15. Molly has a set of cards numbered 1 to 12 . She places eight of them at the vertices of an octagon so that the sum of every pair of numbers at opposite ends of an edge of the octagon is a multiple of 3 .
Which numbers did Molly not place?
A $1,5,9$ and 12
B $3,5,7$ and 9
C $1,2,11$ and 12
D 5,6,7 and 8
E $3,6,9$ and 12
Solution
If one of the numbers at a vertex is a multiple of 3 then the number at every neighbouring vertex must also be a multiple of 3 .

Since we have only 4 multiples of 3 we come to a contradiction. So the numbers $3,6,9,12$ must be removed. The diagram on the right shows one way the remaining numbers could be placed.