03.26.18 What a difference a day makes. Yesterday, Caruana and Karjakin were tied at the top of the Candidates crosstable with 7.0 points apiece. The talk on the street after Round 12 was of deja vu all over again as we had the same scenario at the conclusion of the 2016 Candidates--namely, Caruana and Karjakin tied with Karjakin having the better tiebreaks. Round 13 saw Caruana crush Aronian and Karjakin make a draw with So. That gave Fabi a half-point lead going into the final round. Join us for a mathematical analysis of the possibilities of choosing the next Challenger to Magnus Carlsen.

Here's the deal: Caruana controls his own fate. If he wins tomorrow, he is the next Challenger to the World Champion. Period. End of story.

Life may not be that easy for the prodigy from Brooklyn, New York via Miami, Florida, however. Let's try to demystify the possibilities of the final round of the Candidates tournament.

The Candidates is an eight-player double round robin tournament. In each round, four games are contested. The pairings for Round 14 are as follows:

1. Grischuk-Caruana

2. Aronian-So

3. Karjakin-Ding

4. Kramnik-Mamedyarov

Other than Caruana, there are three players in striking distance of the top of the crosstable: Shakhriyar Mamedyarov, Sergey Karjakin, and Liren Ding. That makes the results on Board 2 irrelevant.

Three things could happen:

Scenario 1: Caruana Wins

Scenario 2: Caruana Draws

Scenario 3: Caruana Loses

Scenario 1: Caruana Wins

We already discussed this. If Fabi wins, he would have 9.0 points and it's all over. Break out the kombucha and celebrate, that is, if you are a Fabi fan.

Scenario 2: Caruana Draws

If Caruana draws, he would have 8.5 points and one thing would be certain: Ding would be eliminated.

That leaves Mamedyarov and Karjakin.

In this scenario, there are only two results in which Caruana would be overtaken. They are:

A) Karjakin defeats Ding

If this happens, Karjakin would have 8.5 points (the same as Caruana).

B) Mamedyarov defeats Kramnik

If this happens, Mamedyarov would have 8.5 points (the same as Caruana).

This would be an opportune time to discuss the tiebreaking system being used at the 2018 Candidates.

The three tiebreaking systems being used in hierarchical order are:

1. Direct Encounter (What are the tied players' scores against each other?)
2. Number of Wins
3. Sonneborn-Berger (This is not a powerful New York City law firm. It's a tiebreaking system calculated by summing the final scores of all the opponents the player defeated and half the final scores of all the opponents with whom the player drew.)

We calculated Sonneborn-Berger tiebreaks based on the 13 rounds played so far.

If either Scenario 2A or Scenario 2B occurs, Caruana will lose on tiebreaks. The calculation goes thusly:

Direct Encounter

Caruana 0.5
Karjakin 1.5

Caruana 1.0
Mamedyarov 1.0

Number of Wins

Caruana 4
Mamedyarov 4

Sonnneborn-Berger (After 13 rounds)

Caruana 35.5
Mamedyarov 39.25

Let's move on to the third scenario.

Scenario 3: Caruana Loses

If Caruana loses, he would remain at 8.0 points.

In this scenario, there are five outcomes that would be of concern:

3A) Karjakin defeats Ding

If this happens, Karjakin would have 8.5 points and Caruana would be eliminated.

3B) Karjakin and Ding draw

If this happens, both Karjakin and Caruana would have 8.0 points. Karjakin would win the Direct Encounter tiebreak and Caruana would be eliminated.

3C) Karjakin loses to Ding

If this happens, Ding would have 8.0 points (the same as Caruana).

The Scenario 3C tiebreaks would look like this:

Direct Encounter

Caruana 1.0
Ding 1.0

Number of Wins

Caruana 4
Ding 2

And Caruana would not have to worry about Ding but would be sweating bullets waiting for the results of Kramnik-Mamedyrov (see below).

3D) Mamedyrov defeats Kramnik

If this happens, Mamedyrov would have 8.5 points and Caruana would be eliminated.

3E) Mamedyrov and Kramnik draw

If this happens, Mamedyrov would have 8.0 points (the same as Caruana).

Here's our last tiebreaking scenario:

Direct Encounter

Caruana 1.0
Mamedyrov 1.0

Number of Wins

Caruana 4
Mamedyrov 3

And Caruana would advance.

There you have it!

Tune in tomorrow at 9:00 a.m. eastern US to witness history in the making!

Standings After Round 13

Official Site of the Candidates Tournament

Full Regulations of the Candidates Tournament

Link to all games played thus far

Official Schedule of the 2018 Candidates Tournament
Most games begin at 10:00 a.m. eastern US time
except those games indicated with ** which begin at 9:00 a.m. eastern US time

March 9 -- Friday Opening Ceremony
March 10 -- Saturday Round 1**
March 11 -- Sunday Round 2
March 12 -- Monday Round 3
March 13 -- Tuesday Rest Day
March 14 -- Wednesday Round 4
March 15 -- Thursday Round 5
March 16 -- Friday Round 6
March 17 -- Saturday Rest Day
March 18 -- Sunday Round 7
March 19 -- Monday Round 8
March 20 -- Tuesday Round 9
March 21 -- Wednesday Rest Day
March 22 -- Thursday Round 10
March 23 -- Friday Round 11
March 24 -- Saturday Round 12
March 25 -- Sunday Rest Day
March 26 -- Monday Round 13**
March 27 -- Tuesday Round 14/Closing Ceremony**
March 28 -- Wednesday Tie Breaks/Closing Ceremony**

Pairings for each round