GK 2017 Q17
17. Tom has a list of nine integers: 1, 2, 3, 4, 5, 6, 7, 8 and 9. He creates a second list by adding 2 to some of the integers in the first list and by adding 5 to all of the other integers in the first list. What is the smallest number of different integers he can obtain in the second list?
- A 5
- B 6
- C 7
- D 8
- E 9
Grey Kangaroo 2017, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 6
Since 5 and 2 differ by 3, Tom can obtain the same integer from two different integers in the first list that also differ by 3 by adding 5 to the smaller integer and adding 2 to the larger integer. In the first list there are six pairs of integers that differ by 3, namely 1 and 4, 2 and 5, 3 and 6, 4 and 7, 5 and 8 and 6 and 9. However, the integers 4, 5 and 6 appear in two of these pairs and hence the same integer in the second list can be obtained from only three pairs of integers from the first list leaving three integers in the first list unpaired. Therefore, the smallest number of different integers Tom can obtain in the second list is six.
Solution © UK Mathematics Trust, from the GK 2017 solutions.