Cincinnati Reds First-Place Eliminiation Scenario
Cincinnati must win 13 more games (69 total) to avoid elimination from first place.
Here is a scenario in which Cincinnati can finish in first place with 69 wins.

To construct the scenario, we start with the current standings in the division:
Team             W	L	GL	PCT	GB
Milwaukee        63	43	56	0.594	0.0
Cincinnati       56	50	56	0.528	7.0
St. Louis        53	52	57	0.505	9.5
Chicago          51	56	55	0.477	12.5
Pittsburgh       40	65	57	0.381	22.5

Next, we show how the remaining series of games plays out in the scenario:

Chicago wins 6 games against Cincinnati and loses 0.
Chicago wins 7 games against Milwaukee and loses 0.
Chicago wins 1 games against Pittsburgh and loses 6.
Chicago wins 0 games against St. Louis and loses 7.
Chicago wins 0 games agains teams outside the division and loses 0.

Cincinnati wins 0 games against Chicago and loses 6.
Cincinnati wins 3 games against Milwaukee and loses 0.
Cincinnati wins 6 games against Pittsburgh and loses 7.
Cincinnati wins 4 games against St. Louis and loses 2.
Cincinnati wins 0 games agains teams outside the division and loses 0.

Milwaukee wins 0 games against Chicago and loses 7.
Milwaukee wins 0 games against Cincinnati and loses 3.
Milwaukee wins 0 games against Pittsburgh and loses 6.
Milwaukee wins 6 games against St. Louis and loses 7.
Milwaukee wins 0 games agains teams outside the division and loses 0.

Pittsburgh wins 6 games against Chicago and loses 1.
Pittsburgh wins 7 games against Cincinnati and loses 6.
Pittsburgh wins 6 games against Milwaukee and loses 0.
Pittsburgh wins 10 games against St. Louis and loses 0.
Pittsburgh wins 0 games agains teams outside the division and loses 0.

St. Louis wins 7 games against Chicago and loses 0.
St. Louis wins 2 games against Cincinnati and loses 4.
St. Louis wins 7 games against Milwaukee and loses 6.
St. Louis wins 0 games against Pittsburgh and loses 10.
St. Louis wins 0 games agains teams outside the division and loses 0.

Combining the current standings with the wins and losses from the scenario results in the following final standings:
Team             W	L	PCT	GB
Cincinnati       69	93	0.426	0.0
Milwaukee        69	93	0.426	0.0
Pittsburgh       69	93	0.426	0.0
St. Louis        69	93	0.426	0.0
Chicago          65	97	0.401	4.0


Next, we explain why Cincinnati cannot win the division with fewer than 69 wins.

There are a total of 78 games left in the following series of games between teams in the National League Central:
Chicago          vs. Cincinnati       6 games
Chicago          vs. Milwaukee        7 games
Chicago          vs. Pittsburgh       7 games
Chicago          vs. St. Louis        7 games
Cincinnati       vs. Milwaukee        3 games
Cincinnati       vs. Pittsburgh       13 games
Cincinnati       vs. St. Louis        6 games
Milwaukee        vs. Pittsburgh       6 games
Milwaukee        vs. St. Louis        13 games
Pittsburgh       vs. St. Louis        10 games

For each team listed above, the table below shows the current number of wins in the (W) column.
The next column shows the maximum number of its remaining games that each team can win assuming that
Cincinnati finishes in first place with a total of 68 wins.

Team             W	Max Additional Wins
Chicago          51	68 - 51 = 17
Cincinnati       56	68 - 56 = 12
Milwaukee        63	68 - 63 = 5
Pittsburgh       40	68 - 40 = 28
St. Louis        53	68 - 53 = 15
Total            263	77

Since the total of the Max Additional Wins column is less than 78, it is not possible for Cincinnati to win the division with 68 (or fewer) total wins.