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# Magic squares watch

1. 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.

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!
2. That rule works for all odd squares, but even squares are much more complicated.
3. Think 3 dimensionally....

(A doughnut shape)
4. (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.
5. 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 such that every pair of 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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