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Squares, subsquares and arrangement puzzle

  1. Aug 6, 2009 #1
    Substitute each of the capital letters in this 3x3 square by a different digit from 1 to 9 such that the sum of digits in each of the four 2x2 subsquares is equal to 6*E.

    A B C
    D E F
    G H I

    What will be the arrangement(s), if keeping all the other conditions unaltered, the sum of digits in each of the four 2x2 subsquares is equal to 5*E?

    Note: The rotations and reflections of a valid arrangement are considered as the same solution.
     
  2. jcsd
  3. Aug 8, 2009 #2
    Part 1 of the puzzle:
    There are 8 solutions, however solution 2-8 are either reflections
    or rotations of solution 1.

    solution1: a=4 b=2 c=7 d=9 e=3 f=6 g=5 h=1 i=8
    solution2: a=4 b=9 c=5 d=2 e=3 f=1 g=7 h=6 i=8
    solution3: a=5 b=1 c=8 d=9 e=3 f=6 g=4 h=2 i=7
    solution4: a=5 b=9 c=4 d=1 e=3 f=2 g=8 h=6 i=7
    solution5: a=7 b=2 c=4 d=6 e=3 f=9 g=8 h=1 i=5
    solution6: a=7 b=6 c=8 d=2 e=3 f=1 g=4 h=9 i=5
    solution7: a=8 b=1 c=5 d=6 e=3 f=9 g=7 h=2 i=4
    solution8: a=8 b=6 c=7 d=1 e=3 f=2 g=5 h=9 i=4


    solution1: a=4 b=2 c=7 d=9 e=3 f=6 g=5 h=1 i=8
    4 2 7
    9 3 6
    5 1 8

    solution2: a=4 b=9 c=5 d=2 e=3 f=1 g=7 h=6 i=8
    4 9 5
    2 3 1
    7 6 8
    (reflection of solution 1 along the AI-line)

    solution3: a=5 b=1 c=8 d=9 e=3 f=6 g=4 h=2 i=7
    5 1 8
    9 3 6
    4 2 7
    (reflection of solution 1 along the DF-line)

    solution4: a=5 b=9 c=4 d=1 e=3 f=2 g=8 h=6 i=7
    5 9 4
    1 3 2
    8 6 7
    (rotation of solution 1 by 90° to the right)

    solution5: a=7 b=2 c=4 d=6 e=3 f=9 g=8 h=1 i=5
    7 2 4
    6 3 9
    8 1 5
    (reflection of solution 1 along the BH-line)

    solution6: a=7 b=6 c=8 d=2 e=3 f=1 g=4 h=9 i=5
    7 6 8
    2 3 1
    4 9 5
    (rotation of solution 1 by 90° to the left)

    solution7: a=8 b=1 c=5 d=6 e=3 f=9 g=7 h=2 i=4
    8 1 5
    6 3 9
    7 2 4
    (rotation of solution 1 by 180°)

    solution8: a=8 b=6 c=7 d=1 e=3 f=2 g=5 h=9 i=4
    8 6 7
    1 3 2
    5 9 4
    (reflection of solution 1 along the CG-line)
     
  4. Aug 8, 2009 #3
    Part 2 of the puzzle:
    Part 2 of puzzle:
    With the 5*E condition we get 16 solutions:
    solution1: a=1 b=6 c=7 d=9 e=4 f=3 g=2 h=5 i=8
    solution2: a=1 b=9 c=2 d=6 e=4 f=5 g=7 h=3 i=8
    solution3: a=2 b=5 c=3 d=9 e=4 f=8 g=6 h=1 i=7
    solution4: a=2 b=5 c=8 d=9 e=4 f=3 g=1 h=6 i=7
    solution5: a=2 b=9 c=1 d=5 e=4 f=6 g=8 h=3 i=7
    solution6: a=2 b=9 c=6 d=5 e=4 f=1 g=3 h=8 i=7
    solution7: a=3 b=5 c=2 d=8 e=4 f=9 g=7 h=1 i=6
    solution8: a=3 b=8 c=7 d=5 e=4 f=1 g=2 h=9 i=6
    solution9: a=6 b=1 c=7 d=9 e=4 f=8 g=2 h=5 i=3
    solution10: a=6 b=9 c=2 d=1 e=4 f=5 g=7 h=8 i=3
    solution11: a=7 b=1 c=6 d=8 e=4 f=9 g=3 h=5 i=2
    solution12: a=7 b=3 c=8 d=6 e=4 f=5 g=1 h=9 i=2
    solution13: a=7 b=6 c=1 d=3 e=4 f=9 g=8 h=5 i=2
    solution14: a=7 b=8 c=3 d=1 e=4 f=5 g=6 h=9 i=2
    solution15: a=8 b=3 c=7 d=5 e=4 f=6 g=2 h=9 i=1
    solution16: a=8 b=5 c=2 d=3 e=4 f=9 g=7 h=6 i=1

    However, the only "real" solutions are solution 1 and 3.
    Solution 2,4,5,12,13,15,16 are either reflections or rotations of solution 1.
    Solution 6,7,8,9,10,11,14 are either reflections or rotations or solution 3.


    solution1: a=1 b=6 c=7 d=9 e=4 f=3 g=2 h=5 i=8
    1 6 7
    9 4 3
    2 5 8


    solution2: a=1 b=9 c=2 d=6 e=4 f=5 g=7 h=3 i=8
    1 9 2
    6 4 5
    7 3 8
    (reflection of solution 1 along the AI-line)

    solution3: a=2 b=5 c=3 d=9 e=4 f=8 g=6 h=1 i=7
    2 5 3
    9 4 8
    6 1 7
    (neither rotation nor reflection of solution 1)

    solution4: a=2 b=5 c=8 d=9 e=4 f=3 g=1 h=6 i=7
    2 5 8
    9 4 3
    1 6 7
    (reflection of solution 1 along the DF-line)

    solution5: a=2 b=9 c=1 d=5 e=4 f=6 g=8 h=3 i=7
    2 9 1
    5 4 6
    8 3 7
    (rotation of solution 1 by 90° to the right)

    solution6: a=2 b=9 c=6 d=5 e=4 f=1 g=3 h=8 i=7
    2 9 6
    5 4 1
    3 8 7
    (neither rotation nor reflection of solution 1)
    (but reflection of solution 3 along the AI-line)

    solution7: a=3 b=5 c=2 d=8 e=4 f=9 g=7 h=1 i=6
    3 5 2
    8 4 9
    7 1 6
    (neither rotation nor reflection of solution 1)
    (but reflection of solution 3 along the BH-line)

    solution8: a=3 b=8 c=7 d=5 e=4 f=1 g=2 h=9 i=6
    3 8 7
    5 4 1
    2 9 6
    (neither rotation nor reflection of solution 1)
    (but rotation of solution 3 by 90° to the left)

    solution9: a=6 b=1 c=7 d=9 e=4 f=8 g=2 h=5 i=3
    6 1 7
    9 4 8
    2 5 3
    (neither rotation nor reflection of solution 1)
    (but reflection of solution 3 along the DF-line)

    solution10: a=6 b=9 c=2 d=1 e=4 f=5 g=7 h=8 i=3
    6 9 2
    1 4 5
    7 8 3
    (neither rotation nor reflection of solution 1)
    (but rotation of solution 3 by 90° to the right)

    solution11: a=7 b=1 c=6 d=8 e=4 f=9 g=3 h=5 i=2
    7 1 6
    8 4 9
    3 5 2
    (neither rotation nor reflection of solution 1)
    (but rotation of solution 3 by 180°)

    solution12: a=7 b=3 c=8 d=6 e=4 f=5 g=1 h=9 i=2
    7 3 8
    6 4 5
    1 9 2
    (rotation of solution 1 by 90° to the right)

    solution13: a=7 b=6 c=1 d=3 e=4 f=9 g=8 h=5 i=2
    7 6 1
    3 4 9
    8 5 2
    (reflection of solution 1 along the BH-line)

    solution14: a=7 b=8 c=3 d=1 e=4 f=5 g=6 h=9 i=2
    7 8 3
    1 4 5
    6 9 2
    (neither rotation nor reflection of solution 1)
    (but reflection of solution 3 along the CG-line)

    solution15: a=8 b=3 c=7 d=5 e=4 f=6 g=2 h=9 i=1
    8 3 7
    5 4 6
    2 9 1
    (reflection of solution 1 along the CG-line)

    solution16: a=8 b=5 c=2 d=3 e=4 f=9 g=7 h=6 i=1
    8 5 2
    3 4 9
    7 6 1
    (rotation of solution 1 by 180°)
     
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