1. Originally Posted by beta12
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

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

2. 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.

3. Originally Posted by beta12
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

4. Hi topsquark,

That is alright

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

5. ## Re: magic square

Some interesting background history on Magic Squares in art, science and culture on this blog: Glenn Westmore -Glenn Westmore

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