What you practice
Each set draws new directed networks from a source S to a sink T, with a capacity on every edge. Work out the capacity of the dashed cut, find the minimum cut and the maximum flow, then read a second drawing that shows one flow: find the missing flow x, the edges at full capacity, the total flow and a path that can take more. Every drawn cut has an edge crossing back towards the source, because counting it is the slip both the HSC and VCE reports name.
Example: the capacity of a cut
In set CUTS23 (HSC) children move through the one-way paths of a maze from S to T. The dashed cut crosses AD (4), BD (10) and BE (4) from the source side, and CA from the sink side back towards the source. Its capacity is 4 + 10 + 4 = 18 children per minute, with CA counting 0. The smallest cut crosses BE (4), CT (6) and DT (5), so the maximum flow is 4 + 6 + 5 = 15 children per minute. Open set CUTS23
Questions
How do you work out the capacity of a cut?
Add the capacities of the edges that cross the cut from the source side to the sink side. An edge that crosses back towards the source counts 0. The 2025 HSC feedback and the VCE examination reports both name counting those backward edges as a common slip.
What is the maximum-flow minimum-cut theorem?
The maximum flow from the source to the sink equals the capacity of the smallest cut. Every cut is an upper limit on the flow, so a flow that equals the capacity of some cut is a maximum flow, and that cut is a minimum cut.
How do you find a missing flow in a network?
At every vertex except the source and the sink, the flow in equals the flow out. Add the flows going in, take away the other flows going out, and what is left is the missing flow. The 2025 HSC paper found x this way.
What does full capacity mean in a flow network?
An edge is at full capacity, or saturated, when its flow equals its capacity. In a maximum flow, every edge that crosses a minimum cut from the source side is at full capacity.
How can the flow be increased?
Look for a path from the source to the sink where every edge still has room, meaning its capacity is more than its flow. The flow can grow by the smallest room on that path. It can never go past the capacity of the minimum cut.
Sources, read October 28, 2026: NESA HSC Mathematics Standard 2 papers, marking guidelines and feedback, 2025, 2024 and 2023; NSW Department of Education, Year 12 Mathematics Standard 2 networks resources; VCE General Mathematics study design; VCAA General Mathematics examinations and reports, 2023 to 2025.