## Problem. Magic
Square (click here to see the problem statement)

__My Solution__
(Basic Square).

### I can use only the whole numbers 1 through
9. I start by making a 3x3 array of those numbers:

### I need to get an idea of how the sum
are interacting, so I add up each row, column, and diagonal. I
find that the middle row, the middle column, and both diagonals
sum to 15:

###

### The sum of the first row is 6 and the
sum of the third row is 24. I need to balance the results, so
I try switching the 2 and the 8 in the middle cloumn:

### I try to balance the first and third
columns more by switching the 4 and 6 in the midle row:

### As 5 is in the middle square and I am
trying to achieve a sum of 15 in all cases, I need to maintain
pairs on the "outer ring" that each add to 10: (1,9)
(2,8) (3,7) (4,6) and that are in opposite positions on the outer
ring. I try rotating the outer ring counter-clockwise:

###

###

### I luck out on my very first rotation.
I try again:

###
then

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###

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then

###

###

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then

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###

###
And the next rotation will bring the ring full-circle.

###

### So it appears that my solutions are:

###

###

###

###

### I now need to consider possibilities
other than rotations, so I can examine Reflections. For example,
I can hold the middle column constant and switch the numbers in
the the first and third columns:

###

###

###

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###

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### If I define this array to
be my base,

###

###

###

###

###

###

###

### then a reflection about
the middle column is:

###

###

###

###

###

###

### Similarly, a reflection about
the middle row

means that I hold the middle row constant

and switch the numbers in the first and third rows:

###

###

###

###

###

###

### So my reflection is:

###

###

### And reflections about each diagonal:

###
gives

###

###
gives

### Last, I need to examine the possibility
of placing numbers other than 5 in the middle square, such as
8

### I can fairly quickly determine that this
is not an option, because I will have 9 somewhere in the outer
ring. As the sum of 8 & 9 is 17, I will have a row or column
greater than 15. I can immediatley extend this to rule out having
9 in the middle square:

#
-&-

#

### For __7 in the middle square__,
I need the sum of all pairs of remaining numbers to equal 8; however
in my set

{1, 2, 3, 4, 5, 6, 8, 9},

I already have 8 and 9, so there will be at least one pair whose
sum is breater than 8.

###

### For __6 in the middle square__,
the sum of all remaining pairs must equal 9;

however my set contains 9, so there is at least one pair whose
sum is greater than 9.

###

### For __4, 3, 2, or 1 in the middle
square__, the sum of all pairs must equal 11, 12, 13, and 14,
respectively;

however in each set we have 1, 2, 3, and 4 requiring the matches
for each to be 10 (or greater),

none of which is included in the set.

###

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