19. Joseph writes the numbers 1 to 12 in the circles so that the numbers in adjacent circles differ by either 1 or 2 . Which pair of numbers does he write in adjacent circles?
A 3 and 4
B 5 and 6
C 6 and 7
D 8 and 9
E 8 and 10
SolUtion
Consider first the placement of the number 12 . Since the only numbers that differ from 12 by 1 or 2 are 10 and 11 , the two numbers on either side of 12 are 10 and 11 . Now consider the number 11. The numbers that differ by 1 or 2 from 11 are 12,10 and 9 and hence, since 12 and 10 are already placed, the number on the other side of 11 from 12 is 9 . Similarly, consider the number 10 . The numbers that differ by 1 or 2 from 10 are $12,11,9$ and 8 and hence, since all of these except 8 have already been placed, the number on the other side of 10 from 12 is 8 . Therefore Joseph writes 8 and 10 in adjacent circles.

The diagram can then be completed in the way shown below, which is unique apart from rotations and reflection about a line of symmetry. From this it can be seen that the only instances where an odd number is adjacent to an even number are when 11 is next to 12 and when 1 is next to 2. Therefore options A to $\mathrm{D}$ are all incorrect.