22. Andrew wants to write the numbers $1,2,3,4,5,6$ and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. Which number should he write in the top circle?
A 2
B 3
C 4
D 5
E 6
Solution
C
Let the numbers in the circles be $p, q, r, s, t, u$ and $v$, as shown. Since these numbers are $1,2,3,4,5,6$ and 7 in some order,
$$
p+q+r+s+t+u+v=1+2+3+4+5+6+7=28
$$

Let the sum of the numbers in each line of three circles be $X$. Consider the five lines of three circles. We have
$$
(p+q+t)+(p+r+u)+(p+s+v)+(q+r+s)+(t+u+v)=5 X
$$
. Therefore
$$
3 p+2 q+2 r+2 s+2 t+2 u+2 v=5 X
$$
and hence
$$
p+2(p+q+r+s+t+u+v)=5 X
$$
. Therefore
$$
5 X=p+2 \times 28=p+56
$$
, and hence $p+56$ is a multiple of 5 . Since $p$ is one of the numbers $1,2,3,4,5,6$ and 7 , the only number for which this is possible is $p=4$.
Note: This shows that if a solution exists, then $p=4$. It can then be shown that such a solution is possible, for example with $p=4, q=3, r=7, s=2, t=5, u=1, v=6$.