Magic Squares - Free Printable

Let's solve both magic square puzzles step by step.

---

Understanding the Problem

A magic square is a grid where the sum of the numbers in each row, each column, and both main diagonals equals the same value — called the magic number.

We are given two 3×3 grids with some numbers filled in and a magic number for each. Our goal is to fill in the missing numbers so that all rows, columns, and diagonals add up to the given magic number.

---

## 🔹 Part A: Magic Number = 9.0

Grid:
```
[ ][ ][3.6]
[1.8][3.0][ ]
[ ][ ][ ]
```

Let’s label the cells as follows:

```
a b 3.6
1.8 3.0 c
d e f
```

We know:
- The sum of each row, column, and diagonal must be 9.0

---

Step 1: Use the second row

Second row:
1.8 + 3.0 + c = 9.0
→ 4.8 + c = 9.0
→ c = 9.0 - 4.8 = 4.2

So now we have:
```
a b 3.6
1.8 3.0 4.2
d e f
```

---

Step 2: Use the third column

Third column:
3.6 + 4.2 + f = 9.0
→ 7.8 + f = 9.0
→ f = 1.2

Now:
```
a b 3.6
1.8 3.0 4.2
d e 1.2
```

---

Step 3: Use the first column

First column:
a + 1.8 + d = 9.0 → a + d = 7.2 ...(1)

---

Step 4: Use the second column

Second column:
b + 3.0 + e = 9.0 → b + e = 6.0 ...(2)

---

Step 5: Use the first row

First row:
a + b + 3.6 = 9.0 → a + b = 5.4 ...(3)

---

Step 6: Use the third row

Third row:
d + e + 1.2 = 9.0 → d + e = 7.8 ...(4)

---

Now we have equations:

(1) a + d = 7.2
(2) b + e = 6.0
(3) a + b = 5.4
(4) d + e = 7.8

Let’s solve this system.

From (3): a = 5.4 - b
Plug into (1):
(5.4 - b) + d = 7.2 → d = 7.2 - 5.4 + b = 1.8 + b

From (2): e = 6.0 - b
Now plug into (4):
d + e = 7.8
→ (1.8 + b) + (6.0 - b) = 7.8
→ 7.8 = 7.8 ✔

It checks out! So the system is consistent, but we need one more constraint.

Let’s use the main diagonal from top-left to bottom-right:
a + 3.0 + 1.2 = 9.0 → a + 4.2 = 9.0 → a = 4.8

Now go back:

From (3): a + b = 5.4 → 4.8 + b = 5.4 → b = 0.6

From (1): a + d = 7.2 → 4.8 + d = 7.2 → d = 2.4

From (2): b + e = 6.0 → 0.6 + e = 6.0 → e = 5.4

Now we have all values:

```
a=4.8 b=0.6 3.6
1.8 3.0 4.2
d=2.4 e=5.4 f=1.2
```

Let’s verify:

- Rows:
- Row 1: 4.8 + 0.6 + 3.6 = 9.0 ✔
- Row 2: 1.8 + 3.0 + 4.2 = 9.0 ✔
- Row 3: 2.4 + 5.4 + 1.2 = 9.0 ✔

- Columns:
- Col 1: 4.8 + 1.8 + 2.4 = 9.0 ✔
- Col 2: 0.6 + 3.0 + 5.4 = 9.0 ✔
- Col 3: 3.6 + 4.2 + 1.2 = 9.0 ✔

- Diagonals:
- Top-left to bottom-right: 4.8 + 3.0 + 1.2 = 9.0 ✔
- Top-right to bottom-left: 3.6 + 3.0 + 2.4 = 9.0 ✔

✔ All verified!

---

## ✔ Answer for Part A:

```
4.8 0.6 3.6
1.8 3.0 4.2
2.4 5.4 1.2
```

---

## 🔹 Part B: Magic Number = 12.6

Grid:
```
[ ][ ][4.8]
[ ][ ][ ]
[3.6][6.6][ ]
```

Label the grid:

```
a b 4.8
c d e
3.6 6.6 f
```

Magic sum = 12.6

---

Step 1: Third row

3.6 + 6.6 + f = 12.6 → 10.2 + f = 12.6 → f = 2.4

Now:
```
a b 4.8
c d e
3.6 6.6 2.4
```

---

Step 2: Third column

4.8 + e + 2.4 = 12.6 → 7.2 + e = 12.6 → e = 5.4

Now:
```
a b 4.8
c d 5.4
3.6 6.6 2.4
```

---

Step 3: Second column

b + d + 6.6 = 12.6 → b + d = 6.0 ...(1)

---

Step 4: First column

a + c + 3.6 = 12.6 → a + c = 9.0 ...(2)

---

Step 5: First row

a + b + 4.8 = 12.6 → a + b = 7.8 ...(3)

---

Step 6: Second row

c + d + 5.4 = 12.6 → c + d = 7.2 ...(4)

Now solve the system:

From (3): a = 7.8 - b
From (2): a + c = 9.0 → (7.8 - b) + c = 9.0 → c = 1.2 + b

From (1): d = 6.0 - b
From (4): c + d = 7.2
→ (1.2 + b) + (6.0 - b) = 7.2 → 7.2 = 7.2 ✔

Again, consistent. Need another constraint.

Use main diagonal (top-left to bottom-right):

a + d + 2.4 = 12.6 → a + d = 10.2

But a = 7.8 - b, d = 6.0 - b
So:
(7.8 - b) + (6.0 - b) = 10.2
13.8 - 2b = 10.2
→ 2b = 3.6 → b = 1.8

Now back-substitute:

- b = 1.8
- a = 7.8 - 1.8 = 6.0
- d = 6.0 - 1.8 = 4.2
- c = 1.2 + 1.8 = 3.0
- e = 5.4 (already known)
- f = 2.4 (known)

Now fill in:

```
6.0 1.8 4.8
3.0 4.2 5.4
3.6 6.6 2.4
```

---

Verify:

Rows:
- Row 1: 6.0 + 1.8 + 4.8 = 12.6 ✔
- Row 2: 3.0 + 4.2 + 5.4 = 12.6 ✔
- Row 3: 3.6 + 6.6 + 2.4 = 12.6 ✔

Columns:
- Col 1: 6.0 + 3.0 + 3.6 = 12.6 ✔
- Col 2: 1.8 + 4.2 + 6.6 = 12.6 ✔
- Col 3: 4.8 + 5.4 + 2.4 = 12.6 ✔

Diagonals:
- Top-left to bottom-right: 6.0 + 4.2 + 2.4 = 12.6 ✔
- Top-right to bottom-left: 4.8 + 4.2 + 3.6 = 12.6 ✔

✔ All good!

---

## ✔ Answer for Part B:

```
6.0 1.8 4.8
3.0 4.2 5.4
3.6 6.6 2.4
```

---

## 🟩 Final Answers:

A.

```
4.8 0.6 3.6
1.8 3.0 4.2
2.4 5.4 1.2
```

B.

```
6.0 1.8 4.8
3.0 4.2 5.4
3.6 6.6 2.4
```

Both satisfy the magic number condition. ✔