Magic Squares 3x3 Worksheet - Free Printable

To solve the 3x3 magic square puzzles, we need to ensure that the sum of each row, column, and diagonal equals the given "magic sum." Let's solve each puzzle step by step.

---

Puzzle 1: Magic Sum = 60

The grid is:
```
| |
---+---+---
18 | 20 | 22
---+---+---
| 24 |
---+---+---
| |
```
with the top middle cell filled with 16.

#### Step 1: Use the magic sum to find missing numbers.
The magic sum is 60. Each row, column, and diagonal must add up to 60.

#### Row 2:
The second row is \(18 + 20 + 22 = 60\). This is already correct.

#### Column 2:
The second column is \(16 + 20 + 24 = 60\). This is already correct.

#### Diagonal (top-left to bottom-right):
The diagonal is \( \text{top-left} + 20 + \text{bottom-right} = 60 \).

#### Row 1:
Let the top-left cell be \(a\) and the top-right cell be \(b\). The first row is:
\[ a + 16 + b = 60 \]
\[ a + b = 44 \quad \text{(Equation 1)} \]

#### Row 3:
Let the bottom-left cell be \(c\) and the bottom-right cell be \(d\). The third row is:
\[ c + 24 + d = 60 \]
\[ c + d = 36 \quad \text{(Equation 2)} \]

#### Column 1:
The first column is:
\[ a + 18 + c = 60 \]
\[ a + c = 42 \quad \text{(Equation 3)} \]

#### Column 3:
The third column is:
\[ b + 22 + d = 60 \]
\[ b + d = 38 \quad \text{(Equation 4)} \]

#### Diagonal (top-left to bottom-right):
The diagonal is:
\[ a + 20 + d = 60 \]
\[ a + d = 40 \quad \text{(Equation 5)} \]

#### Solve the system of equations:
From Equation 1: \(a + b = 44\)
From Equation 2: \(c + d = 36\)
From Equation 3: \(a + c = 42\)
From Equation 4: \(b + d = 38\)
From Equation 5: \(a + d = 40\)

- From Equation 5: \(d = 40 - a\)
- Substitute \(d = 40 - a\) into Equation 2:
\[ c + (40 - a) = 36 \]
\[ c = a - 4 \]

- Substitute \(c = a - 4\) into Equation 3:
\[ a + (a - 4) = 42 \]
\[ 2a - 4 = 42 \]
\[ 2a = 46 \]
\[ a = 23 \]

- Now, find \(c\):
\[ c = a - 4 = 23 - 4 = 19 \]

- Find \(d\):
\[ d = 40 - a = 40 - 23 = 17 \]

- Find \(b\):
\[ a + b = 44 \]
\[ 23 + b = 44 \]
\[ b = 21 \]

#### Fill in the grid:
```
23 | 16 | 21
---+----+----
18 | 20 | 22
---+----+----
19 | 24 | 17
```

---

Puzzle 2: Magic Sum = 39

The grid is:
```
| 9 |
---+----+---
11 | | 15
---+----+---
12 | 17 |
```

#### Step 1: Use the magic sum to find missing numbers.
The magic sum is 39.

#### Row 2:
The second row is \(11 + x + 15 = 39\).
\[ 11 + x + 15 = 39 \]
\[ x = 13 \]

#### Column 1:
The first column is \(a + 11 + 12 = 39\).
\[ a + 23 = 39 \]
\[ a = 16 \]

#### Column 3:
The third column is \(b + 15 + c = 39\).

#### Diagonal (top-left to bottom-right):
The diagonal is \(16 + 13 + c = 39\).
\[ 16 + 13 + c = 39 \]
\[ 29 + c = 39 \]
\[ c = 10 \]

#### Row 3:
The third row is \(12 + 17 + 10 = 39\). This is already correct.

#### Column 3:
The third column is \(b + 15 + 10 = 39\).
\[ b + 25 = 39 \]
\[ b = 14 \]

#### Fill in the grid:
```
16 | 9 | 14
---+----+----
11 | 13 | 15
---+----+----
12 | 17 | 10
```

---

Puzzle 3: Magic Sum = 45

The grid is:
```
18 | | 16
---+----+----
| 15 |
---+----+----
| 19 | 12
```

#### Step 1: Use the magic sum to find missing numbers.
The magic sum is 45.

#### Row 1:
The first row is \(18 + x + 16 = 45\).
\[ 18 + x + 16 = 45 \]
\[ x = 11 \]

#### Column 2:
The second column is \(11 + 15 + 19 = 45\). This is already correct.

#### Row 3:
The third row is \(y + 19 + 12 = 45\).
\[ y + 31 = 45 \]
\[ y = 14 \]

#### Column 1:
The first column is \(18 + z + 14 = 45\).
\[ 32 + z = 45 \]
\[ z = 13 \]

#### Column 3:
The third column is \(16 + w + 12 = 45\).
\[ 28 + w = 45 \]
\[ w = 17 \]

#### Fill in the grid:
```
18 | 11 | 16
---+----+----
13 | 15 | 17
---+----+----
14 | 19 | 12
```

---

Puzzle 4: Magic Sum = 51

The grid is:
```
20 | | 18
---+----+----
15 | |
---+----+----
| 21 | 14
```

#### Step 1: Use the magic sum to find missing numbers.
The magic sum is 51.

#### Row 1:
The first row is \(20 + x + 18 = 51\).
\[ 20 + x + 18 = 51 \]
\[ x = 13 \]

#### Column 3:
The third column is \(18 + y + 14 = 51\).
\[ 18 + y + 14 = 51 \]
\[ y = 19 \]

#### Row 3:
The third row is \(z + 21 + 14 = 51\).
\[ z + 35 = 51 \]
\[ z = 16 \]

#### Column 1:
The first column is \(20 + 15 + 16 = 51\). This is already correct.

#### Column 2:
The second column is \(13 + w + 21 = 51\).
\[ 34 + w = 51 \]
\[ w = 17 \]

#### Fill in the grid:
```
20 | 13 | 18
---+----+----
15 | 17 | 19
---+----+----
16 | 21 | 14
```

---

Final Answers:

1. Puzzle 1:
```
23 | 16 | 21
---+----+----
18 | 20 | 22
---+----+----
19 | 24 | 17
```
\boxed{
\begin{array}{|c|c|c|}
\hline
23 & 16 & 21 \\
\hline
18 & 20 & 22 \\
\hline
19 & 24 & 17 \\
\hline
\end{array}
}

2. Puzzle 2:
```
16 | 9 | 14
---+----+----
11 | 13 | 15
---+----+----
12 | 17 | 10
```
\boxed{
\begin{array}{|c|c|c|}
\hline
16 & 9 & 14 \\
\hline
11 & 13 & 15 \\
\hline
12 & 17 & 10 \\
\hline
\end{array}
}

3. Puzzle 3:
```
18 | 11 | 16
---+----+----
13 | 15 | 17
---+----+----
14 | 19 | 12
```
\boxed{
\begin{array}{|c|c|c|}
\hline
18 & 11 & 16 \\
\hline
13 & 15 & 17 \\
\hline
14 & 19 & 12 \\
\hline
\end{array}
}

4. Puzzle 4:
```
20 | 13 | 18
---+----+----
15 | 17 | 19
---+----+----
16 | 21 | 14
```
\boxed{
\begin{array}{|c|c|c|}
\hline
20 & 13 & 18 \\
\hline
15 & 17 & 19 \\
\hline
16 & 21 & 14 \\
\hline
\end{array}
}