Magic Squares (E) Puzzle | FREE Teaching Resources - Free Printable

Let’s solve each magic square one by one.

Remember the rule:
Each row, column, and diagonal adds up to three times the middle number.

That means if we can find the middle number, we multiply it by 3 to get the “magic sum” — the total every row/column/diagonal must add to.

---

Magic Square 1 (Top Left)

Given:
```
1 ? 5
? ? ?
3 ? 7
```

We don’t know the middle yet. But notice: the corners are given. In a 3x3 magic square, the center is always the average of opposite corners.

Check diagonals:

- Top-left to bottom-right: 1 + ? + 7 → should equal 3 × center
- Top-right to bottom-left: 5 + ? + 3 → should also equal 3 × center

But we can use another trick: in any 3x3 magic square, the center = (sum of all four corners) ÷ 4? Not exactly. Better way:

Actually, let’s use this fact: The center number is the average of any two opposite numbers that pass through the center.

Look at top row: 1 and 5 → their average is 3 → maybe center is 3? Let’s test.

If center = 3, then magic sum = 3 × 3 = 9.

Check bottom row: 3 + ? + 7 = 9 → so middle of bottom row = 9 - 3 - 7 = -1? That seems odd, but possible.

Wait — let’s try using the diagonal.

Diagonal from top-left to bottom-right: 1 + center + 7 = 3 × center
→ 1 + 7 + center = 3×center
→ 8 + center = 3×center
→ 8 = 2×center
→ center = 4

Ah! So center = 4 → magic sum = 3 × 4 = 12.

Now fill in:

Top row: 1 + ? + 5 = 12 → ? = 6
Bottom row: 3 + ? + 7 = 12 → ? = 2
Middle row left: ? + 4 + ? = 12 → we need more info.

Left column: 1 + ? + 3 = 12 → ? = 8
Right column: 5 + ? + 7 = 12 → ? = 0
Middle row: 8 + 4 + 0 = 12 ✔️

Check other diagonal: 5 + 4 + 3 = 12 ✔️

So completed square:

```
1 6 5
8 4 0
3 2 7
```

✔ All rows, columns, diagonals = 12.

---

Magic Square 2 (Top Right)

Given:
```
9 ? 3
? ? ?
7 ? 1
```

Use same method.

Diagonal: 9 + center + 1 = 3 × center
→ 10 + center = 3×center
→ 10 = 2×center
→ center = 5 → magic sum = 15

Top row: 9 + ? + 3 = 15 → ? = 3
Bottom row: 7 + ? + 1 = 15 → ? = 7
Left column: 9 + ? + 7 = 15 → ? = -1
Right column: 3 + ? + 1 = 15 → ? = 11
Middle row: -1 + 5 + 11 = 15 ✔️

Other diagonal: 3 + 5 + 7 = 15 ✔️

Completed:

```
9 3 3
-1 5 11
7 7 1
```

Wait — top row has two 3s? That’s okay, magic squares can repeat unless specified otherwise. But let’s double-check.

Actually, wait — top row: 9 + 3 + 3 = 15 ✔️
But right column: 3 + 11 + 1 = 15 ✔️
Left column: 9 + (-1) + 7 = 15 ✔️
Middle row: -1 + 5 + 11 = 15 ✔️
Diagonals: 9+5+1=15, 3+5+7=15 ✔️

It works. Even with repeats.

But let me check if there’s a better way — maybe I made an error.

Alternative: Use the fact that in magic square, center = average of top and bottom middle? We don’t have them.

Our calculation is correct based on diagonal.

So accept it.

---

Magic Square 3 (Middle Left)

Given:
```
? 5 ?
11 ? 19
? 25 ?
```

Notice: middle column: 5, ?, 25 → if this is arithmetic sequence, center might be average: (5+25)/2 = 15 → let’s assume center = 15 → magic sum = 45

Check middle row: 11 + 15 + 19 = 45 ✔️

Now top row: ? + 5 + ? = 45 → sum of ends = 40
Bottom row: ? + 25 + ? = 45 → sum of ends = 20

Left column: ? + 11 + ? = 45 → sum of top and bottom left = 34
Right column: ? + 19 + ? = 45 → sum of top and bottom right = 26

Also, diagonals:

Top-left to bottom-right: ? + 15 + ? = 45 → sum = 30
Top-right to bottom-left: ? + 15 + ? = 45 → sum = 30

Let’s denote:

Let A = top-left, B = top-right
C = bottom-left, D = bottom-right

From top row: A + B = 40
From bottom row: C + D = 20
From left col: A + C = 34
From right col: B + D = 26
From diag1: A + D = 30
From diag2: B + C = 30

Now solve:

From A + B = 40 and A + C = 34 → subtract: B - C = 6 → B = C + 6

From B + C = 30 → (C+6) + C = 30 → 2C = 24 → C = 12 → B = 18

Then A = 40 - B = 22
D = 20 - C = 8

Check diag1: A + D = 22 + 8 = 30 ✔️
diag2: B + C = 18 + 12 = 30 ✔️

All good.

Completed:

```
22 5 18
11 15 19
12 25 8
```

Check rows:
22+5+18=45, 11+15+19=45, 12+25+8=45
Cols: 22+11+12=45, 5+15+25=45, 18+19+8=45
Diags: 22+15+8=45, 18+15+12=45 ✔️

Perfect.

---

Magic Square 4 (Middle Right)

Given:
```
? 28 ?
11 ? 21
? 4 ?
```

Again, look at middle column: 28, ?, 4 → if arithmetic, center = (28+4)/2 = 16 → magic sum = 48

Check middle row: 11 + 16 + 21 = 48 ✔️

Now top row: ? + 28 + ? = 48 → sum of ends = 20
Bottom row: ? + 4 + ? = 48 → sum of ends = 44

Left column: ? + 11 + ? = 48 → sum of top and bottom left = 37
Right column: ? + 21 + ? = 48 → sum of top and bottom right = 27

Diagonals:

Top-left to bottom-right: ? + 16 + ? = 48 → sum = 32
Top-right to bottom-left: ? + 16 + ? = 48 → sum = 32

Let A = top-left, B = top-right
C = bottom-left, D = bottom-right

A + B = 20
C + D = 44
A + C = 37
B + D = 27
A + D = 32
B + C = 32

From A + B = 20 and A + C = 37 → subtract: B - C = -17 → B = C - 17

From B + C = 32 → (C - 17) + C = 32 → 2C = 49 → C = 24.5 → B = 7.5

Then A = 20 - B = 12.5
D = 44 - C = 19.5

Check diag1: A + D = 12.5 + 19.5 = 32 ✔️
diag2: B + C = 7.5 + 24.5 = 32 ✔️

All good.

Completed:

```
12.5 28 7.5
11 16 21
24.5 4 19.5
```

Check sums:

Rows:
12.5+28+7.5=48, 11+16+21=48, 24.5+4+19.5=48
Cols: 12.5+11+24.5=48, 28+16+4=48, 7.5+21+19.5=48
Diags: 12.5+16+19.5=48, 7.5+16+24.5=48 ✔️

Good.

---

Magic Square 5 (Bottom Left)

Given:
```
-9 ? -8
? ? ?
-6 ? -5
```

Use diagonal: -9 + center + (-5) = 3 × center
→ -14 + center = 3×center
→ -14 = 2×center
→ center = -7 → magic sum = 3 × (-7) = -21

Top row: -9 + ? + (-8) = -21 → ? = -21 + 17 = -4
Bottom row: -6 + ? + (-5) = -21 → ? = -21 + 11 = -10
Left column: -9 + ? + (-6) = -21 → ? = -21 + 15 = -6
Right column: -8 + ? + (-5) = -21 → ? = -21 + 13 = -8
Middle row: -6 + (-7) + (-8) = -21 ✔️

Other diagonal: -8 + (-7) + (-6) = -21 ✔️

Completed:

```
-9 -4 -8
-6 -7 -8
-6 -10 -5
```

Wait — right column: -8 + (-8) + (-5) = -21? -8-8-5=-21 ✔️
But middle row right is -8, and top right is -8 — again, repeats allowed.

Check all:

Rows:
-9-4-8=-21, -6-7-8=-21, -6-10-5=-21
Cols: -9-6-6=-21, -4-7-10=-21, -8-8-5=-21
Diags: -9-7-5=-21, -8-7-6=-21 ✔️

Good.

---

Magic Square 6 (Bottom Right)

Given:
```
-2.8 ? -0.3
? ? ?
-2.1 ? 0.4
```

Diagonal: -2.8 + center + 0.4 = 3 × center
→ -2.4 + center = 3×center
→ -2.4 = 2×center
→ center = -1.2 → magic sum = 3 × (-1.2) = -3.6

Top row: -2.8 + ? + (-0.3) = -3.6 → ? = -3.6 + 3.1 = -0.5
Bottom row: -2.1 + ? + 0.4 = -3.6 → ? = -3.6 + 1.7 = -1.9
Left column: -2.8 + ? + (-2.1) = -3.6 → ? = -3.6 + 4.9 = 1.3
Right column: -0.3 + ? + 0.4 = -3.6 → ? = -3.6 - 0.1 = -3.7
Middle row: 1.3 + (-1.2) + (-3.7) = -3.6 ✔️

Other diagonal: -0.3 + (-1.2) + (-2.1) = -3.6 ✔️

Completed:

```
-2.8 -0.5 -0.3
1.3 -1.2 -3.7
-2.1 -1.9 0.4
```

Check all:

Rows:
-2.8-0.5-0.3=-3.6, 1.3-1.2-3.7=-3.6, -2.1-1.9+0.4=-3.6
Cols: -2.8+1.3-2.1=-3.6, -0.5-1.2-1.9=-3.6, -0.3-3.7+0.4=-3.6
Diags: -2.8-1.2+0.4=-3.6, -0.3-1.2-2.1=-3.6 ✔️

Perfect.

---

## Final Answers (Completed Magic Squares):

Square 1:
```
1 6 5
8 4 0
3 2 7
```

Square 2:
```
9 3 3
-1 5 11
7 7 1
```

Square 3:
```
22 5 18
11 15 19
12 25 8
```

Square 4:
```
12.5 28 7.5
11 16 21
24.5 4 19.5
```

Square 5:
```
-9 -4 -8
-6 -7 -8
-6 -10 -5
```

Square 6:
```
-2.8 -0.5 -0.3
1.3 -1.2 -3.7
-2.1 -1.9 0.4
```

Final Answer:
The six completed magic squares are as shown above. Each satisfies the condition that every row, column, and diagonal sums to three times the center number.