SOLUTION: Create a 3-by-3 magic square using nine of the ten numbers 20, 21, 22, 23, 24, 25, 26, 27, 28, and 29. Explain your solution and reasoning. List the strategies you have used.

In base 3 there are only 3 digits, 0, 1, and 2

I can easily make a magic square using those 3 digits on
each column and row, like this:

0 2 1    
2 1 0
1 0 2 

The rows, columns and diagonals all add up to 3 

And if I rotate it 90° clockwise I have this magic square,
where all of the digits changed position except the middle number.

1 2 0
0 1 2
2 0 1

And if I put the digits of those two magic squares together,
I get this:

01 22 10
20 11 02
12 00 21

I notice that all those are different integers in base 3

Now if I take those and convert them to base 10 numbers, I will
get a magic square because the 3's digits and the 1's digits
add to the same number in each row. column, and diagonal.

0×3+1  2×3+2  1×3+0
2×3+0  1×3+1  0×3+2
1×3+2  0×3+0  2×3+1

which is

1 8 3
6 4 2
5 0 7

As we see, all the rows, columns and diagonals add up to 12. 
So that's a magic square with the integers from 0 through 8.

So I can make that into a magic square with the integers from 
20 through 28 just by adding 20 to each number:

1+20=21 8+20=28 3+20=23
6+20=26 4+20=24 2+20=22
5+20=25 0+20=20 7+20=27

which is this magic square

21 28 23
26 24 22
25 20 27

All the rows, columns and diagonals add up to 72.

-------

Or I could have made that into a magic square with the integers from 21 
through 29 by adding 21 to each number instead of 20. That would have 
just made each number 1 more than they are in the above magic square:

22 29 24
27 25 23
26 21 28

And all the rows, columns and diagonals add up to 75.

Edwin

21 28 23
26 24 22
25 20 27

and

22 29 24
27 25 23
26 21 28

Here is my strategy:

In base 3 there are only 3 digits, 0, 1, and 2

I can easily make a magic square using those 3 digits on
each column and row, like this:

0 2 1    
2 1 0
1 0 2 

The rows, columns and diagonals all add up to 3 

And if I rotate it 90° clockwise I have this magic square,
where all of the digits changed position except the middle number.

1 2 0
0 1 2
2 0 1

And if I put the digits of those two magic squares together,
I get this:

01 22 10
20 11 02
12 00 21

I notice that all those are different integers in base 3

Now if I take those and convert them to base 10 numbers, I will
get a magic square because the 3's digits and the 1's digits
add to the same number in each row. column, and diagonal.

0×3+1  2×3+2  1×3+0
2×3+0  1×3+1  0×3+2
1×3+2  0×3+0  2×3+1

which is

1 8 3
6 4 2
5 0 7

As we see, all the rows, columns and diagonals add up to 12. 
So that's a magic square with the integers from 0 through 8.

So I can make that into a magic square with the integers from 
20 through 28 just by adding 20 to each number:

1+20=21 8+20=28 3+20=23
6+20=26 4+20=24 2+20=22
5+20=25 0+20=20 7+20=27

which is this magic square

21 28 23
26 24 22
25 20 27

All the rows, columns and diagonals add up to 72.

-------

Or I could have made that into a magic square with the integers from 21 
through 29 by adding 21 to each number instead of 20. That would have 
just made each number 1 more than they are in the above magic square:

22 29 24
27 25 23
26 21 28

And all the rows, columns and diagonals add up to 75.

Edwin