Detroit Tigers First-Place Eliminiation Scenario
Detroit must win 12 more games (63 total) to avoid elimination from first place.
Here is a scenario in which Detroit can finish in first place with 63 wins.

To construct the scenario, we start with the current standings in the division:
Team             W	L	GL	PCT	GB
Chicago          62	44	56	0.585	0.0
Cleveland        51	51	60	0.500	9.0
Detroit          51	57	54	0.472	12.0
Kansas City      45	59	58	0.433	16.0
Minnesota        44	62	56	0.415	18.0

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

Chicago wins 0 games against Cleveland and loses 5.
Chicago wins 1 games against Detroit and loses 5.
Chicago wins 0 games against Kansas City and loses 6.
Chicago wins 0 games against Minnesota and loses 3.
Chicago wins 0 games agains teams outside the division and loses 0.

Cleveland wins 5 games against Chicago and loses 0.
Cleveland wins 3 games against Detroit and loses 3.
Cleveland wins 4 games against Kansas City and loses 6.
Cleveland wins 0 games against Minnesota and loses 10.
Cleveland wins 0 games agains teams outside the division and loses 0.

Detroit wins 5 games against Chicago and loses 1.
Detroit wins 3 games against Cleveland and loses 3.
Detroit wins 0 games against Kansas City and loses 3.
Detroit wins 4 games against Minnesota and loses 0.
Detroit wins 0 games agains teams outside the division and loses 0.

Kansas City wins 6 games against Chicago and loses 0.
Kansas City wins 6 games against Cleveland and loses 4.
Kansas City wins 3 games against Detroit and loses 0.
Kansas City wins 0 games against Minnesota and loses 6.
Kansas City wins 0 games agains teams outside the division and loses 0.

Minnesota wins 3 games against Chicago and loses 0.
Minnesota wins 10 games against Cleveland and loses 0.
Minnesota wins 0 games against Detroit and loses 4.
Minnesota wins 6 games against Kansas City and loses 0.
Minnesota 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
Detroit          63	99	0.389	0.0
Chicago          63	99	0.389	0.0
Cleveland        63	99	0.389	0.0
Minnesota        63	99	0.389	0.0
Kansas City      60	102	0.370	3.0


Next, we explain why Detroit cannot win the division with fewer than 63 wins.

There are a total of 59 games left in the following series of games between teams in the American League Central:
Chicago          vs. Cleveland        5 games
Chicago          vs. Detroit          6 games
Chicago          vs. Kansas City      6 games
Chicago          vs. Minnesota        3 games
Cleveland        vs. Detroit          6 games
Cleveland        vs. Kansas City      10 games
Cleveland        vs. Minnesota        10 games
Detroit          vs. Kansas City      3 games
Detroit          vs. Minnesota        4 games
Kansas City      vs. Minnesota        6 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
Detroit finishes in first place with a total of 62 wins.

Team             W	Max Additional Wins
Chicago          62	62 - 62 = 0
Cleveland        51	62 - 51 = 11
Detroit          51	62 - 51 = 11
Kansas City      45	62 - 45 = 17
Minnesota        44	62 - 44 = 18
Total            253	57

Since the total of the Max Additional Wins column is less than 59, it is not possible for Detroit to win the division with 62 (or fewer) total wins.