PK 2020 Q20
20. Kanga labelled the vertices of a square-based pyramid using 1, 2, 3, 4 and 5 once each. For each face Kanga calculated the sum of the numbers on its vertices. Four of these sums equalled 7, 8, 9 and 10. What is the sum for the fifth face?
- A 11
- B 12
- C 13
- D 14
- E 15
Pink Kangaroo 2020, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 13
One face has total 7 which can only be obtained from the given numbers by adding 1, 2 and 4. Therefore it is a triangular face, and so the label of the top vertex, \(x\) say, is one of these three values. Hence the square face has 5 at one vertex. So the smallest possible face total for the square is 5 + 1 + 2 + 3 = 11. Therefore the four face totals given, 7, 8, 9 and 10, must be the face totals of the triangular faces and their sum is 34. Note that each vertex except the top belongs to two triangles; and the top belongs to all four. So the sum 34 is twice the sum of all the labels plus an extra \(2x\); that is \(34 = 2(1 + 2 + 3 + 4 + 5) + 2x = 30 + 2x\). Hence \(x = 2\) and the face total for the square face is 1 + 3 + 4 + 5 = 13.
Solution © UK Mathematics Trust, from the PK 2020 solutions.