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Magic Bitboards is probably the fastest approach to generate sliding piece attacks on recent architectures with fast 64-bit multiplication. It performs a mask/mul/shift->lookup to gain all the attacked squares of a bishop or rook.

Trial and Error

So far the only approach to find 64-bit magic factors is trial and error. One has the idea, how many bits are needed for a perfect hashing - and about constrains of least and most significant one-bit inside the magic factor. In September 2017, Niklas Fiekas introduced a technique based on the least common multiple of the periods for all relevant occupancies to reduce the range for magic candidates, specially to find magics with one bit less than the number of relevant occupancies to half the individual tables sizes as mentioned in Best Magics so far ^[1].

                                        any consecutive
relevant occupancy                      combination of
bishop b1, 5 bits                       the masked bits
. . . . . . . .     . . . . . . . .     . . .[C D E F G]
. . . . . . . .     . 1 . . . . . .     . . . . . . . .
. . . . . . G .     . 1 . . . . . .     . . . . . . . .
. . . . . F . .     . 1 . . . . . .     . . . . . . . .
. . . . E . . .  *  . 1 . . . . . .  =  . . garbage . .    >> (64- 5)
. . . D . . . .     . 1 . . . . . .     . . . . . . . .
. . C . . . . .     . . . . . . . .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .
 
                                        any consecutive
relevant occupancy                      combination of
bishop d4, 9 bits                       the masked bits
. . . . . . . .     . . . . . . . .     2 4 5 B C E F G]
. . . . . . G .     . . .some . . .     . . . . . . .[1
. 5 . . . F . .     . . . . . . . .     . . . . . . . .
. . 4 . E . . .     . . .magic. . .     . . . . . . . .
. . . . . . . .  *  . . . . . . . .  =  . . garbage . .    >> (64- 9)
. . C . 2 . . .     . . .bits . . .     . . . . . . . .
. B . . . 1 . .     . . . . . . . .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .
 
                                        any consecutive
relevant occupancy                      combination of
rook d4, 10 bits                        the masked bits
. . . . . . . .     . . . . . . . .     4 5 6 B C E F G]
. . . 6 . . . .     . . .some . . .     . . . . . .[1 2
. . . 5 . . . .     . . . . . . . .     . . . . . . . .
. . . 4 . . . .     . . .magic. . .     . . . . . . . .
. B C . E F G .  *  . . . . . . . .  =  . . garbage . .    >> (64-10)
. . . 2 . . . .     . . .bits . . .     . . . . . . . .
. . . 1 . . . .     . . . . . . . .     . . . . . . . .
. . . . . . . .     . . . . . . . .     . . . . . . . .
 
                                        any consecutive
relevant occupancy                      combination of
rook a1, 12 bits                        the masked bits
. . . . . . . .     . . . . . . . .     5 6 B C D E F G]
6 . . . . . . .     . . .some . . .     . . . .[1 2 3 4
5 . . . . . . .     . . . . . . . .     . . . . . . . .
4 . . . . . . .     . . .magic. . .     . . . . . . . .
3 . . . . . . .  *  . . . . . . . .  =  . . garbage . .    >> (64-12)
2 . . . . . . .     . . .bits . . .     . . . . . . . .
1 . . . . . . .     . . . . . . . .     . . . . . . . .
. B C D E F G .     . . . . . . . .     . . . . . . . .

Feeding in Randoms

Tord Romstad's proposal to find magics ^[2] :

Just trying out random numbers with a low number of nonzero bits until you find a number which works is by far the fastest and easiest way to generate the magic numbers, in my experience. On my Core Duo 2.8 GHz, it takes less than a second to find magic numbers for rooks and bishops for all squares (and I have made no attempt to optimize the code, it should be easy to make it much faster). Here is the source code:

#include <stdio.h>
#include <stdlib.h>
 
#define USE_32_BIT_MULTIPLICATIONS
 
typedef unsigned long long uint64;
 
uint64 random_uint64() {
  uint64 u1, u2, u3, u4;
  u1 = (uint64)(random()) & 0xFFFF; u2 = (uint64)(random()) & 0xFFFF;
  u3 = (uint64)(random()) & 0xFFFF; u4 = (uint64)(random()) & 0xFFFF;
  return u1 | (u2 << 16) | (u3 << 32) | (u4 << 48);
}
 
uint64 random_uint64_fewbits() {
  return random_uint64() & random_uint64() & random_uint64();
}
 
int count_1s(uint64 b) {
  int r;
  for(r = 0; b; r++, b &= b - 1);
  return r;
}
 
const int BitTable[64] = {
  63, 30, 3, 32, 25, 41, 22, 33, 15, 50, 42, 13, 11, 53, 19, 34, 61, 29, 2,
  51, 21, 43, 45, 10, 18, 47, 1, 54, 9, 57, 0, 35, 62, 31, 40, 4, 49, 5, 52,
  26, 60, 6, 23, 44, 46, 27, 56, 16, 7, 39, 48, 24, 59, 14, 12, 55, 38, 28,
  58, 20, 37, 17, 36, 8
};
 
int pop_1st_bit(uint64 *bb) {
  uint64 b = *bb ^ (*bb - 1);
  unsigned int fold = (unsigned) ((b & 0xffffffff) ^ (b >> 32));
  *bb &= (*bb - 1);
  return BitTable[(fold * 0x783a9b23) >> 26];
}
 
uint64 index_to_uint64(int index, int bits, uint64 m) {
  int i, j;
  uint64 result = 0ULL;
  for(i = 0; i < bits; i++) {
    j = pop_1st_bit(&m);
    if(index & (1 << i)) result |= (1ULL << j);
  }
  return result;
}
 
uint64 rmask(int sq) {
  uint64 result = 0ULL;
  int rk = sq/8, fl = sq%8, r, f;
  for(r = rk+1; r <= 6; r++) result |= (1ULL << (fl + r*8));
  for(r = rk-1; r >= 1; r--) result |= (1ULL << (fl + r*8));
  for(f = fl+1; f <= 6; f++) result |= (1ULL << (f + rk*8));
  for(f = fl-1; f >= 1; f--) result |= (1ULL << (f + rk*8));
  return result;
}
 
uint64 bmask(int sq) {
  uint64 result = 0ULL;
  int rk = sq/8, fl = sq%8, r, f;
  for(r=rk+1, f=fl+1; r<=6 && f<=6; r++, f++) result |= (1ULL << (f + r*8));
  for(r=rk+1, f=fl-1; r<=6 && f>=1; r++, f--) result |= (1ULL << (f + r*8));
  for(r=rk-1, f=fl+1; r>=1 && f<=6; r--, f++) result |= (1ULL << (f + r*8));
  for(r=rk-1, f=fl-1; r>=1 && f>=1; r--, f--) result |= (1ULL << (f + r*8));
  return result;
}
 
uint64 ratt(int sq, uint64 block) {
  uint64 result = 0ULL;
  int rk = sq/8, fl = sq%8, r, f;
  for(r = rk+1; r <= 7; r++) {
    result |= (1ULL << (fl + r*8));
    if(block & (1ULL << (fl + r*8))) break;
  }
  for(r = rk-1; r >= 0; r--) {
    result |= (1ULL << (fl + r*8));
    if(block & (1ULL << (fl + r*8))) break;
  }
  for(f = fl+1; f <= 7; f++) {
    result |= (1ULL << (f + rk*8));
    if(block & (1ULL << (f + rk*8))) break;
  }
  for(f = fl-1; f >= 0; f--) {
    result |= (1ULL << (f + rk*8));
    if(block & (1ULL << (f + rk*8))) break;
  }
  return result;
}
 
uint64 batt(int sq, uint64 block) {
  uint64 result = 0ULL;
  int rk = sq/8, fl = sq%8, r, f;
  for(r = rk+1, f = fl+1; r <= 7 && f <= 7; r++, f++) {
    result |= (1ULL << (f + r*8));
    if(block & (1ULL << (f + r * 8))) break;
  }
  for(r = rk+1, f = fl-1; r <= 7 && f >= 0; r++, f--) {
    result |= (1ULL << (f + r*8));
    if(block & (1ULL << (f + r * 8))) break;
  }
  for(r = rk-1, f = fl+1; r >= 0 && f <= 7; r--, f++) {
    result |= (1ULL << (f + r*8));
    if(block & (1ULL << (f + r * 8))) break;
  }
  for(r = rk-1, f = fl-1; r >= 0 && f >= 0; r--, f--) {
    result |= (1ULL << (f + r*8));
    if(block & (1ULL << (f + r * 8))) break;
  }
  return result;
}
 
 
int transform(uint64 b, uint64 magic, int bits) {
#if defined(USE_32_BIT_MULTIPLICATIONS)
  return
    (unsigned)((int)b*(int)magic ^ (int)(b>>32)*(int)(magic>>32)) >> (32-bits);
#else
  return (int)((b * magic) >> (64 - bits));
#endif
}
 
uint64 find_magic(int sq, int m, int bishop) {
  uint64 mask, b[4096], a[4096], used[4096], magic;
  int i, j, k, n, fail;
 
  mask = bishop? bmask(sq) : rmask(sq);
  n = count_1s(mask);
 
  for(i = 0; i < (1 << n); i++) {
    b[i] = index_to_uint64(i, n, mask);
    a[i] = bishop? batt(sq, b[i]) : ratt(sq, b[i]);
  }
  for(k = 0; k < 100000000; k++) {
    magic = random_uint64_fewbits();
    if(count_1s((mask * magic) & 0xFF00000000000000ULL) < 6) continue;
    for(i = 0; i < 4096; i++) used[i] = 0ULL;
    for(i = 0, fail = 0; !fail && i < (1 << n); i++) {
      j = transform(b[i], magic, m);
      if(used[j] == 0ULL) used[j] = a[i];
      else if(used[j] != a[i]) fail = 1;
    }
    if(!fail) return magic;
  }
  printf("***Failed***\n");
  return 0ULL;
}
 
int RBits[64] = {
  12, 11, 11, 11, 11, 11, 11, 12,
  11, 10, 10, 10, 10, 10, 10, 11,
  11, 10, 10, 10, 10, 10, 10, 11,
  11, 10, 10, 10, 10, 10, 10, 11,
  11, 10, 10, 10, 10, 10, 10, 11,
  11, 10, 10, 10, 10, 10, 10, 11,
  11, 10, 10, 10, 10, 10, 10, 11,
  12, 11, 11, 11, 11, 11, 11, 12
};
 
int BBits[64] = {
  6, 5, 5, 5, 5, 5, 5, 6,
  5, 5, 5, 5, 5, 5, 5, 5,
  5, 5, 7, 7, 7, 7, 5, 5,
  5, 5, 7, 9, 9, 7, 5, 5,
  5, 5, 7, 9, 9, 7, 5, 5,
  5, 5, 7, 7, 7, 7, 5, 5,
  5, 5, 5, 5, 5, 5, 5, 5,
  6, 5, 5, 5, 5, 5, 5, 6
};
 
int main() {
  int square;
 
  printf("const uint64 RMagic[64] = {\n");
  for(square = 0; square < 64; square++)
    printf("  0x%llxULL,\n", find_magic(square, RBits[square], 0));
  printf("};\n\n");
 
  printf("const uint64 BMagic[64] = {\n");
  for(square = 0; square < 64; square++)
    printf("  0x%llxULL,\n", find_magic(square, BBits[square], 1));
  printf("};\n\n");
 
  return 0;
}

Fixed Shift Magics

Another application is Looking for overlapping Magics for a fixed shift approach, as recently proposed by Volker Annuss ^[3] .

See also

• Best Magics so far
• Traversing Subsets of a Set

Forum Posts

• Magic Move Generation by Colin, CCC, February 18, 2008
• Magics revisited by Richard Pijl, CCC, January 04, 2009
• magic search by Daniel Shawul, CCC, March 19, 2010
• Build magics on the fly by Marco Costalba, CCC, June 07, 2011
• Super Fast 'Looking for magics' 1.0 by Syed Fahad, CCC, October 25, 2014
• question about magic keys generation time by Fermin Serrano, CCC, March 19, 2015
• Magic number generation taking 17 seconds or more by Kalyankumar Ramaseshan, CCC, November 29, 2015
• Re: Some questions from a beginner by Tim Hagen, CCC, April 04, 2016
  Re: Some questions from a beginner by Gerd Isenberg, CCC, April 05, 2016
  Re: Some questions from a beginner by Mincho Georgiev, CCC, April 06, 2016
• Looking for dense magics by Lucas Braesch, CCC, July 11, 2017
• Disproving the existence of some magics by Niklas Fiekas, CCC, September 16, 2017
• No bishop magics with fixed shift 8 by Niklas Fiekas, CCC, April 09, 2018

External Links

• Chess Programming | Generating Magic Multipliers by Steve Maughan

References

1. ^ Disproving the existence of some magics by Niklas Fiekas, CCC, September 16, 2017
2. ^ Re: Magic Move Generation by Tord Romstad, CCC, February 20, 2008
3. ^ Fixed shift magics with 800KB lookup table by Volker Annuss, Winboard Forum, September, 05, 2010

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