Magic Squares worksheet with four puzzles to solve, each with a target sum for rows, columns, and diagonals.

Let's solve each magic square puzzle step by step. In a magic square, each row, column, and both main diagonals must add up to the same given sum.

We'll go through each of the four grids one at a time.

---

🔷 Grid 1: Sum = 34

```
| | | | 1 |
|---|---|---|----|
| | 11| | 14 |
|---|---|---|----|
| 3 | 10| | |
|---|---|---|----|
| | 5| 9| 4 |
```

We need to fill in the blanks so that every row, column, and diagonal adds to 34.

Let’s label the grid:

```
a b c 1
d 11 e 14
3 10 f g
h 5 9 4
```

We’ll use logic and equations.

---

#### Step 1: Use Row 4

Row 4: h + 5 + 9 + 4 = 34 → h + 18 = 34 → h = 16

So bottom-left is 16

Now update:
```
a b c 1
d 11 e 14
3 10 f g
16 5 9 4
```

---

#### Step 2: Column 1

Column 1: a + d + 3 + 16 = 34 → a + d = 34 - 19 = 15

So: a + d = 15 → (Eq 1)

---

#### Step 3: Column 4

Column 4: 1 + 14 + g + 4 = 34 → 19 + g = 34 → g = 15

So right-bottom is 15

Update:
```
a b c 1
d 11 e 14
3 10 f 15
16 5 9 4
```

---

#### Step 4: Row 3

Row 3: 3 + 10 + f + 15 = 34 → 28 + f = 34 → f = 6

Update:
```
a b c 1
d 11 e 14
3 10 6 15
16 5 9 4
```

---

#### Step 5: Column 3

Column 3: c + e + 6 + 9 = 34 → c + e = 19 → (Eq 2)

---

#### Step 6: Row 2

Row 2: d + 11 + e + 14 = 34 → d + e = 9 → (Eq 3)

---

#### Step 7: From Eq 1: a + d = 15
From Eq 3: d + e = 9 → e = 9 - d

Plug into Eq 2: c + e = 19 → c + (9 - d) = 19 → c = 10 + d

Now let’s look at Diagonal: top-left to bottom-right: a + 11 + 6 + 4 = 34?
Wait: a + 11 + 6 + 4 = a + 21 = 34 → a = 13

Now from Eq 1: a + d = 15 → 13 + d = 15 → d = 2

Then from Eq 3: d + e = 9 → 2 + e = 9 → e = 7

From earlier: c = 10 + d = 10 + 2 = 12

Now check Row 1: a + b + c + 1 = 34 → 13 + b + 12 + 1 = 34 → b = 34 - 26 = 8

Now we have all values.

Check Row 1: 13 + 8 + 12 + 1 = 34 ✔
Row 2: 2 + 11 + 7 + 14 = 34 ✔
Row 3: 3 + 10 + 6 + 15 = 34 ✔
Row 4: 16 + 5 + 9 + 4 = 34 ✔

Columns:
- Col 1: 13+2+3+16 = 34 ✔
- Col 2: 8+11+10+5 = 34 ✔
- Col 3: 12+7+6+9 = 34 ✔
- Col 4: 1+14+15+4 = 34 ✔

Diagonals:
- Top-left to bottom-right: 13 + 11 + 6 + 4 = 34 ✔
- Top-right to bottom-left: 1 + 11 + 10 + 16 = 38 ✘ → Wait! That’s not 34!

Wait — that’s a problem.

Wait: top-right to bottom-left:
Position: (1,4)=1, (2,3)=7, (3,2)=10, (4,1)=16 → 1 + 7 + 10 + 16 = 34 ✔

Yes! 1+7=8, +10=18, +16=34 ✔

All good!

✔ Final Grid 1:

```
13 8 12 1
2 11 7 14
3 10 6 15
16 5 9 4
```

---

🔷 Grid 2: Sum = 102

```
| 48 | | | |
|----|---|---|---|
| | 30| 33| 24 |
|----|---|---|---|
| | | | 36 |
|----|---|---|---|
| | 45| 42| 3 |
```

Label:

```
48 a b c
d 30 33 24
e f g 36
h 45 42 3
```

Sum = 102

---

#### Step 1: Row 4

h + 45 + 42 + 3 = 102 → h + 90 = 102 → h = 12

Update:
```
48 a b c
d 30 33 24
e f g 36
12 45 42 3
```

---

#### Step 2: Column 1

48 + d + e + 12 = 102 → d + e = 102 - 60 = 42 → (Eq 1)

---

#### Step 3: Column 4

c + 24 + 36 + 3 = 102 → c + 63 = 102 → c = 39

Update:
```
48 a b 39
d 30 33 24
e f g 36
12 45 42 3
```

---

#### Step 4: Row 1

48 + a + b + 39 = 102 → a + b = 102 - 87 = 15 → (Eq 2)

---

#### Step 5: Row 2

d + 30 + 33 + 24 = 102 → d + 87 = 102 → d = 15

From Eq 1: d + e = 42 → 15 + e = 42 → e = 27

Now update:
```
48 a b 39
15 30 33 24
27 f g 36
12 45 42 3
```

---

#### Step 6: Column 2

a + 30 + f + 45 = 102 → a + f = 102 - 75 = 27 → (Eq 3)

---

#### Step 7: Column 3

b + 33 + g + 42 = 102 → b + g = 102 - 75 = 27 → (Eq 4)

---

From Eq 2: a + b = 15
From Eq 3: a + f = 27 → f = 27 - a
From Eq 4: b + g = 27 → g = 27 - b

Now consider diagonal: top-left to bottom-right
48 + 30 + g + 3 = 102 → 81 + g = 102 → g = 21

Then from Eq 4: b + 21 = 27 → b = 6

From Eq 2: a + 6 = 15 → a = 9

From Eq 3: 9 + f = 27 → f = 18

Now we have all values.

Check Row 1: 48 + 9 + 6 + 39 = 102 ✔
Row 2: 15 + 30 + 33 + 24 = 102 ✔
Row 3: 27 + 18 + 21 + 36 = 102 ✔
Row 4: 12 + 45 + 42 + 3 = 102 ✔

Columns:
- Col 1: 48+15+27+12 = 102 ✔
- Col 2: 9+30+18+45 = 102 ✔
- Col 3: 6+33+21+42 = 102 ✔
- Col 4: 39+24+36+3 = 102 ✔

Diagonals:
- 48 + 30 + 21 + 3 = 102 ✔
- 39 + 33 + 18 + 12 = 102 ✔

Perfect!

✔ Final Grid 2:

```
48 9 6 39
15 30 33 24
27 18 21 36
12 45 42 3
```

---

🔷 Grid 3: Sum = 136

```
| | | 48| 4 |
|---|---|---|---|
| 8 | | | 56 |
|---|---|---|---|
| | 40| | |
|---|---|---|---|
| | 20| 36| 16 |
```

Label:

```
a b 48 4
8 c d 56
e 40 f g
h 20 36 16
```

Sum = 136

---

#### Step 1: Row 4

h + 20 + 36 + 16 = 136 → h + 72 = 136 → h = 64

Update:
```
a b 48 4
8 c d 56
e 40 f g
64 20 36 16
```

---

#### Step 2: Column 1

a + 8 + e + 64 = 136 → a + e = 136 - 72 = 64 → (Eq 1)

---

#### Step 3: Column 4

4 + 56 + g + 16 = 136 → g + 76 = 136 → g = 60

Update:
```
a b 48 4
8 c d 56
e 40 f 60
64 20 36 16
```

---

#### Step 4: Row 3

e + 40 + f + 60 = 136 → e + f = 36 → (Eq 2)

---

#### Step 5: Column 3

48 + d + f + 36 = 136 → d + f = 52 → (Eq 3)

---

#### Step 6: Row 2

8 + c + d + 56 = 136 → c + d = 72 → (Eq 4)

---

#### Step 7: Row 1

a + b + 48 + 4 = 136 → a + b = 84 → (Eq 5)

---

#### Step 8: Column 2

b + c + 40 + 20 = 136 → b + c = 76 → (Eq 6)

Now solve system:

From Eq 1: a + e = 64 → a = 64 - e
From Eq 2: e + f = 36 → f = 36 - e
From Eq 3: d + f = 52 → d = 52 - f = 52 - (36 - e) = 16 + e
From Eq 4: c + d = 72 → c = 72 - d = 72 - (16 + e) = 56 - e
From Eq 6: b + c = 76 → b = 76 - c = 76 - (56 - e) = 20 + e
From Eq 5: a + b = 84 → (64 - e) + (20 + e) = 84 → 84 = 84 ✔

Always true — consistent.

Now pick a value for e? We need another constraint.

Try diagonal: top-left to bottom-right:
a + c + f + 16 = 136
Substitute:
a = 64 - e
c = 56 - e
f = 36 - e
So: (64 - e) + (56 - e) + (36 - e) + 16 = 136
→ 172 - 3e = 136 → 3e = 36 → e = 12

Now back-substitute:

- e = 12
- a = 64 - 12 = 52
- f = 36 - 12 = 24
- d = 16 + 12 = 28
- c = 56 - 12 = 44
- b = 20 + 12 = 32

Now check:

Row 1: 52 + 32 + 48 + 4 = 136 ✔
Row 2: 8 + 44 + 28 + 56 = 136 ✔
Row 3: 12 + 40 + 24 + 60 = 136 ✔
Row 4: 64 + 20 + 36 + 16 = 136 ✔

Columns:
- Col 1: 52+8+12+64 = 136 ✔
- Col 2: 32+44+40+20 = 136 ✔
- Col 3: 48+28+24+36 = 136 ✔
- Col 4: 4+56+60+16 = 136 ✔

Diagonals:
- Main: 52 + 44 + 24 + 16 = 136 ✔
- Anti-diag: 4 + 28 + 40 + 64 = 136 ✔

Perfect!

✔ Final Grid 3:

```
52 32 48 4
8 44 28 56
12 40 24 60
64 20 36 16
```

---

🔷 Grid 4: Sum = 102

```
| | 42| | |
|---|---|---|---|
| 36| 21| 18| |
|---|---|---|---|
| 24| 33| | |
|---|---|---|---|
| 39| | | 48 |
```

Label:

```
a 42 b c
36 21 18 d
24 33 e f
39 g h 48
```

Sum = 102

---

#### Step 1: Row 1

a + 42 + b + c = 102 → a + b + c = 60 → (Eq 1)

#### Step 2: Row 2

36 + 21 + 18 + d = 102 → 75 + d = 102 → d = 27

Update:
```
a 42 b c
36 21 18 27
24 33 e f
39 g h 48
```

#### Step 3: Row 3

24 + 33 + e + f = 102 → e + f = 45 → (Eq 2)

#### Step 4: Row 4

39 + g + h + 48 = 102 → g + h = 15 → (Eq 3)

#### Step 5: Column 1

a + 36 + 24 + 39 = 102 → a + 99 = 102 → a = 3

From Eq 1: 3 + b + c = 60 → b + c = 57 → (Eq 4)

#### Step 6: Column 2

42 + 21 + 33 + g = 102 → 96 + g = 102 → g = 6

From Eq 3: g + h = 15 → 6 + h = 15 → h = 9

#### Step 7: Column 3

b + 18 + e + h = 102 → b + e + 9 = 102 → b + e = 93 → (Eq 5)

#### Step 8: Column 4

c + 27 + f + 48 = 102 → c + f = 27 → (Eq 6)

Now from Eq 4: b + c = 57
From Eq 5: b + e = 93 → e = 93 - b
From Eq 2: e + f = 45 → f = 45 - e = 45 - (93 - b) = b - 48
From Eq 6: c + f = 27 → c + (b - 48) = 27 → c + b = 75

But from Eq 4: b + c = 57 → contradiction! 57 vs 75?

Wait — conflict.

Wait: from Eq 6: c + f = 27
And f = b - 48
So c + (b - 48) = 27 → c + b = 75 → but earlier we had b + c = 57 → impossible.

Contradiction!

Wait — did I make an error?

Let’s double-check.

From earlier:

- a = 3
- g = 6
- h = 9

Now check column 3:

b + 18 + e + h = 102 → b + 18 + e + 9 = 102 → b + e = 75 → (Eq 5)

Earlier I said 93 — that was wrong.

I wrote: "b + e = 93" — no:

b + 18 + e + 9 = 102 → b + e = 102 - 27 = 75 → (Eq 5)

Corrected.

From Eq 4: b + c = 57
From Eq 6: c + f = 27
From Eq 2: e + f = 45 → f = 45 - e
From Eq 5: e = 75 - b

So f = 45 - (75 - b) = b - 30

Now plug into Eq 6: c + f = 27 → c + (b - 30) = 27 → c + b = 57 → matches Eq 4 ✔

So consistent.

Now we can solve.

From Eq 4: c = 57 - b
From Eq 5: e = 75 - b
From Eq 2: f = 45 - e = 45 - (75 - b) = b - 30

Now check diagonal: top-left to bottom-right

a + 21 + e + 48 = 102 → 3 + 21 + e + 48 = 102 → 72 + e = 102 → e = 30

Now from Eq 5: e = 75 - b → 30 = 75 - b → b = 45

Then:
- c = 57 - 45 = 12
- f = b - 30 = 45 - 30 = 15
- e = 30

Now check Row 1: a + 42 + b + c = 3 + 42 + 45 + 12 = 102 ✔
Row 2: 36 + 21 + 18 + 27 = 102 ✔
Row 3: 24 + 33 + 30 + 15 = 102 ✔
Row 4: 39 + 6 + 9 + 48 = 102 ✔

Columns:
- Col 1: 3+36+24+39 = 102 ✔
- Col 2: 42+21+33+6 = 102 ✔
- Col 3: 45+18+30+9 = 102 ✔
- Col 4: 12+27+15+48 = 102 ✔

Diagonals:
- Main: 3 + 21 + 30 + 48 = 102 ✔
- Anti-diag: 12 + 18 + 33 + 39 = 102 ✔

Perfect!

✔ Final Grid 4:

```
3 42 45 12
36 21 18 27
24 33 30 15
39 6 9 48
```

---

✔ Final Answers:

#### Grid 1 (Sum 34):
```
13 8 12 1
2 11 7 14
3 10 6 15
16 5 9 4
```

#### Grid 2 (Sum 102):
```
48 9 6 39
15 30 33 24
27 18 21 36
12 45 42 3
```

#### Grid 3 (Sum 136):
```
52 32 48 4
8 44 28 56
12 40 24 60
64 20 36 16
```

#### Grid 4 (Sum 102):
```
3 42 45 12
36 21 18 27
24 33 30 15
39 6 9 48
```

All solved correctly! 🎉