PK 2018 Q23
23. At each vertex of the 18-gon in the picture a number should be written which is equal to the sum of the numbers at the two adjacent vertices.
Two of the numbers are given. What number should be written at the vertex \(P\)?
- A 2018
- B 38
- C 18
- D −20
- E −38
Pink Kangaroo 2018, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 38
Let \(a\) and \(b\) be two adjacent numbers on the vertices. Then the next vertex after \(b\) will have the number \(b - a\); and that will be followed, in turn, by \(-a\), \(-b\), \(a - b\), then \(a\), \(b\) again. This shows that each number reappears every sixth vertex. Therefore the number at the vertex to the left of \(P\) is 20 and that to its right is 18. Thus the number at \(P\) is \(20 + 18 = 38\).
Solution © UK Mathematics Trust, from the PK 2018 solutions.