NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Spatial problem-solving algorithms

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A city planner is designing a new pedestrian pathway network in Melbourne that connects five key locations: the Royal Botanic Gardens, Federation Square, Melbourne Cricket Ground, Flinders Street Station, and the Melbourne Arts Centre. The planner uses a digital mapping tool to model the network as a weighted graph, where edge weights represent walking distances in metres.

The direct walking distances (in metres) between the locations are as follows: - Royal Botanic Gardens to Federation Square: 1200 - Federation Square to Melbourne Cricket Ground: 1400 - Melbourne Cricket Ground to Flinders Street Station: 1100 - Flinders Street Station to Melbourne Arts Centre: 1300 - Melbourne Arts Centre to Royal Botanic Gardens: 1500 - Federation Square to Melbourne Arts Centre: 1000 - Melbourne Cricket Ground to Flinders Street Station: 1200 - Royal Botanic Gardens to Melbourne Arts Centre: 1600 - Federation Square to Flinders Street Station: 1300 - Royal Botanic Gardens to Melbourne Cricket Ground: 1800 The planner wants to find the shortest path from the Royal Botanic Gardens to Flinders Street Station that visits no location more than once. Using a computational approach to test all valid paths, what is the exact length in metres of the shortest such path?

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Worked solution

  1. The problem requires finding the shortest path from Royal Botanic Gardens to Flinders Street Station, visiting no location more than once. The network is modelled as a weighted graph with 5 nodes and given edge weights in metres.
  2. List all valid paths from Royal Botanic Gardens to Flinders Street Station that visit each location at most once. The path must not revisit any node.
  3. Enumerate all simple paths (no repeated nodes) from Royal Botanic Gardens to Flinders Street Station. The nodes are: R (Royal Botanic Gardens), F (Federation Square), M (Melbourne Cricket Ground), S (Flinders Street Station), A (Melbourne Arts Centre).
  4. List all valid 4-edge paths (since 5 nodes, a path visits 4 edges) from R to S, with no repeated nodes. Possible paths are determined by permutations of the intermediate nodes that maintain connectivity via given edges.
  5. Path 1: R → F → M → S. Edges: R-F (1200), F-M (1400), M-S (1100). Sum: 1200 + 1400 + 1100 = 3700 metres.
  6. Path 2: R → F → A → S. Edges: R-F (1200), F-A (1000), A-S (1300). Sum: 1200 + 1000 + 1300 = 3500 metres.
  7. Path 3: R → F → S. Edges: R-F (1200), F-S (1300). Sum: 1200 + 1300 = 2500 metres. This path visits only 3 nodes, but all 5 are to be considered. However, the problem does not require visiting all nodes, only that no node is visited more than once. This path is valid.
  8. Path 4: R → A → F → S. Edges: R-A (1600), A-F (1000), F-S (1300). Sum: 1600 + 1000 + 1300 = 3900 metres.
  9. Path 5: R → A → S. Edges: R-A (1600), A-S (1300). Sum: 1600 + 1300 = 2900 metres.
  10. Path 6: R → M → S. Edges: R-M (1800), M-S (1100). Sum: 1800 + 1100 = 2900 metres.
  11. Path 7: R → F → M → A → S. Edges: R-F (1200), F-M (1400), M-A (1200), A-S (1300). Sum: 1200 + 1400 + 1200 + 1300 = 5100 metres.
  12. Compare all valid paths: shortest is R → F → S = 2500 metres. All other paths are longer. No shorter path exists under the constraints.

Reference answer: 2500

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