2018 Berlin Candidates Chess Tournament

Kramnik is not eliminated!

Out of the 3**8 = 6561 possible outcomes for the tournament, there are still 6 outcomes in which Kramnik wins. Kramnik must win both his games, Karjakin must lose both his games, and Grischuk must lose to Mamedyarov and win against Caruana. Caruana can either draw with or lose to Aronian, but must not win. This would throw the tournament into a big tie for first place, at 7.5 points (also known as "plus 1"):

If Caruana and Aronian draw, it's a 4-way tie, and Kramnik wins the tiebreak with the only winning record in games played against the other three players involved in the tie (the "Direct" column in the charts below).

If Aronian beats Caruana, it's a 5-way tie, and all five tied players will have an overall even score in games played against the other four. The tiebreak goes to most wins (fewest draws), and Kramnik wins that easily with 5 wins.

Following is the list of combinations resulting in Kramnik winning, with the 8-letter string indicating the results of the following respective games: (D=Draw W=White wins B=Black wins)

Mamedyarov must beat Grischuk
Ding Liren must lose to Kramnik
So must beat Karjakin
Caruana must not beat Aronian
Grischuk must beat Caruana
Aronian So
Karjakin must lose to Ding Liren
Kramnik must beat Mamedyarov

                Score      Direct  Wins  S-B
WBWDWDBW
 1    Kramnik  7.5 /14    4.0 / 8    5  50.00
 2 Mamedyarov  7.5 /14    4.0 / 8    3  52.00
 3   Grischuk  7.5 /14    4.0 / 8    3  51.75
 4    Caruana  7.5 /14    4.0 / 8    3  50.75
 5  DingLiren  7.5 /14    4.0 / 8    2  52.00
 6   Karjakin  7.0 /14    0.0 / 0    4  49.00
 7         So  6.5 /14    0.0 / 0    2  44.50
 8    Aronian  5.0 /14    0.0 / 0    1  36.50

WBWDWWBW
 1    Kramnik  7.5 /14    4.0 / 8    5  50.50
 2 Mamedyarov  7.5 /14    4.0 / 8    3  52.00
 3   Grischuk  7.5 /14    4.0 / 8    3  51.50
 4    Caruana  7.5 /14    4.0 / 8    3  50.75
 5  DingLiren  7.5 /14    4.0 / 8    2  52.00
 6   Karjakin  7.0 /14    0.0 / 0    4  49.00
 7         So  6.0 /14    0.0 / 0    2  42.50
 8    Aronian  5.5 /14    0.0 / 0    2  39.25

WBWDWBBW
 1    Kramnik  7.5 /14    4.0 / 8    5  49.50
 2 Mamedyarov  7.5 /14    4.0 / 8    3  52.00
 3   Grischuk  7.5 /14    4.0 / 8    3  52.00
 4    Caruana  7.5 /14    4.0 / 8    3  50.75
 5  DingLiren  7.5 /14    4.0 / 8    2  52.00
 6   Karjakin  7.0 /14    1.0 / 2    4  49.00
 7         So  7.0 /14    1.0 / 2    3  46.00
 8    Aronian  4.5 /14    0.0 / 0    1  33.25

WBWBWDBW
 1    Kramnik  7.5 /14    3.5 / 6    5  50.75
 2 Mamedyarov  7.5 /14    3.0 / 6    3  52.00
 3  DingLiren  7.5 /14    3.0 / 6    2  52.00
 4   Grischuk  7.5 /14    2.5 / 6    3  51.50
 5   Karjakin  7.0 /14    1.5 / 2    4  48.75
 6    Caruana  7.0 /14    0.5 / 2    3  48.75
 7         So  6.5 /14    0.0 / 0    2  45.00
 8    Aronian  5.5 /14    0.0 / 0    2  39.75

WBWBWWBW
 1    Kramnik  7.5 /14    3.5 / 6    5  51.25
 2 Mamedyarov  7.5 /14    3.0 / 6    3  52.00
 3  DingLiren  7.5 /14    3.0 / 6    2  52.00
 4   Grischuk  7.5 /14    2.5 / 6    3  51.25
 5   Karjakin  7.0 /14    1.5 / 2    4  48.75
 6    Caruana  7.0 /14    0.5 / 2    3  48.50
 7    Aronian  6.0 /14    1.0 / 2    3  42.50
 8         So  6.0 /14    1.0 / 2    2  42.75

WBWBWBBW
 1    Kramnik  7.5 /14    3.5 / 6    5  50.25
 2 Mamedyarov  7.5 /14    3.0 / 6    3  52.00
 3  DingLiren  7.5 /14    3.0 / 6    2  52.00
 4   Grischuk  7.5 /14    2.5 / 6    3  51.75
 5   Karjakin  7.0 /14    2.5 / 4    4  48.75
 6    Caruana  7.0 /14    2.0 / 4    3  49.00
 7         So  7.0 /14    1.5 / 4    3  46.75
 8    Aronian  5.0 /14    0.0 / 0    2  36.50

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