#### QUESTION

I have some interesting questions that have to do with the rating changes and calculations for the Anand-Carlsen world championship.

First, some preliminary information:

Carlsen's live rating: **2870**, Anand's live rating: **2775**

If Carlsen wins a game, he gains **3.7 points**; draw **-1.3 points**, loss **-6.3 points**.

If Anand wins a game, he gains **6.3 points**; draw **1.3 points**, loss **-3.7 points**.

The match will be 12 games, with the winner being the first to 6 points. Therefore, **the number of standard rated games in the match can vary** based on all the **ordered** possibilities of wins, draws, and losses.

My questions are:

**1. What are the possibilities for Carlsen to cross 2900 in live rating?**

**2. What are the possibilities for Anand to cross 2800 in live rating?**

**3. What are the upper and lower bounds for Carlsen's rating in the match, given the stipulations?**

**4. What are the upper and lower bounds for Anand's rating in the match, given the stipulations?**

Thanks!

#### ANSWER

First, note that the final ratings depend entirely on the score of the match, not on the particular games. For example, if they each win a game, their ratings will change by the same amount as if they draw both games. This may seem like a coincidence but it's just a consequence of the way that FIDE ratings work.

Here is the entire table of possible results. The columns are Anand's match score, Carlsen's match score, Anand's post-match rating, and Carlsen's post-match rating. The results are sorted by post-match rating.

```
A C A_rat C_rat
0.0 7.0 2749.1 2895.9
0.5 6.5 2754.1 2890.9
1.0 7.0 2755.4 2889.6
1.5 6.5 2760.4 2884.6
2.0 7.0 2761.7 2883.3
2.5 6.5 2766.7 2878.3
3.0 7.0 2768.0 2877.0
3.5 6.5 2773.0 2872.0
4.0 7.0 2774.3 2870.7
4.5 6.5 2779.3 2865.7
5.0 7.0 2780.6 2864.4
5.5 6.5 2785.6 2859.4
6.0 6.0 2790.6 2854.4
6.5 5.5 2795.6 2849.4
6.5 4.5 2799.3 2845.7
7.0 5.0 2800.6 2844.4
6.5 3.5 2803.0 2842.0
7.0 4.0 2804.3 2840.7
6.5 2.5 2806.7 2838.3
7.0 3.0 2808.0 2837.0
6.5 1.5 2810.4 2834.6
7.0 2.0 2811.7 2833.3
6.5 0.5 2814.1 2830.9
7.0 1.0 2815.4 2829.6
7.0 0.0 2819.1 2825.9
```

Therefor the answers to your questions are

- None.
- He needs to win the match with 7 points, or with 6.5 points with Carlsen scoring at most 3.5.
- 2895.9, 2825.9.
- 2819.1, 2749.1.