Answer:
,-3
A magic square is a square grid where the numbers in each row, column, and diagonal add up to the same total. In this case, we have 8 numbers to fill in a 3x3 grid, so we'll start with the number in the middle cell, which must be 1 to balance the negative numbers around it:
|- - -|
|- 1 -|
|- - -|
Now let's fill in the other cells one at a time, following the rule that each row, column, and diagonal must add up to the same total (which we'll call S):
|-2 - -|
|- 1 -|
|- - -|
We can't put -3 in the top left corner, since that would leave us with two negative numbers in the top row. Instead, we'll put -3 in the bottom right corner:
|-2 - -|
|- 1 -|
|- -3 -|
Now we need to find a number to put in the top row that will make it add up to S. The sum of the top row so far is -2, and we need it to be S/3, since there are 3 cells in the row. So we need to find a number x that satisfies:
-2 + x = S/3
Multiplying by 3 and adding 2 to both sides, we get:
3x = S + 6
So the number we need to put in the top row is (S+6)/3. We'll call this number y:
|-2 y -|
|- 1 -|
|- -3 -|
Now let's find a number to put in the bottom row. The sum of the bottom row so far is -3, and we need it to be S/3. So we need to find a number z that satisfies:
-3 + z = S/3
Multiplying by 3 and adding 3 to both sides, we get:
3z = S + 9
So the number we need to put in the bottom row is (S+9)/3. We'll call this number w:
|-2 y -|
|- 1 -|
|w -3 z|
Finally, let's find a number to put in the top right corner. The sum of the diagonal from top right to bottom left is -2+1-z, which must be equal to S. So we have:
S = -2 + 1 - z
Simplifying, we get:
S = -1 - z
Since we know that the sum of each row is S, we can add up the numbers in the top row and subtract that from S to get:
S -2 - y = -1 - z
Simplifying, we get:
S = z - y + 1
Now we can substitute in expressions for S, y, and z to get an equation for w:
(z + 6)/3 - 2 - (S+6)/3 = -3(z + 9)/3 + 1
Multiplying by 3 to clear the fractions, we get:
z + 6 - 2(S+6) = -3z - 24 + 3
Simplifying, we get:
2S = 2z - 27
So the value of w must satisfy:
(w + 9)/3 + (S+9)/3 = -3z - 24 + 1
Multiplying by 3 to clear the fractions, we get:
w + 9 + S + 9 = -9z - 69
Substituting in expressions for S and z, we get:
w + 9 + (z - y + 1) + 9 = -9z - 69
Simplifying, we get:
w = -10z - 88
Now we can plug in values for z