Answer :
We are given a magic square where every row, column, and main diagonal adds up to the same sum, which we call [tex]$S$[/tex]. The square is:
[tex]$$
\begin{array}{|c|c|c|}
\hline
25 & X & 65 \\
\hline
60 & 20 & Y \\
\hline
C & B & A \\
\hline
\end{array}
$$[/tex]
We want to compute the value of the expression
[tex]$$
X - Y - A + B - C.
$$[/tex]
Let’s work through the steps:
1. From the first row, the sum is:
[tex]$$
25 + X + 65 = S \quad \Longrightarrow \quad X + 90 = S. \tag{1}
$$[/tex]
2. From the second row, the sum is:
[tex]$$
60 + 20 + Y = S \quad \Longrightarrow \quad Y + 80 = S. \tag{2}
$$[/tex]
Since both expressions equal [tex]$S$[/tex], we set them equal:
[tex]$$
X + 90 = Y + 80 \quad \Longrightarrow \quad X = Y - 10. \tag{3}
$$[/tex]
3. From the first column:
[tex]$$
25 + 60 + C = S \quad \Longrightarrow \quad 85 + C = S.
$$[/tex]
Substituting [tex]$S = X + 90$[/tex] from (1):
[tex]$$
85 + C = X + 90 \quad \Longrightarrow \quad C = X + 5. \tag{4}
$$[/tex]
4. From the main diagonal (top left to bottom right):
[tex]$$
25 + 20 + A = S \quad \Longrightarrow \quad A = S - 45.
$$[/tex]
Using (1) [tex]$S=X+90$[/tex], we have:
[tex]$$
A = (X + 90) - 45 = X + 45. \tag{5}
$$[/tex]
5. From the second column:
[tex]$$
X + 20 + B = S \quad \Longrightarrow \quad B = S - X - 20.
$$[/tex]
Again using (1):
[tex]$$
B = (X + 90) - X - 20 = 70. \tag{6}
$$[/tex]
6. From the third column:
[tex]$$
65 + Y + A = S \quad \Longrightarrow \quad A = S - 65 - Y.
$$[/tex]
Substituting [tex]$S = X + 90$[/tex] as before:
[tex]$$
A = (X + 90) - 65 - Y = X + 25 - Y. \tag{7}
$$[/tex]
7. Now equate the two expressions for [tex]$A$[/tex] from (5) and (7):
[tex]$$
X + 45 = X + 25 - Y.
$$[/tex]
Subtract [tex]$X$[/tex] from both sides:
[tex]$$
45 = 25 - Y \quad \Longrightarrow \quad Y = 25 - 45 = -20. \tag{8}
$$[/tex]
8. Substitute [tex]$Y = -20$[/tex] into (3) to find [tex]$X$[/tex]:
[tex]$$
X = (-20) - 10 = -30. \tag{9}
$$[/tex]
9. Now, use (1) to determine [tex]$S$[/tex]:
[tex]$$
S = X + 90 = -30 + 90 = 60. \tag{10}
$$[/tex]
10. Find [tex]$A$[/tex] using (5):
[tex]$$
A = X + 45 = -30 + 45 = 15. \tag{11}
$$[/tex]
11. Find [tex]$C$[/tex] using (4):
[tex]$$
C = X + 5 = -30 + 5 = -25. \tag{12}
$$[/tex]
12. We already have [tex]$B = 70$[/tex] from (6).
Now we substitute all these values into the expression:
[tex]$$
X - Y - A + B - C = (-30) - (-20) - 15 + 70 - (-25).
$$[/tex]
Simplify step by step:
[tex]\[
\begin{aligned}
X - Y - A + B - C &= -30 + 20 - 15 + 70 + 25 \\
&= (-10) - 15 + 70 + 25 \\
&= -25 + 70 + 25 \\
&= 70.
\end{aligned}
\][/tex]
Thus, the value of the expression [tex]$$X - Y - A + B - C$$[/tex] is [tex]$$\boxed{70}.$$[/tex]
[tex]$$
\begin{array}{|c|c|c|}
\hline
25 & X & 65 \\
\hline
60 & 20 & Y \\
\hline
C & B & A \\
\hline
\end{array}
$$[/tex]
We want to compute the value of the expression
[tex]$$
X - Y - A + B - C.
$$[/tex]
Let’s work through the steps:
1. From the first row, the sum is:
[tex]$$
25 + X + 65 = S \quad \Longrightarrow \quad X + 90 = S. \tag{1}
$$[/tex]
2. From the second row, the sum is:
[tex]$$
60 + 20 + Y = S \quad \Longrightarrow \quad Y + 80 = S. \tag{2}
$$[/tex]
Since both expressions equal [tex]$S$[/tex], we set them equal:
[tex]$$
X + 90 = Y + 80 \quad \Longrightarrow \quad X = Y - 10. \tag{3}
$$[/tex]
3. From the first column:
[tex]$$
25 + 60 + C = S \quad \Longrightarrow \quad 85 + C = S.
$$[/tex]
Substituting [tex]$S = X + 90$[/tex] from (1):
[tex]$$
85 + C = X + 90 \quad \Longrightarrow \quad C = X + 5. \tag{4}
$$[/tex]
4. From the main diagonal (top left to bottom right):
[tex]$$
25 + 20 + A = S \quad \Longrightarrow \quad A = S - 45.
$$[/tex]
Using (1) [tex]$S=X+90$[/tex], we have:
[tex]$$
A = (X + 90) - 45 = X + 45. \tag{5}
$$[/tex]
5. From the second column:
[tex]$$
X + 20 + B = S \quad \Longrightarrow \quad B = S - X - 20.
$$[/tex]
Again using (1):
[tex]$$
B = (X + 90) - X - 20 = 70. \tag{6}
$$[/tex]
6. From the third column:
[tex]$$
65 + Y + A = S \quad \Longrightarrow \quad A = S - 65 - Y.
$$[/tex]
Substituting [tex]$S = X + 90$[/tex] as before:
[tex]$$
A = (X + 90) - 65 - Y = X + 25 - Y. \tag{7}
$$[/tex]
7. Now equate the two expressions for [tex]$A$[/tex] from (5) and (7):
[tex]$$
X + 45 = X + 25 - Y.
$$[/tex]
Subtract [tex]$X$[/tex] from both sides:
[tex]$$
45 = 25 - Y \quad \Longrightarrow \quad Y = 25 - 45 = -20. \tag{8}
$$[/tex]
8. Substitute [tex]$Y = -20$[/tex] into (3) to find [tex]$X$[/tex]:
[tex]$$
X = (-20) - 10 = -30. \tag{9}
$$[/tex]
9. Now, use (1) to determine [tex]$S$[/tex]:
[tex]$$
S = X + 90 = -30 + 90 = 60. \tag{10}
$$[/tex]
10. Find [tex]$A$[/tex] using (5):
[tex]$$
A = X + 45 = -30 + 45 = 15. \tag{11}
$$[/tex]
11. Find [tex]$C$[/tex] using (4):
[tex]$$
C = X + 5 = -30 + 5 = -25. \tag{12}
$$[/tex]
12. We already have [tex]$B = 70$[/tex] from (6).
Now we substitute all these values into the expression:
[tex]$$
X - Y - A + B - C = (-30) - (-20) - 15 + 70 - (-25).
$$[/tex]
Simplify step by step:
[tex]\[
\begin{aligned}
X - Y - A + B - C &= -30 + 20 - 15 + 70 + 25 \\
&= (-10) - 15 + 70 + 25 \\
&= -25 + 70 + 25 \\
&= 70.
\end{aligned}
\][/tex]
Thus, the value of the expression [tex]$$X - Y - A + B - C$$[/tex] is [tex]$$\boxed{70}.$$[/tex]
