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    My dad taught me this method for making magic squares about 10 years ago and I was wondering if some clever person could help me explain the maths behind it.

    The rules will be hard to explain but I'll do my best!

    Here are the rules:

    -Put a 1 in the top-middle square of a square grid which has an odd number of rows/columns.

    -Proceed by putting consecutive numbers 1,2,3... into squares that are north east of the previous square.

    Here comes the complicated part:

    -If there is no square in the north-east direction, move north-east out of the grid (with your finger,mind etc.) and you should be at the start/end of a row/column. Move to the last square of this row/column and put the next number in this square.

    -If you have got to the north-eastern most square or you are in a position where there is already a number occupying the next square to move to, move down one square and place your next number in this square.

    I think that's all the rules.

    e.g.

    \[ \left( \begin{array}{ccc}

 & 1 &  \\

 &  &  \\

 &  &  \end{array} \right)\] \[ \left( \begin{array}{ccc}

 & 1 &  \\

 &  &  \\

 &  & 2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 & 1 &  \\

 3  &  \\

 &  & 2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 & 1 &  \\

 3  &  \\

 4  & &2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 & 1 & 6 \\

 3  &5 & \\

 4  & &2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 & 1 &6  \\

 3  &5  &7\\

 4  & &2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 8 &1 &6  \\

 3 & 5 & 7\\

 4  & &2 \end{array} \right)\] \[ \left( \begin{array}{ccc}

 8 &1 &6  \\

 3  &5 & 7\\

 4 & 9 &2 \end{array} \right)\]

    If you do it correctly then you'll end up in the middle-bottom square. All rows, columns and diagonals should add up to the middle number multiplied by the row length.

    I remember creating a 23x23 grid once... Remember, I was 8 .

    I don't expect anyone to tell me straight away how it works unless you've seen it before. But if someone could tell me how I could start a proof for any size grid, I'd be very grateful. Alternatively, you could just marvel at its magnificance!
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    That rule works for all odd squares, but even squares are much more complicated.
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    Think 3 dimensionally....

    (A doughnut shape)
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    (Original post by Maths Buster)
    That rule works for all odd squares, but even squares are much more complicated.
    Do you know why the rule works? I have no idea how to even start explaining why the method works.
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    I remember that something very much like this came up in the context of the Chinese remainder theorem... something about showing that there is a unique solution to a set of modular-arithmetic equations if each equation is mod n_i such that every pair of n_i is coprime...? I forget. But I remember part of the proof involved filling up a table in the manner you describe - as if it were a torus.
 
 
 
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