Magic bitboards are a relatively recent development used in most modern chess engines. They are a bit difficult to understand though. Bitboards use an integer (usually a 64-bit unsigned) to store 64 individual boolean values. For chess they are perfect, because a single integer can story information about the entire board. For example, an integer could store information about which squares contain a white piece.

Bitboards have been used for a while, but one of the problems is looping over the board. One has to find the next set bit in a bitboard. Modern processors have instructions for finding the highest (or lowest) set bit, but this usually requires processor dependent inline assembly. Earlier chess engines used a combination of logical AND’s with lookup tables. For example (bitboard & 0xFF) quickly determines whether the lowest 8 bits contain at least a 1. A 256 byte lookup table contains the pre-computed position of the lowest bit.

A faster way is using magic bitboards. We’ll skip the mathematical details for now, but the idea is to multiply sparse bitboards (bitboards with a low number of 1’s) by a ‘magic’ constant. This constant is chosen, such that the information about the bit pattern is shifted to the highest bits of the result value. Shifting this to the right results in a small unique value that can be used to index a lookup table.

As an example, here’s how to compute the index of the lowest bit of a bitboard:

// Find index of lowest bit in a bitboard final long MAGIC = 0x021bb2bf47316a4fL; const unsigned int lowBitTable[64] = { 0, 1, 2, 7, 3, 44, 8, 34, 4, 55, 20, 45, 40, 9, 35, 58, 5, 32, 53, 56, 51, 21, 46, 23, 41, 17, 48, 10, 36, 13, 59, 25, 63, 6, 43, 33, 54, 19, 39, 57, 31, 52, 50, 22, 16, 47, 12, 24, 62, 42, 18, 38, 30, 49, 15, 11, 61, 37, 29, 14, 60, 28, 27, 26 }; int lowBitIndex(long b) { return lowBitTable[(int)(((b&-b)*MAGIC) >>> 58)]; }

The ‘b&-b’ trick isolates the lowest bits, the multiplication gives a unique 6-bit value in the highest 6 bits for all 64-bit integers with a single bit set. More details about the mathematics of magic bitboards in a later post.