Consider the following table
\begin{array}{c|c|c|c|c|c|c|lcr} &the&A.J.& team &the &G.N.&team \\ \hline & 1 & 2 & John & 1 &2& Mark \\ \hline Butterfly & 5 & 3 & 1&6&4&2 \\ Backstroke & 6 & 5 & 2&4&3&1 \\ Breaststroke & 5 & 4 &1 &6&3&2 \end{array}
And this one
\begin{array}{c|c|c|lcr} & \text{Butterfly} & \text{Backstroke} & \text{Breaststroke} \\ \hline Butterfly & 1 & -1 & \\ Backstroke & & & \\ Breaststroke & & & \end{array}
- Where 1 came from:
(I think in these 3 equations Mark and John participate doing all styles)
6-3=3 (As both participated, John won 5 points,Mark won 3 points and '2 swimmer' won 1 point, hence 5+1=6 less 3 because this is from the oponent team)
3-6=-3
5-4=1
Then 3-3+1=1
And -1 came from
3-6=-3 (I think this was because John did not participate and Mark did)
5-4=1 (I think this was because John did participate and Mark didn't)
5-4=1 (I think this was because John and Mark participated both)
Then -3+1+1=-1
Knowing this information and the points given (see the statement below) what should be the next number in the same row?
And then in the rest of the table?
(The original problem started with this: The A. J. Swim Team soon will have an important swim meet with the G. N. Swim Team. Each team has a star swimmer (John and Mark, respectively) who can swim very well in the 100- yard butterfly, backstroke, and breaststroke events. However, the rules prevent them from being used in more than two of these events.
Therefore, their coaches now need to decide how to use them to maximum advantage. Each team will enter three swimmers per event (the maximum allowed). For each event, the following table gives the best time previously achieved by John and Mark as well as the best time for each of the other swimmers who will definitely enter that event. (Whichever event John or Mark does not swim, his team’s third entry for that event will be slower than the two shown in the table (first table).)
The points awarded are 5 points for first place, 3 points for second place, 1 point for third place, and none for lower places. Both coaches believe that all swimmers will essentially equal their best times in this meet. Thus, John and Mark each will definitely be entered in two of these three events.
Each coach has three strategies-which event to hold the star swimmer out of. The second table should show the result of the meet based on the strategy each coach selects. In the upper left, both stars sit out the butterfly, so the order of finish is AJ2, GN2, AJ3, which gives AJ six points for first and third and GN three points for second. In the backstroke, GN places first and third and takes six points to AJ's three. In the breaststroke AJ wins for five and GN takes second and third for four. The overall match is $1$ point in AJ's favor, which is the $1$.
You can go through the table with this same approach. Find the top three swimmers in each race given the participation or not of the stars. See who wins the meet. I assume the next question is what strategy each coach should follow.