IMC 2016 Q13
13. The diagram shows a heptagon with a line of three circles on each side. Each circle is to contain exactly one number. The numbers 8 to 14 are distributed as shown and the numbers 1 to 7 are to be distributed to the remaining circles. The total of the numbers in each of the lines of three circles is to be the same.
What is this total?
- A 18
- B 19
- C 20
- D 21
- E 22
Intermediate Mathematical Challenge 2016, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 19
In the seven lines each of the integers from 1 to 7 is used twice and each of the integers from 8 to 14 is used once. So the sum of the numbers in the seven lines is (1 + 2 + … + 14) + (1 + 2 + … + 7) = 105 + 28 = 133. Therefore the total of the numbers in each line is 133 ÷ 7 = 19.
It is left as an exercise for the reader to show that it is possible to complete the diagram so that the total of the three numbers in each line is indeed 19.
Solution © UK Mathematics Trust, from the IMC 2016 solutions. Longer versions with investigations: Solutions and investigations (PDF).