Explain why Eulerian circuit impossible

3 A local council is responsible for gritting roads. The diagram shows the length, in miles, of the roads that have to be gritted. \includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-03_671_686_488_669} Total length \(= 87\) miles The gritter is based at \(A\), and must travel along all the roads, at least once, before returning to \(A\).

1. Explain why it is not possible to start from \(A\) and, by travelling along each road only once, return to \(A\).
2. Find an optimal 'Chinese postman' route around the network, starting and finishing at \(A\). State the length of your route.

1 The diagram shows the layout of a Mediterranean garden. Thick lines represent paths. \includegraphics[max width=\textwidth, alt={}, center]{aac29742-fee8-48a9-896c-e96696742251-2_961_1093_440_468}

1. Draw a graph to represent this information using the vertices listed below, and with arcs representing the 18 paths. Vertices: patio (pa); pool (po); top steps (ts); orange tree (or); fig tree (fi); pool door (pd); back door (bd); front door (fd); front steps (fs); gate (gat); olive tree (ol); garage (gar). [2] Joanna, the householder, wants to walk along all of the paths.
2. Explain why she cannot do this without repeating at least one path.
3. Write down a route for Joanna to walk along all of the paths, repeating exactly one path. Write down the path which must be repeated. Joanna has a new path constructed which links the pool directly to the top steps.
4. Describe how this affects Joanna's walk, and where she can start and finish. (You are not required to give a new route.)

1 The design for the lines on a playing area for a game is shown below. The letters are not part of the design. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-2_350_855_388_605} Priya paints the lines by pushing a machine. When she is pushing the machine she is about a metre behind the point being painted. She must not duplicate any line by painting it twice.

• To relocate the machine, it must be stopped and then started again to continue painting the lines.
• When the machine is being relocated it must still be pushed along the lines of the design, and not 'cut across' on a diagonal for example.
• The machine can be turned through \(90 ^ { \circ }\) without having to be stopped.
  1. What is the minimum number of times that the machine will need to be started to paint the design?

The design is horizontally and vertically symmetric. $$\mathrm { AB } = 6 \text { metres, } \mathrm { AE } = 26 \text { metres, } \mathrm { AF } = 1.5 \text { metres and } \mathrm { AS } = 9 \text { metres. }$$

(a) Find the minimum distance that Priya needs to walk to paint the design. You should show enough working to make your reasoning clear but you do not need to use an algorithmic method.
(b) Why, in practice, will the distance be greater than this?
(c) What additional information would you need to calculate a more accurate shortest distance?

7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-007_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).

1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.

7 Stella is visiting Tijuana on a day trip. The diagram shows the lengths, in metres, of the roads near the bus station. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-06_1539_1162_495_424} Stella leaves the bus station at \(A\). She decides to walk along all of the roads at least once before returning to \(A\).

1. Explain why it is not possible to start from \(A\), travel along each road only once and return to \(A\).
2. Find the length of an optimal 'Chinese postman' route around the network, starting and finishing at \(A\).
3. At each of the 9 places \(B , C , \ldots , J\), there is a statue. Find the number of times that Stella will pass a statue if she follows her optimal route.

Norris delivers newspapers to houses on an estate. The network shows the streets on the estate. The number on each edge shows the length of the street, in metres. Norris starts from the newsagents located at vertex \(A\), and he must walk along all the streets at least once before returning to the newsagents. \includegraphics{figure_5} The total length of the streets is 1215 metres.

1. Give a reason why it is not possible to start at \(A\), walk along each street once only, and return to \(A\). [1]
2. Find the length of an optimal Chinese postman route around the estate, starting and finishing at \(A\). [5]
3. For an optimal Chinese postman route, state:
   1. the number of times that the vertex \(F\) would occur; [1]
   2. the number of times that the vertex \(H\) would occur. [1]