# Talk:Peaceful chess queen armies

## Original Python exhaustive search

I was experimenting with various things when doing the Python. This is the original:

Exhaustive search. <lang python>from itertools import combinations, count from functools import lru_cache, reduce

- n-by-n board

n = 5

def _2d(n=n):

for i in range(n): print(' '.join(f'{i},{j}' for j in range(n)))

def _1d(n=n):

for i in range(0, n*n, n): print(', '.join(f'{i+j:2}' for j in range(n)))

_bbullet, _wbullet = '\u2022\u25E6'

- _bqueen, _wqueen = 'BW'

_bqueen, _wqueen = '\u265B\u2655' _bqueenh, _wqueenh = '♛', '♕' _or = set.__or__

def place(m, n):

"Place m black and white queens, peacefully, on an n-by-n board" # 2-D Board as 1-D array: 2D(x, y) == 1D(t%n, t//n) board = set(range(n*n))

#placements = list(combinations(board, m)) placements = {frozenset(c) for c in combinations(board, m)} for blacks in placements: black_attacks = reduce(_or, (queen_attacks_from(pos, n) for pos in blacks), set()) #for whites in placements: for whites in {frozenset(c) for c in combinations(board - black_attacks, m)}: if not black_attacks & whites: return blacks, whites return set(), set()

@lru_cache(maxsize=None) def queen_attacks_from(pos, n=n):

a = set([pos]) # Its position a.update(range(pos//n*n, pos//n*n+n)) # Its row a.update(range(pos%n, n*n, n)) # Its column # Diagonals x0, y0 = pos%n, pos//n for x1 in range(n): # l-to-r diag y1 = y0 -x0 +x1 if 0 <= y1 < n: a.add(x1 + y1 * n) # r-to-l diag y1 = y0 +x0 -x1 if 0 <= y1 < n: a.add(x1 + y1 * n) return a

def pboard(black_white=None, n=n):

if black_white is None: blk, wht = set(), set() else: blk, wht = black_white print(f"## {len(blk)} black and {len(wht)} white queens " f"on a {n}-by-{n} board:", end=)for xy in range(n*n): if xy %n == 0: print() ch = ('?' if xy in blk and xy in wht else _bqueen if xy in blk else _wqueen if xy in wht else _bbullet if (xy%n + xy//n)%2 else _wbullet) print('%s' % ch, end=)print()

def hboard(black_white=None, n=n):

if black_white is None: blk, wht = set(), set() else: blk, wht = black_white out = (f"## {len(blk)} black and {len(wht)} white queens " f"on a {n}-by-{n} board:

\n")

out += "

\n " tbl = for xy in range(n*n): if xy %n == 0: tbl += '\n \n' ch = ('?' if xy in blk and xy in wht else _bqueenh if xy in blk else _wqueenh if xy in wht else "") bg = "" if (xy%n + xy//n)%2 else ' bgcolor="silver"' tbl += f' \n'out += tbl[7:]out += '\n

{ch} |

\n

\n'

return out

if __name__ == '__main__':

n=2 html = for n in range(2, 7): print() queen_attacks_from.cache_clear() # memoization cache # for m in count(1): ans = place(m, n) if ans[0]: pboard(ans, n) html += hboard(ans, n) else: comment = f"# Can't place {m}+ queens on a {n}-by-{n} board" print (comment) html += f"{comment}

\n\n" break print('\n') html += '

\n' # m, n = 5, 7 queen_attacks_from.cache_clear() ans = place(m, n) pboard(ans, n) html += hboard(ans, n) with open('peaceful_queen_armies.htm', 'w') as f: f.write(html)</lang>

- Output:

The console output Unicode queen characters display wider than other characters in monospace font so the alternative HTML output is shown below.

**# Can't place 1+ queens on a 2-by-2 board**

**## 1 black and 1 white queens on a 3-by-3 board:**

♛ | ||

♕ | ||

**# Can't place 2+ queens on a 3-by-3 board**

**## 1 black and 1 white queens on a 4-by-4 board:**

♛ | |||

♕ | |||

**## 2 black and 2 white queens on a 4-by-4 board:**

♛ | ♛ | ||

♕ | |||

♕ |

**# Can't place 3+ queens on a 4-by-4 board**

**## 1 black and 1 white queens on a 5-by-5 board:**

♛ | ||||

♕ |

**## 2 black and 2 white queens on a 5-by-5 board:**

♛ | ||||

♛ | ||||

♕ | ||||

♕ |

**## 3 black and 3 white queens on a 5-by-5 board:**

♕ | ♕ | |||

♛ | ||||

♕ | ||||

♛ | ♛ |

**## 4 black and 4 white queens on a 5-by-5 board:**

♕ | ♕ | |||

♕ | ||||

♛ | ♛ | |||

♕ | ||||

♛ | ♛ |

**# Can't place 5+ queens on a 5-by-5 board**

**## 1 black and 1 white queens on a 6-by-6 board:**

♕ | |||||

♛ |

**## 2 black and 2 white queens on a 6-by-6 board:**

♛ | |||||

♕ | ♕ | ||||

♛ | |||||

**## 3 black and 3 white queens on a 6-by-6 board:**

♛ | |||||

♕ | |||||

♛ | ♛ | ||||

♕ | ♕ |

**## 4 black and 4 white queens on a 6-by-6 board:**

♕ | ♕ | ||||

♛ | |||||

♕ | ♕ | ||||

♛ | |||||

♛ | |||||

♛ |

**## 5 black and 5 white queens on a 6-by-6 board:**

♛ | ♛ | ♛ | |||

♛ | ♛ | ||||

♕ | ♕ | ♕ | |||

♕ | ♕ |

**# Can't place 6+ queens on a 6-by-6 board**

**## 5 black and 5 white queens on a 7-by-7 board:**

♕ | ♕ | |||||

♛ | ||||||

♛ | ||||||

♛ | ||||||

♛ | ||||||

♛ | ||||||

♕ | ♕ | ♕ |

## Error in solution?

No solutions for {8,9},{10,14} and some other boards. For {9, 12} correctly:

12 black and 12 white queens on a 9 x 9 board: B * x * B * x * B * x W x * x W x * B * x * B * x * B * x W x * x W x * B * x * B * x * B * x W x * x W x * B * x * B * x * B * x W x * x W x * x W x W x W x W x

I checked C and C++ codes and compare results from https://oeis.org/A250000

Hi, Please sign your contribution bove, thanks. --Paddy3118 (talk) 10:58, 19 January 2020 (UTC)