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Math Help - magic square

  1. #1
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    Question magic square

    1) Suppose that S is a square such that the sum of the entries in each row is some number R, and the sum of the entries in each column is some number C.
    Prove that S is in fact a magic square , i.e. R=C.

    2) Prove that every regular square is necessarily magic.

    How should I prove these two questions?
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  2. #2
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    Quote Originally Posted by beta12 View Post
    1) Suppose that S is a square such that the sum of the entries in each row is some number R, and the sum of the entries in each column is some number C.
    Prove that S is in fact a magic square , i.e. R=C.
    If each row sums to R then the total sum is nR, where n is the number
    of rows. Similarly if each col sums to C and there are m cols, then the total
    sum is mC. But we have a square so n=m, so the total sum=nR=nC,
    which implies R=C.

    2) Prove that every regular square is necessarily magic.
    What is a regular square?

    RonL
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    Hi Captain black,

    I don't quite get the meaning of " regular squares" as well as. If you can figure it out from the below question, please do teach me. Thank you very much.

    Question
    We say that n x n square is regular provided that:
    a) Each of the integers from 0 to n^2 - 1 appears in exactly one cell, and each cell contains only one integer.
    b) If we express the entries in base-n form , each base-n digit occurs exactly once in the units' position, and exactly once in the n's position.

    For example:

    7 3 2
    5 1 6
    0 8 4

    expressed base 3 is

    21 10 02
    12 01 20
    00 22 11

    The square is regular because each of the ternary digits 0, 1, 2 appears exactly once in the units' and 3's in each row and each column.

    * construct an example of 4 x 4 regular square.
    * express your answer in both decimal and base-4 notation.

    Do you know how to construct such 4 x 4 regular square in both decimal and base - 4 notation?
    Last edited by beta12; September 27th 2006 at 05:47 AM. Reason: better outcome
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  4. #4
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    Quote Originally Posted by beta12 View Post
    Prove that every regular square is necessarily magic.
    If I understand what you posted below for a regular square, this can't be proved. For example:

    0 1 2
    3 4 5
    6 7 8

    is a regular square by your definition, and it certainly isn't magic.

    -Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    Hi Captain black,

    I don't quite get the meaning of " regular squares" as well as. If you can figure it out from the below question, please do teach me. Thank you very much.

    Question
    We say that n x n square is regular provided that:
    a) Each of the integers from 0 to n^2 - 1 appears in exactly one cell, and each cell contains only one integer.
    b) If we express the entries in base-n form , each base-n digit occurs exactly once in the units' position, and exactly once in the n's position.

    For example:

    7 3 2
    5 1 6
    0 8 4

    expressed base 3 is

    21 10 02
    12 01 20
    00 22 11

    The square is regular because each of the ternary digits 0, 1, 2 appears exactly once in the units' and 3's in each row and each column.

    * construct an example of 4 x 4 regular square.
    * express your answer in both decimal and base-4 notation.

    Do you know how to construct such 4 x 4 regular square in both decimal and base - 4 notation?
    There are methods to produce many types of magic squares. This one will produce a 4 x 4 magic square:

    We want to write out the numbers 0 through 15 in order in a 4 x 4 matrix, but we only want certain positions filled. The pattern is as follows:

    0 _ _ 3
    _ 5 6 _
    _ 9 10 _
    12 _ _ 15

    Now we want to do the same thing backward, that is starting from the lower right corner list the numbers 0 through 15 again, filling in the blank spots. This gives you a square:

    0 14 13 3
    11 5 6 8
    7 9 10 4
    12 2 1 15
    Hopefully this will look okay when I post it. (Waaa! I want LaTeX back!)

    This square is magic, as you can verify for yourself.

    -Dan
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    Quote Originally Posted by topsquark View Post
    If I understand what you posted below for a regular square, this can't be proved. For example:

    0 1 2
    3 4 5
    6 7 8

    is a regular square by your definition, and it certainly isn't magic.

    -Dan
    Write your square out in ternery and you have:

    00 01 02
    10 11 12
    20 21 22

    and since col-1 contains 0 in the units poition more than once this is
    not regular.

    RonL
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Write your square out in ternery and you have:

    00 01 02
    10 11 12
    20 21 22

    and since col-1 contains 0 in the units poition more than once this is
    not regular.

    RonL
    Ahhh, I see it now!

    -Dan
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  8. #8
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    Quote Originally Posted by beta12 View Post
    Hi Captain black,

    I don't quite get the meaning of " regular squares" as well as. If you can figure it out from the below question, please do teach me. Thank you very much.

    Question
    We say that n x n square is regular provided that:
    a) Each of the integers from 0 to n^2 - 1 appears in exactly one cell, and each cell contains only one integer.
    b) If we express the entries in base-n form , each base-n digit occurs exactly once in the units' position, and exactly once in the n's position.

    For example:

    7 3 2
    5 1 6
    0 8 4

    expressed base 3 is

    21 10 02
    12 01 20
    00 22 11

    The square is regular because each of the ternary digits 0, 1, 2 appears exactly once in the units' and 3's in each row and each column.

    * construct an example of 4 x 4 regular square.
    * express your answer in both decimal and base-4 notation.

    Do you know how to construct such 4 x 4 regular square in both decimal and base - 4 notation?
    The sum of a row is (1+..+n-1)n+(1+..+n), and the same for any col,
    hence the square in magic (this is the sum of the digits in the units position
    plus n times the sum of the digits in the n position).

    RonL
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    0 14 13 3
    11 5 6 8
    7 9 10 4
    12 2 1 15
    Hopefully this will look okay when I post it. (Waaa! I want LaTeX back!)

    This square is magic, as you can verify for yourself.

    =======================

    Hi topsquark,

    I know how to construct any n x n magic squares.

    But I don't understand how to express my answer ( for example the above 4 x 4 magic square) into both decimal and base-4 notation.

    Can you explain to me how
    7 3 2
    5 1 6
    0 8 4

    changed to be base 3

    21 10 02
    12 01 20
    00 22 11

    Thks
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  10. #10
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    Quote Originally Posted by beta12 View Post
    0 14 13 3
    11 5 6 8
    7 9 10 4
    12 2 1 15
    Hopefully this will look okay when I post it. (Waaa! I want LaTeX back!)

    This square is magic, as you can verify for yourself.

    =======================

    Hi topsquark,

    I know how to construct any n x n magic squares.

    But I don't understand how to express my answer ( for example the above 4 x 4 magic square) into both decimal and base-4 notation.

    Can you explain to me how
    7 3 2
    5 1 6
    0 8 4

    changed to be base 3

    21 10 02
    12 01 20
    00 22 11

    Thks
    The digits d(0)..d(k) of the base n representation of N are non-negative
    integers form {0, 1, .., (n-1)}, d(k) != 0, such that:

    N=sum d(r) n^r

    where d(0) is the least significant digit and d(k) the most significant.

    Then it is clear that if we define

    N(0)=N

    d(r)=mod(N(r),n)
    N(r)=[N(r-1)-d(r-1)]/n

    Then if we stop when N(k)=0 that d(0), d(1), .., d(k) are the base n
    digits of the base n representation of N.

    Example N=27, n=5

    N(0)=27

    d(0)=mod(27,5)=2
    N(1)=5

    d(1)=mod(5,5)=0
    N(2)=1

    d(2)=mod(1,5)=1
    N(3)=0

    so the base 5 representation of 27 is (1)(0)(2).

    RonL
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  11. #11
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    Hi CaptainBlack,

    So how so I express

    0 14 13 3
    11 5 6 8
    7 9 10 4
    12 2 1 15

    in both decimal and base-4 notation?

    I am still looking at your explaination about the tenary digit. Not quite get it at this moment....
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  12. #12
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    0 14 13 3
    11 5 6 8
    7 9 10 4
    12 2 1 15
    Hopefully this will look okay when I post it. (Waaa! I want LaTeX back!)

    This square is magic, as you can verify for yourself.

    =======================

    Hi topsquark,

    I know how to construct any n x n magic squares.

    But I don't understand how to express my answer ( for example the above 4 x 4 magic square) into both decimal and base-4 notation.

    Can you explain to me how
    7 3 2
    5 1 6
    0 8 4

    changed to be base 3

    21 10 02
    12 01 20
    00 22 11

    Thks
    I know how to construct any n x n magic squares.
    Really? Do you know how to do a 6 x 6? I don't know how to do those.

    We wish to convert the numbers 0 - 8 to base 3. The simplest way is just to count them.
    0 -- 00
    1 -- 01
    2 -- 02
    3 -- 10
    4 -- 11
    5 -- 12
    6 -- 20
    7 -- 21
    8 -- 22
    (Where the first column is in base 10 and the second column is its equivalent in base 3.)

    The base 4 counting works the same way:
    0 -- 00
    1 -- 01
    2 -- 02
    3 -- 03
    4 -- 10
    5 -- 11
    6 -- 12
    7 -- 13
    8 -- 20
    9 -- 21
    10 -- 22
    11 -- 23
    12 -- 30
    13 -- 31
    14 -- 32
    15 -- 33
    And just make the replacements.

    So the 4 x 4 square I showed you will become:

    00 32 31 03
    23 11 12 20
    13 21 22 10
    30 02 01 33

    So if I (finally) have the definition of a regular square correct, this square is not regular.

    -Dan

    The flaw in my construction method is probably that the methods I know also cause the diagonals to have the same sum as the rows and columns. I don't know if that can be done for a regular square.
    Last edited by topsquark; September 28th 2006 at 04:48 AM. Reason: Counted wrong
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  13. #13
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    Quote Originally Posted by topsquark View Post
    Really? Do you know how to do a 6 x 6? I don't know how to do those.

    Please take a look of this web for constructing 4n+2 magic squares.
    The Other Series Solution


    ======================================
    So the 4 x 4 square I showed you will become:

    00 32 31 03
    23 11 12 20
    13 21 22 10
    30 02 01 33

    So if I (finally) have the definition of a regular square correct, this square is not regular.

    -Dan

    The flaw in my construction method is probably that the methods I know also cause the diagonals to have the same sum as the rows and columns. I don't know if that can be done for a regular square.
    Yes, I have just found out from the book that ==whenever the uniform step method gives a filled magic square, that square is regular.

    I think we should use uniform step method to construct this 4 x 4 regular square.
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  14. #14
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    Quote Originally Posted by beta12 View Post
    Yes, I have just found out from the book that ==whenever the uniform step method gives a filled magic square, that square is regular.

    I think we should use uniform step method to construct this 4 x 4 regular square.
    Fascinating. The last I knew the 6 x 6 and 10 x 10, etc. had been solved (for at least several solutions) but not that a method for constructing them had been found. Thank you! (The geek in me is truely satisfied. )

    The site you mentioned has the methods that I know for how to construct a 4n x 4n square and this is obviously not the "uniform step method" construction as my 4 x 4 wasn't regular. Would you be willing to share this construction method?

    -Dan
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  15. #15
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    Hi Topsquark,

    The website is very good!

    I used the uniform step method but the square which it produced is not magic.

    Uniform step method does not gurantee can product a filled magic square.

    Here is the example of application of U S M of 5 x 5 magic square with magic sum 60:

    19_7_20_13_1
    10_3_16_9_22
    6_24_12_0_18
    2_15_8_21_14
    23_11_4_17_5

    you need to set the value of a , b , c, d, e, f,
    a = 4 ( since x0=4)
    b = 3 (since y0=3)
    c=1 (xj move RHS by 1)
    d = 2(yj move up by 2)
    e = 1 ( xj move RHS by 1 from 0 to 5)
    f = 3 (yj move up by 3 from 0 to 5)

    note:
    you can put the initial number (0) anywhere in the cell. so a, b is changing according to the position of the initial number.

    c , d is fixed

    e and f apply when
    0 jump to 5,
    5 jump to 10
    10 jump to 15
    15 jump to 20

    Hope this information is helpful to you.
    ================================
    But this one will work as a regular square
    0_5_10_15
    11_14_1_4
    13_8_7_2
    6_3_12_9

    Finally, I solved it!

    ===========================
    Can anyone construct the below 9 x 9 square?
    # Construct a 9 x 9 filled, magic square using the integers from 0 to 80. Your magic square should additionally have the property that when it is divided into ninths, each 3 x 3 subsquare is also magic.
    Last edited by beta12; September 28th 2006 at 08:52 AM. Reason: adding more information
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