Bundesliga: The final day permutations

The relegation battle - who will survive?

14. Augsburg (away at Hoffenheim) will be safe if: 

… they avoid defeat.
… they lose but Hamburg don't beat Wolfsburg.
… they lose but Hamburg beat Wolfsburg by an bigger margin, thus sending Wolfsburg into the play-off.

15. Wolfsburg (away at Hamburg) will be safe if:

… they avoid defeat.
… they lose, but Augsburg lose by a greater margin against Hoffenheim.

16. Hamburg (at home to Wolfsburg):

… must win. Any other result will see them face a third play-off in four years, most likely against Eintracht Braunschweig. 

Europa League 

The team which finishes 5th in the Bundesliga goes straight into next season's Europa League group stage. The team which finishes 6th must first navigate a qualifying round. Seventh place could also be enough, providing Borussia Dortmund beat Eintracht Frankfurt in the German Cup final on May 28, since Dortmund have already qualified for Europe.

5. Hertha Berlin (at home to Bayer Leverkusen)

… will finish at least 6th if they win or draw against Leverkusen.
… will finish at least 6th if they lose but Cologne and Frankfurt don't win.

6. Freiburg (away at Bayern Munich)

… will finish at least 6th if they beat Bayern.
… will finish at least 6th if they draw with Bayern and Cologne don't beat Mainz.
… will finish at least 6th if they lose against Bayern but neither Cologne nor Bremen win.

7. Cologne (at home to Mainz)

… will finish at least 7th if they win.
… will finish 6th if they win and Freiburg don't win OR Hertha lose.
… will finish 7th if they draw and Bremen don't win (assuming Borussia Mönchengladbach don't beat Darmstadt by eleven goals - see below).
… will finish 7th if they lose but neither Bremen nor Gladbach win.

8. Werder Bremen (away at Borussia Dortmund)

… will finish 6th if they win, Freiburg lose and Cologne don't win.
… will finish 7th if they draw and Freiburg don't lose and Cologne win.

9. Borussia Mönchengladbach (at home to Darmstadt)

… can finish 7th if they win, Bremen don't win and Cologne lose.

If Cologne draw, Gladbach would have to win by eleven goals - which isn't actually unheard of. On 29 April 1978, second-placed Gladbach beat Borussia Dortmund 12-0 on the final day of the season, but it still wasn't enough as Cologne beat St Pauli 5-0 to win the title by 3 goals.

Automatic Champions League qualification

The team which finishes third in the Bundesliga qualifies directly for the Champions League group stage. The team which finishes fourth must first navigate a qualifying round - which could feature the likes of English sides Manchester City, Arsenal or Liverpool. 

3. Borussia Dortmund (at home to Werder Bremen) will finish third if:

… they win and Hoffenheim don't beat Augsburg by five goals more.
… they draw and Hoffenheim don't win.
… both they and Hoffenheim lose (assuming Hoffenheim don't make up five goals in the process)
… points and goal difference are equal, since Dortmund have the better head-to-head record.

4. TSG 1899 Hoffenheim (at home to Augsburg) will finish third if:

… they beat Augsburg by five goals more than Dortmund beat Bremen.
… they win and Dortmund don't win.
… they draw and Dortmund lose.
… they lose, but Dortmund lose to Bremen by five goals more.

Here is the table as it stands going into Matchday 34, so you can work the rest out for yourself!