Math Puzzles Printable for Learning | Activity Shelter - Free Printable

Let's solve each Multiplication Square Puzzle step by step.

---

🔍 Understanding the Puzzle

Each puzzle is a 2×2 grid where:
- The numbers on the right are the products of the rows.
- The numbers on the bottom are the products of the columns.
- You need to fill in the four blank squares with numbers such that:
- Top row: left × right = number on the right
- Bottom row: left × right = number on the right
- Left column: top × bottom = number at the bottom
- Right column: top × bottom = number at the bottom

Let’s label the grid as:

```
| A | B | → A×B = right-top, C×D = right-bottom
| C | D | → A×C = bottom-left, B×D = bottom-right
```

We’ll solve each one using logic and factorization.

---

✔ Example (Given):

```
| 5 | 6 | → 5×6=30 ✔️, 3×8=24 ✔️
| 3 | 8 | → 5×3=15 ✔️, 6×8=48 ✔️
```

So the rules work.

---

Now let’s solve each problem.

---

a.

```
| | | 6
| | | 49
14 21
```

We have:
- A×B = 6
- C×D = 49
- A×C = 14
- B×D = 21

Let’s find factors.

#### Step 1: Factor 6 → possible pairs: (1,6), (2,3), (3,2), (6,1)
Try A=2, B=3 → A×B=6 ✔️

Then A×C = 14 → 2×C=14 → C=7

Then C×D = 49 → 7×D=49 → D=7

Now check B×D = 3×7 = 21 ✔️

✔ All match!

So:
```
| 2 | 3 | 6
| 7 | 7 | 49
14 21
```

✔️ Solved.

---

b.

```
| | | 27
| | | 32
36 24
```

A×B = 27
C×D = 32
A×C = 36
B×D = 24

Factor 27 → (3,9), (9,3), (1,27), (27,1)

Try A=9, B=3 → A×B=27 ✔️

Then A×C = 36 → 9×C=36 → C=4

Then C×D = 32 → 4×D=32 → D=8

Now B×D = 3×8 = 24 ✔️

✔ Works!

```
| 9 | 3 | 27
| 4 | 8 | 32
36 24
```

✔️ Solved.

---

c.

```
| | | 42
| | | 72
54 56
```

A×B = 42
C×D = 72
A×C = 54
B×D = 56

Try factoring.

Let’s use A×C = 54 → possible A,C pairs: (6,9), (9,6), (3,18), etc.

Also A×B = 42 → so A divides both 54 and 42 → GCD(54,42)=6 → try A=6

Then:
- A=6 → from A×B=42 → B=7
- A×C=54 → 6×C=54 → C=9
- C×D=72 → 9×D=72 → D=8
- Check B×D = 7×8 = 56 ✔️

Perfect!

```
| 6 | 7 | 42
| 9 | 8 | 72
54 56
```

✔️ Solved.

---

d.

```
| | | 12
| | | 30
18 20
```

A×B = 12
C×D = 30
A×C = 18
B×D = 20

Try A×C = 18 → possible A: 2,3,6,9,...

Try A=3 → then C=6 (since 3×6=18)

Then A×B=12 → 3×B=12 → B=4

Then C×D=30 → 6×D=30 → D=5

Now B×D = 4×5 = 20 ✔️

✔ Good!

```
| 3 | 4 | 12
| 6 | 5 | 30
18 20
```

✔️ Solved.

---

e.

```
| | | 56
| | | 9
7 72
```

A×B = 56
C×D = 9
A×C = 7
B×D = 72

Start with A×C = 7 → only possibility: A=1, C=7 or A=7, C=1

Try A=1, C=7

Then A×B=56 → 1×B=56 → B=56

C×D=9 → 7×D=9 → D=9/7 ✘ not integer

Try A=7, C=1

Then A×B=56 → 7×B=56 → B=8

C×D=9 → 1×D=9 → D=9

Now B×D = 8×9 = 72 ✔️

Perfect!

```
| 7 | 8 | 56
| 1 | 9 | 9
7 72
```

✔️ Solved.

---

f.

```
| | | 9
| | | 35
21 15
```

A×B = 9
C×D = 35
A×C = 21
B×D = 15

Try A×C = 21 → possible: (3,7), (7,3), (1,21), (21,1)

Try A=3, C=7

Then A×B=9 → 3×B=9 → B=3

C×D=35 → 7×D=35 → D=5

Then B×D = 3×5 = 15 ✔️

✔ Good!

```
| 3 | 3 | 9
| 7 | 5 | 35
21 15
```

✔️ Solved.

---

g.

```
| | | 24
| | | 36
16 54
```

A×B = 24
C×D = 36
A×C = 16
B×D = 54

Try A×C = 16 → possible A,C: (2,8), (4,4), (8,2), (1,16), (16,1)

Try A=4, C=4

Then A×B=24 → 4×B=24 → B=6

C×D=36 → 4×D=36 → D=9

Then B×D = 6×9 = 54 ✔️

Perfect!

```
| 4 | 6 | 24
| 4 | 9 | 36
16 54
```

✔️ Solved.

---

h.

```
| | | 63
| | | 2
18 7
```

A×B = 63
C×D = 2
A×C = 18
B×D = 7

C×D = 2 → possibilities: (1,2), (2,1)

Try C=1, D=2

Then A×C=18 → A×1=18 → A=18

Then A×B=63 → 18×B=63 → B=63/18 = 3.5 ✘

Try C=2, D=1

Then A×C=18 → A×2=18 → A=9

Then A×B=63 → 9×B=63 → B=7

Now B×D = 7×1 = 7 ✔️

✔ Perfect!

```
| 9 | 7 | 63
| 2 | 1 | 2
18 7
```

✔️ Solved.

---

i.

```
| | | 48
| | | 18
36 24
```

A×B = 48
C×D = 18
A×C = 36
B×D = 24

Try A×C = 36 → possible A,C: (6,6), (4,9), (9,4), (3,12), (12,3), etc.

Try A=6, C=6

Then A×B=48 → 6×B=48 → B=8

C×D=18 → 6×D=18 → D=3

Then B×D = 8×3 = 24 ✔️

✔ Good!

```
| 6 | 8 | 48
| 6 | 3 | 18
36 24
```

✔️ Solved.

---

j.

```
| | | 40
| | | 64
40 64
```

A×B = 40
C×D = 64
A×C = 40
B×D = 64

Interesting! A×C = 40, A×B = 40 → So A×B = A×C → B=C (if A≠0)

Similarly, B×D = 64, C×D = 64 → so B=C

So B=C

Let’s set B=C

From A×B=40, and A×C=40 → same

And C×D=64 → B×D=64 → same

So we can pick any factor pair.

Try A=8 → then B=5 (since 8×5=40)

Then C=B=5

Then C×D=64 → 5×D=64 → D=12.8 ✘ not integer

Try A=10 → B=4 → C=4 → D=16 (since 4×16=64) → B×D=4×16=64 ✔️

Check: A×B=10×4=40 ✔️
C×D=4×16=64 ✔️
A×C=10×4=40 ✔️
B×D=4×16=64 ✔️

Perfect!

```
|10 | 4 | 40
| 4 |16 | 64
40 64
```

✔️ Solved.

---

k.

```
| | | 7
| | | 6
42 1
```

A×B = 7
C×D = 6
A×C = 42
B×D = 1

Note: B×D = 1 → so B=1, D=1

Then A×B=7 → A×1=7 → A=7

Then A×C=42 → 7×C=42 → C=6

Then C×D=6×1=6 ✔️

✔ Perfect!

```
| 7 | 1 | 7
| 6 | 1 | 6
42 1
```

✔️ Solved.

---

✔ Final Answers:

Here are all the completed puzzles:

#### a.
```
| 2 | 3 | 6
| 7 | 7 | 49
14 21
```

#### b.
```
| 9 | 3 | 27
| 4 | 8 | 32
36 24
```

#### c.
```
| 6 | 7 | 42
| 9 | 8 | 72
54 56
```

#### d.
```
| 3 | 4 | 12
| 6 | 5 | 30
18 20
```

#### e.
```
| 7 | 8 | 56
| 1 | 9 | 9
7 72
```

#### f.
```
| 3 | 3 | 9
| 7 | 5 | 35
21 15
```

#### g.
```
| 4 | 6 | 24
| 4 | 9 | 36
16 54
```

#### h.
```
| 9 | 7 | 63
| 2 | 1 | 2
18 7
```

#### i.
```
| 6 | 8 | 48
| 6 | 3 | 18
36 24
```

#### j.
```
|10 | 4 | 40
| 4 |16 | 64
40 64
```

#### k.
```
| 7 | 1 | 7
| 6 | 1 | 6
42 1
```

---

🧠 Strategy Summary:

To solve these:
1. Look at the row products and column products.
2. Use common factors between them.
3. Try small values or factor pairs.
4. Work backwards from known relationships (like A×C = X and A×B = Y → divide to get B/C).
5. Always verify all four conditions.

Let me know if you'd like this as a printable PDF or want explanations for any specific one!