GK 2018 Q15
15. James wrote a different integer from 1 to 9 in each cell of a table. He then calculated the sum of the integers in each of the rows and in each of the columns of the table. Five of his answers were 12, 13, 15, 16 and 17, in some order. What was his sixth answer?
- A 17
- B 16
- C 15
- D 14
- E 13
Grey Kangaroo 2018, Question 15. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 17
The total of the sums of the three rows of the table is equal to the sum of all the digits from 1 to 9, which is 45. Similarly, the total of the sums of the three columns of the table is also equal to 45. Hence James' six answers add to 45 + 45 = 90. The sum of the five answers given is 12 + 13 + 15 + 16 + 17 = 73 and hence his sixth answer is 90 – 73 = 17.
[It is left as an exercise to find a possible arrangement of the digits 1 to 9 that actually gives these six sums.]
Solution © UK Mathematics Trust, from the GK 2018 solutions.