>The players in the baseball lineup wear uniform numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9.
>
>Each of the uniform numbers of the first four batters totals 21.
>
>Batting between players 1 and 2, are players whose uniforms total 18.
>
>Between players 2 and 3, the uniforms total 24.
>
>Between 3 and 4, the total is 14, and
>
>Between 4 and 5, the total is 9.

I'm not going to give the solution (yet), but for me I see a lovely site, if I put it in a (simple) graph

^      /\            /\            /\
|     /  \    /\    /  \    /\    /  \
|    /    \/\/  \/\/    \/\/  \/\/    \

Where
  /\
 /  \
/    \ = 7, 8 or 9

  /\
 /  \  = 5 or 6

  /\   = 1,2,3 or 4

Regards,