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Unfilled grid (Posted on 2006-05-23) Difficulty: 3 of 5
Fill as much of a 6x6 grid with the letters A, B, C, D, E, F so no two of the same letter are in the same row, column or diagonal.

It is impossible to entirely fill the grid, but what is the largest number of letters that may be placed?

See The Solution Submitted by Jer    
Rating: 4.1429 (7 votes)

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Hints/Tips computer program still running, but so far... | Comment 1 of 5

The program has a lot of possibilities to try and it's still running, but so far it has found a solution with 33 letters filled in.  They are the lower-case letters in the grid below.  While there are no conflicts within a row or column, the capital letters conflict with a correctly placed letter on a diagonal, and so should not actually be placed--consider them blank:

afebdc
bdcfae
ceAdbf
dbfcea
eCdafb
fabecD

The program was written to assume the left-hand column goes from a to f from top to bottom. Every possibility is tried for every position thereafter so long as there is no conflict within row or column. The letters for any position are tried in reverse order, to minimize initial diagonal conflicts between the first row and the first column.

DECLARE SUB place (row!, col!)
CLEAR , , 9999
DIM SHARED g$(6, 6), bad(6, 6), badCt, l$, leastCt
l$ = "abcdef": leastCt = 999
FOR i = 1 TO 6
  g$(i, 1) = MID$(l$, i, 1)
NEXT

place 1, 2

SUB place (row, col)
 FOR i = 6 TO 1 STEP -1
  lt$ = MID$(l$, i, 1)
  good = 1
  FOR c = 1 TO col - 1
   IF g$(row, c) = lt$ THEN good = 0: EXIT FOR
  NEXT
  IF good THEN
    FOR r = 1 TO row - 1
     IF g$(r, col) = lt$ THEN good = 0: EXIT FOR
    NEXT
  END IF
  IF good THEN
    g$(row, col) = lt$
    conflict = 0
    FOR r = 1 TO row - 1
     c = col - (row - r)
     IF c > 0 THEN IF g$(r, c) = lt$ THEN conflict = 1: EXIT FOR
     c = col + (row - r)
     IF c < 7 THEN IF g$(r, c) = lt$ THEN conflict = 1: EXIT FOR
    NEXT
    bad(row, col) = conflict
    IF conflict THEN badCt = badCt + 1

    IF row = 6 AND col = 6 THEN
      IF badCt < leastCt THEN
        leastCt = badCt
        PRINT leastCt
        FOR r = 1 TO 6
          FOR c = 1 TO 6
           IF bad(r, c) THEN
            PRINT UCASE$(g$(r, c));
           ELSE
            PRINT g$(r, c);
           END IF
          NEXT
          PRINT
        NEXT
        PRINT
      END IF
    ELSE
      c = col + 1: r = row
      IF c > 6 THEN c = 2: r = r + 1
      place r, c
    END IF

    IF conflict THEN bad(row, col) = 0: badCt = badCt - 1
  END IF
 NEXT i
END SUB

 


  Posted by Charlie on 2006-05-23 09:41:16
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