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Square Patterns (Posted on 2007-04-20) Difficulty: 3 of 5
Consider a 2x2 board in which each square can be black or white. Obviously there are 16 different 2x2 patterns, counting reflections and rotations.
A 5x5 board has 16 different 2x2 subsquares. Can the squares of a 5x5 board be colored so that each 2x2 subsquare has a different pattern?

See The Solution Submitted by Brian Smith    
Rating: 2.0000 (1 votes)

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Solution 50 unique solutions Comment 3 of 3 |
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None of the above is trivially related to any other, as

1. All have more Black than white squares.
2. The all-black square of 4 is in the upper left quadrant.
3. Each of those meeting those two criteria was checked to be sure it was not a diagonal reflection of a previous solution.

Thus, what originally were 800 solutions became 50.

I've bolded the solution that's the diagonal reflection of Penny's solution.

DECLARE SUB build (row!, col!)
CLEAR , , 9999

DIM SHARED used(15), board(5, 5), wCt, hist(111, 5, 5)
DIM SHARED blkCt, row15, col15


build 1, 1



SUB build (row, col)
 board(row, col) = 0
 GOSUB tryIt
 board(row, col) = 1
 blkCt = blkCt + 1
 GOSUB tryIt
 blkCt = blkCt - 1

   idx = -1
   IF row > 1 AND col > 1 THEN
     idx = 8 * board(row - 1, col - 1) + 4 * board(row - 1, col) + 2 * board(row, col - 1) + board(row, col)
     IF used(idx) THEN RETURN
     used(idx) = 1
     IF idx = 15 THEN row15 = row: col15 = col
   r = row: c = col + 1
   IF c > 5 THEN
     c = 1: r = r + 1
   IF r < 6 THEN
     build r, c
     IF blkCt > 12 AND row15 <= 3 AND col15 <= 3 THEN
       good = 1
       FOR h = 1 TO wCt
         match = 1
         FOR r1 = 1 TO 5
          FOR c1 = 1 TO 5
            IF board(r1, c1) <> hist(h, c1, r1) THEN match = 0: EXIT FOR
          IF match = 0 THEN EXIT FOR
         IF match THEN good = 0: EXIT FOR
       NEXT h
       IF good THEN
         pRow = wCt 10
         pCol = wCt MOD 10
         pRow = pRow * 6: pCol = pCol * 7
         wCt = wCt + 1
         FOR r1 = 1 TO 5
          FOR c1 = 1 TO 5
            LOCATE pRow + r1, pCol + c1
            IF board(r1, c1) THEN PRINT "B"; :  ELSE PRINT "w";
            hist(wCt, r1, c1) = board(r1, c1)
       END IF
     END IF
   IF idx >= 0 THEN used(idx) = 0


Edited on April 20, 2007, 11:19 pm
  Posted by Charlie on 2007-04-20 22:30:52

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