# The number of starting positions in Freestyle Chess

Chess960 (or Fischer random chess, or Freestyle chess), is a variant of chess where the eight pawns start on the second rank/row as usual but the rest of the pieces (king, queen, two rooks, two knights, and two bishops) on the first rank are randomly arranged. Here is an example of a Chess960 game's starting position.


First introduced by Bobby Fischer in the 1990s, Chess960 has gained popularity in recent years in both online and in-person play. The best few players of Chess960 are GM Wesley So, GM Hikaru Nakamura, and of course GM Magnus Carlsen.

For a starting position to be valid, we must ensure that the two bishops are of opposite colours. Also, the king must be placed between the two rooks to allow for castling. The 960 in the name signifies the number of valid starting positions. In this note, we shall verify this number.

We denote the number of possible arrangements for each piece by its chess notation, i.e. king $K$, rook $R$, queen $Q$, bishop $B$ and knight $N$.

We start with the bishops. They must be on different coloured squares. There are 4 light and dark squares each in one row, so each bishop have 4 choices. So, we have

$$B_\text{light} = B_{\text{dark}} = 4.$$

After placing the bishops, we can imagine that we remove the now-occupied pair of white and black squares. This leaves us with 6 squares, 3 of each colour.

We now place the king and rooks together. Assume that the king is on the $i$-th square (counting from left). Note that we must obey $2 \leq i \leq 5$, as we may not be place the king on the leftmost or rightmost square to allow for castling. Then, we have for the left and right rooks that

$$R_l(i) = i-1, \quad R_r(i) = 6-i.$$

Therefore, given the king position $i$ we have the rook placement combinations

$$R(i) = R_l(i) \times R_r(i) = (i-1)\,(-i+6) = -i^2 + 7i - 6.$$

Lastly, we consider placing the queen. Given there 3 squares left, we simply have

$$Q = 3.$$

We need not consider the knights because they will simply go into the two remaining squares, i.e. they have no choice. Further, the two knights are indistinguishable, so there are no permutations.

The final number is therefore

$$
\begin{align}
& B_w \times B_b \times Q \times \sum_{i=2}^{5} R(i) \\
&= 4 \times 4 \times 3 \times (-54 + 98 - 24)\\
&= 4 \times 4 \times 3 \times 20\\
&= 960.
\end{align}
$$

## Some thoughts

The variant's popularity has a lot to do with the way chess is being played nowadays. Today, any respectable online chess website provides a large database with powerful engines for analysis. You can play a game and immediately analyse it afterwards. To contrast, this was barely possible just 30 years ago.

It is not uncommon for the average club player to memorize theory/openings beyond 10 moves. Therefore, it can be argued that the game really only begins after 10 moves or so, when the theory ends. For world class players, this is certainly the case --- memorizing theory up to 20 30 moves is standard affair. GMs and their team of trainers work on this heavily in hopes to secure an early advantage.

Chess960 throws most of this out the window. Opening theory is mostly inapplicable, especially if one merely memorizes them by rote. Chess960 forces players to really think from the get-go and truly 'play chess'.

Fun fact: in official tournaments, Chess960 is played mostly on rapid time controls, meaning about 15 minutes for each side with some time increment. For example, 15+10 means 15 minutes per side and 10 seconds of increment after each move is made. Further, the starting position is announced maybe 15 minutes before the game starts and in that short amount of time the players can prepare.