NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
← Back to practice

Networks and connectedness

Wide tables scroll horizontally. For detailed working, you may prefer a larger screen.

Required skills (click to reveal)

A walking route network in Melbourne connects 35 distinct park entrances (nodes) and 37 walking trails (edges) between them. Each trail connects exactly two entrances, and no trail is duplicated.

The network is designed so that every entrance is connected to exactly 2 other entrances via trails. Using the principle that the sum of all degrees in a network equals twice the number of edges, determine the exact number of entrances (nodes) in this network.

The sum of degrees is 2 × 37 = 74. Since each entrance has degree 2, the number of entrances is 74 / 2 = 37.

However, this contradicts the given 35 entrances. Therefore, the network cannot have 35 entrances and 37 edges with all nodes degree 2.

Re-evaluate: if 37 edges exist, sum of degrees is 2 × 37 = 74. If each of the 35 entrances has degree 2, sum of degrees would be 35 × 2 = 70, which is less than 74.

Hence, not all nodes can have degree 2. But the problem states every entrance is connected to exactly 2 other entrances.

So degree of each node is 2. Then sum of degrees = 2 × number of nodes = 2 × 35 = 70.

But sum of degrees must also equal 2 × number of edges = 2 × 37 = 74. 70 ≠ 74, contradiction. Therefore, the network as described is impossible.

But the question asks for the exact number of entrances given 37 edges and each entrance degree 2. So solve: sum of degrees = 2 × 37 = 74.

Each node has degree 2. Number of nodes = 74 / 2 = 37.

Thus, the number of entrances must be 37. The initial 35 is a distractor.

The correct number of entrances is 37.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Degree counts incident edges. This connected path has edges AB and BC. The degrees are 1 at A, 2 at B and 1 at C. Every pair of vertices is connected by a path.ABCdegree 1degree 2degree 1
This connected path has edges AB and BC. The degrees are 1 at A, 2 at B and 1 at C. Every pair of vertices is connected by a path.

Press Start when ready. Time includes time spent in another tab. Submit your answer before opening a help panel for a timed rank.

00:00
0 wordsOnly your numerical final answer or selected MCQ option is checked. Scratch working is not graded.

Report a problem or suggest an improvement

My notes — optional, not graded 0 words

Notes and scratch working stay in this browser and are linked to this question version. They are not graded or included in feedback. On shared devices, other people using this browser may see them.

For this question only. Your full notebook is on the dashboard.

Notes save in this browser. Download a copy to keep them elsewhere.

Report a problem or suggest an improvement

Your observations help improve this question. You can report an issue or recommend a better explanation, visual or experience.

0 words

The report includes your message and question/version details. Saved profile details, answers, notes and attempt history are not automatically attached. Leave out names and contact details, and check screenshots for personal information and embedded metadata.

Attachments may contain personal information. Review them before sharing. Logs collect only event codes and timestamps after you enable them. Download the ZIP to keep a copy, then attach it to an email if you choose.

Formula and explanation (click to reveal)

Opening this before your first submission makes the attempt practice only.

Formula

Explanation

Worked solution and answer (click to reveal)

Opening this before your first submission makes the attempt practice only.

Worked solution

  1. The network has 37 walking trails, each representing an edge in the graph. The sum of the degrees of all nodes is equal to twice the number of edges, according to the handshaking lemma.
  2. Substitute the number of edges into the formula: sum of degrees = 2 × 37 = 74.
  3. Each park entrance (node) is connected to exactly 2 other entrances, so each node has degree 2.
  4. Let N be the number of entrances (nodes). Since each node has degree 2, the sum of degrees is 2 × N.
  5. Set up the equation: 2 × N = 74.
  6. Solve for N: N = 74 ÷ 2 = 37.
  7. The initial claim of 35 entrances contradicts the degree sum, as 35 × 2 = 70 ≠ 74.
  8. Therefore, the actual number of entrances must be 37 to satisfy both the edge count and the degree condition.
  9. Check: 37 nodes × degree 2 = 74; 2 × 37 edges = 74. The values match.
  10. The network is consistent only when there are 37 entrances.
  11. The correct number of entrances is 37.
  12. Final answer confirmed by degree sum and edge count consistency.

Reference answer: 37

Alternative methods (click to reveal)

Shortcuts are shown only when applicable. Check the conditions before using one.

Earlier attempts for this question

#Date / timeYour answerResultSecondsHelp revealedRank
History stays in this browser when storage is available. Export it to your shared Drive folder and import it on another computer.
Worksheet details

Naming pattern: Topic_Grade_Complexity_YYYYMMDD_HH24MISS.html

Validation compares your answer with the generated reference. The model's reference can be wrong. Time ranks use an estimated student solving time, not a measured benchmark or a comparison with other students. An incorrect answer receives no speed rank.

NavGanitMate privacy & saved work

Saved in this browser. Your profile preferences, answers, practice history and notes are stored here when browser storage is available. They are not automatically uploaded to our servers. People sharing this browser may see saved work; changing the learner name does not create a separate private account. A nickname is enough.

Keep copies of important work. Clearing browser data, private browsing or changing browser or device can mean losing saved work. Download notes and export history separately. Downloads are copies of the selected content, not a complete account backup; not every format can be imported to restore your work. Keep these files somewhere private.

Visiting the website. Cloudflare helps deliver and protect this site and processes connection information such as IP addresses and requested pages. Server error logs may also contain request details. Security features may use cookies. This is separate from your browser-saved learning records. Cloudflare privacy information.

A parent-connection reminder uses a local list of up to five completed question IDs for this browser profile and a tab-session dismissal marker. Reminder activity is not uploaded and practice remains available.

Sharing feedback is your choice. A report includes your message and question/version details. Your saved profile, answers and notes are not automatically attached. Files you select can reveal personal details or contain embedded metadata: check and redact them before sharing. Diagnostic event codes and timestamps are included only after you enable logging. Sending email shares your message and attachments with the recipient and email providers. A download alone sends nothing to us. Website feedback delivery may be unavailable; only a confirmed receipt indicates it was received.