Place eight 'A's and eight 'B's in a 4x4 grid such that each 'A' is orthogonally bordered by 1 or 3 'A's and each 'B' is orthogonally bordered by 0, 2, or 4 'B's.
Can a 6x6 grid be filled with 18 'A's and 18 'B's in this manner?
Note: 'orthogonally' means horizontally or vertically
(In reply to
solution of first part by Robby Goetschalckx)
DECLARE FUNCTION count! (r!, c!, p!)
DECLARE SUB place (r!, c!)
CLEAR , , 44444
DIM SHARED sz
sz = 4
DIM SHARED g(sz + 1, sz + 1), noRem(1)
CLS
FOR r = 0 TO sz + 1 STEP sz + 1
FOR c = 0 TO sz + 1
g(r, c) = 1
g(c, r) = 1
NEXT
NEXT
noRem(0) = sz * sz / 2: noRem(1) = sz * sz / 2
place 1, 1
END
FUNCTION count (r, c, pc)
p = g(r, c)
ct = (g(r  1, c) = p) + (g(r, c  1) = p) + (g(r, c + 1) = p) + (g(r + 1, c) = p)
count = ABS(ct)
END FUNCTION
SUB place (r, c)
FOR pc = 0 TO 1
IF noRem(pc) > 0 THEN
good = 1
g(r, c) = pc
noRem(pc) = noRem(pc)  1
IF r > 1 THEN
n = count(r  1, c, pc)
IF n MOD 2 = g(r  1, c) THEN good = 0
END IF
IF r = sz AND c > 1 THEN
n = count(r, c  1, pc)
IF n MOD 2 = g(r, c  1) THEN good = 0
END IF
IF r = sz AND c = sz THEN
n = count(r, c, pc)
IF n MOD 2 = g(r, c) THEN good = 0
IF good THEN
FOR row = 1 TO sz
FOR col = 1 TO sz
PRINT g(row, col);
NEXT
PRINT
NEXT
END IF
ELSE
IF good THEN
cl = c + 1: rw = r
IF cl > sz THEN
rw = rw + 1: cl = 1
END IF
place rw, cl
END IF
END IF
noRem(pc) = noRem(pc) + 1
END IF
NEXT
END SUB
finds
1 0 0 1
0 1 1 0
0 1 1 0
1 0 0 1
with 0's representing A's and 1's representing B's.
When sz is set to 6 it finds no solutions.

Posted by Charlie
on 20070808 11:48:06 