PK 2019 Q20
20. A network consists of 16 vertices and 24 edges that connect them, as shown. An ant begins at the vertex labelled Start. Every minute, it walks from one vertex to a neighbouring vertex, crawling along a connecting edge. At which of the vertices labelled \(P\), \(Q\), \(R\), \(S\), \(T\) can the ant be after 2019 minutes?
- A only \(P\), \(R\) or \(S\),
- B not \(Q\)
- C only \(Q\)
- D only \(T\)
- E all of the vertices are possible
Pink Kangaroo 2019, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C only \(Q\)
Labelling vertices alternately 0/1 leads to the labelling shown. After an odd number of steps, the ant is always on a vertex labelled 1. The only such vertex labelled with a letter is \(Q\).
Solution © UK Mathematics Trust, from the PK 2019 solutions.