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Math Help - magic square

  1. #16
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    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.
    Thanks for the info, I appreciate it.

    There are a few methods to fill in the larger magic squares. This one I came up with on my own (though admittedly it isn't THAT big of a conceptual leap...) I'm not going to produce the 9 x 9 here, which would be messy without LaTeX, but I can easily describe how to do it.

    Split the numbers 0 - 80 into groups of 9, 0 - 8, 9 - 17, etc.

    Now split the 9 x 9 grid into blocks of 3 x 3.

    You recall the 3 x 3 magic square?
    7 3 2
    5 1 6
    0 8 4

    Place the numbers 0 - 8 into the 3 x 3 block (of the 9 x 9) represented by the number "0" in the 3 x 3 magic square:

    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    ~~~~~~~~~~~~~
    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    ~~~~~~~~~~~~~
    7 3 2 | _ _ _ | _ _ _
    5 1 6 | _ _ _ | _ _ _
    0 8 4 | _ _ _ | _ _ _

    Now place the next set of numbers, 9 - 17 into the "1" position in the 3 x 3:

    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    _ _ _ | _ _ _ | _ _ _
    ~~~~~~~~~~~~~
    _ _ _ | 16 12 11 | _ _ _
    _ _ _ | 14 10 15 | _ _ _
    _ _ _ | 9 17 13 | _ _ _
    ~~~~~~~~~~~~~
    7 3 2 | _ _ _ | _ _ _
    5 1 6 | _ _ _ | _ _ _
    0 8 4 | _ _ _ | _ _ _

    etc.

    Since each 3 x 3 block is magic, and the pattern of the 3 x 3 blocks is positioned "magically," the 9 x 9 square will also be magic.

    Heck it might even be regular, though I'm not going to place bets on that.

    -Dan
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  2. #17
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    Hi Topsquark,

    Thank you very much. Your approach is excellent. I can easily construct this 9 x 9 square with magic subsquares.

    Also, do you know how to solve the below question?

    Question:
    We want to describe via a picture a set of subsets of a square which are something like diagonals, but are not quite the same. We'll call them steep diagonals. One of them, labelled e, is illustrated in the square below; the other 6 are parallel to it

    x_e_x_x_x_x_x
    x_x_x_x_e_x_x
    e_x_x_x_x_x_x
    x_x_x_e_x_x_x
    x_x_x_x_x_x_e
    x_x_e_x_x_x_x
    x_x_x_x_x_e_x

    State and prove a theorem about under what conditions we can expect that the sums on the positive ( or negative ) steep diagonals are constant, when we're dealing with a square full of consecutive integers starting at 0.
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  3. #18
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by beta12 View Post
    Hi Topsquark,

    Thank you very much. Your approach is excellent. I can easily construct this 9 x 9 square with magic subsquares.

    Also, do you know how to solve the below question?

    Question:
    We want to describe via a picture a set of subsets of a square which are something like diagonals, but are not quite the same. We'll call them steep diagonals. One of them, labelled e, is illustrated in the square below; the other 6 are parallel to it

    x_e_x_x_x_x_x
    x_x_x_x_e_x_x
    e_x_x_x_x_x_x
    x_x_x_e_x_x_x
    x_x_x_x_x_x_e
    x_x_e_x_x_x_x
    x_x_x_x_x_e_x

    State and prove a theorem about under what conditions we can expect that the sums on the positive ( or negative ) steep diagonals are constant, when we're dealing with a square full of consecutive integers starting at 0.
    I'm afraid this one's going to be a bit beyond me.

    -Dan
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  4. #19
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    Hi topsquark,

    That is alright

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



    Anyone has idea about this question?
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