I saw the boolean expressions for the N Queens problem from here .
My modified N queens rules are simpler:
For ap*p chessboard I want to place N queens in such a way so that
For example, say N = 17, then we need a 5*5 chessboard and the placement will be:
Q_Q_Q_Q_Q
Q_Q_Q_Q_Q
Q_Q_Q_Q_Q
Q_Q_*_*_*
*_*_*_*_*
The question is I am trying to come up with a boolean expression for this problem .
This problem can be solved using the Python packages humanize
and omega
.
"""Solve variable size square fitting."""
import humanize
from omega.symbolic.fol import Context
def pick_chessboard(q):
ctx = Context()
# compute size of chessboard
#
# picking a domain for `p`
# requires partially solving the
# problem of computing `p`
ctx.declare(p=(0, q))
s = f'''
(p * p >= {q}) # chessboard fits the queens, and
/\ ((p - 1) * (p - 1) < {q}) # is the smallest such board
'''
u = ctx.add_expr(s)
d, = list(ctx.pick_iter(u)) # assert unique solution
p = d['p']
print(f'chessboard size: {p}')
# compute number of full rows
ctx.declare(x=(0, p))
s = f'x = {q} / {p}' # integer division
u = ctx.add_expr(s)
d, = list(ctx.pick_iter(u))
r = d['x']
print(f'{r} rows are full')
# compute number of queens on the last row
s = f'x = {q} % {p}' # modulo
u = ctx.add_expr(s)
d, = list(ctx.pick_iter(u))
n = d['x']
k = r + 1
kword = humanize.ordinal(k)
print(f'{n} queens on the {kword} row')
if __name__ == '__main__':
q = 10 # number of queens
pick_chessboard(q)
Representing multiplication (and integer division and modulo) with binary decision diagrams has complexity exponential in the number of variables, as proved in: https://doi.org/10.1109/12.73590
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