3,2,1,2,3), value 5; source outgoing cut capacity 5 proves maximal.
Run the round
Draw directed arrows S→A capacity 3, S→B capacity 2, A→B capacity 1, A→T capacity 2 and B→T capacity 3. S supplies and T receives; A and B cannot store anything between rounds. Capacity is a maximum per shared round, not a cost or distance.
Assign nonnegative whole-number flow to every arrow, no larger than its label. At A and B total inflow must equal total outflow. Add the flow entering T and try to make it as large as possible. Erase and revise together; no real water or physical racing is involved.
Send 3 on S→A,2 on S→B,1 on A→B,2 on A→T and 3 on B→T. A receives 3 and sends 1+2. B receives 2+1 and sends 3. T receives 5. Sending 3 on A→T would violate its capacity 2 even if supply were available.
Show why five is maximal: only 3+2 can leave S. A feasible flow reaching that upper bound proves this example’s maximum. Drawing a promising route alone does not prove a maximum. Finish after a partner checks every capacity and both balance equations.
Thirty learners can form ten trios with allocator, accountant and checker. Supply a table for arithmetic and keep arrows prominent. The technical reference concerns network flow; our small paper network and upper-primary grouping are a separate teaching adaptation.
Related activities
Activity library · Weighted bridge network: rules and printable materials