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Chess - Programming - Magic BitBoards

Magic BitBoard Principle

The basic principle is:

Isolate the squares that a given piece may reach from a given position, without taking board occupancy into account. This gives us a 64-bit bitmask of potential target squares. Let's call it T (for Target).
Perform a bitwise AND of T with the bitmask of occupied squares on the board. Let's call the latter O (for Occupied).
Multiply the result by a magic value M and shift the result to the right by a magic amount S. This gives us I (for Index).
Use I as an index in a lookup table to retrieve the bitmask of squares that can actually be reached with this configuration.

To sum it up:

I = ((T & O) * M) >> S

NOTE:

T, M, S and lookup are all pre-computed and depend on the position of the piece (P = 0 … 63).
So, a more accurate formula would be:
```
I = ((T[P] & O) * M[P]) >> S[P]
```
- reachable_squares = lookup[P][I]

The purpose of step #3 is to transform the 64-bit value T & O into a much smaller one, so that a table of a reasonable size can be used.

What we get by computing ((T & O) * M) » S is essentially a random sequence of bits, and we want to map each of these sequences to a unique bitmask of reachable target squares.

The 'magic' part in this algorithm is to determine the M and S values that will produce a collision-free lookup table as small as possible.
This is a Perfect Hash Function problem.
However, no perfect hashing has been found for magic bitboards so far, which means that the lookup tables in use typically contain many unused 'holes'.
Most of the time, they are built by running an extensive brute-force search.

┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │ Q │   │   │ k │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘ 
  7   6   5   4   3   2   1   0

NOTE: Here, the position of the piece is in [0 … 7] and the occupancy bitmask is in [0x00 … 0xFF] (as it is 8-bit wide).

Therefore, it is entirely feasible to build a direct lookup table based on the position and the current board without applying the magic part.

Obviously we cannot do that for chess, as it would contain:
```
64 * 2^64 = 1,180,591,620,717,411,303,424 entries
```

Hence the need for the magic multiply and shift operations to store the data in a more compact manner.