Free and fun math worksheets with puzzles and riddles - Free Printable

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

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🧩 How the puzzle works:

Each 2x2 grid has:
- Top row: two numbers → their product is shown on the right side (top-right cell).
- Bottom row: two numbers → their product is shown on the right side (bottom-right cell).
- Left column: two numbers → their product is shown at the bottom (bottom-left cell).
- Right column: two numbers → their product is shown at the bottom (bottom-right cell).

So, for a square:

```
[ A ][ B ] → A×B = right-top
[ C ][ D ] → C×D = right-bottom
↓ ↓
A×C B×D
←left-bottom ←right-bottom
```

We need to find integers A, B, C, D such that:
- A × B = right-top
- C × D = right-bottom
- A × C = bottom-left
- B × D = bottom-right

---

We’ll solve each puzzle one by one.

---

## 🔹 a.
```
? ? | 6
? ? |49
-----
14 21
```

We need:
- Top row: A × B = 6
- Bottom row: C × D = 49
- Left col: A × C = 14
- Right col: B × D = 21

Try factor pairs:

- 6 = 1×6, 2×3
- 49 = 7×7
- 14 = 2×7
- 21 = 3×7

Try A=2, then from A×C=14 → C=7
Then from C×D=49 → 7×D=49 → D=7
Then from B×D=21 → B×7=21 → B=3
Check A×B=2×3=6 ✔

✔ So:
Top row: 2, 3
Bottom row: 7, 7

> Answer a:
```
2 3
7 7
```

---

## 🔹 b.
```
? ? |27
? ? |32
-----
36 24
```

Need:
- A×B=27
- C×D=32
- A×C=36
- B×D=24

Factor pairs:

- 27: 1×27, 3×9
- 32: 1×32, 2×16, 4×8
- 36: 4×9, 6×6, 3×12, etc.
- 24: 3×8, 4×6, etc.

Try A=9 → A×C=36 → C=4
Then C×D=32 → 4×D=32 → D=8
Then B×D=24 → B×8=24 → B=3
Check A×B=9×3=27 ✔

✔ So:
Top row: 9, 3
Bottom row: 4, 8

> Answer b:
```
9 3
4 8
```

---

## 🔹 c.
```
? ? |42
? ? |72
-----
54 56
```

Need:
- A×B=42
- C×D=72
- A×C=54
- B×D=56

Factor pairs:

- 42: 6×7, 3×14
- 72: 8×9, 6×12
- 54: 6×9, 3×18
- 56: 7×8, 4×14

Try A=6 → A×C=54 → C=9
C×D=72 → 9×D=72 → D=8
B×D=56 → B×8=56 → B=7
A×B=6×7=42 ✔

✔ So:
Top row: 6, 7
Bottom row: 9, 8

> Answer c:
```
6 7
9 8
```

---

## 🔹 d.
```
? ? |12
? ? |30
-----
18 20
```

Need:
- A×B=12
- C×D=30
- A×C=18
- B×D=20

Try A=3 → A×C=18 → C=6
C×D=30 → 6×D=30 → D=5
B×D=20 → B×5=20 → B=4
A×B=3×4=12 ✔

✔ So:
Top row: 3, 4
Bottom row: 6, 5

> Answer d:
```
3 4
6 5
```

---

## 🔹 e.
```
? ? |56
? ? |9
-----
7 72
```

Need:
- A×B=56
- C×D=9
- A×C=7
- B×D=72

Note: 7 is prime → A×C=7 → possibilities: A=1,C=7 or A=7,C=1

Try A=7 → A×C=7 → C=1
C×D=9 → 1×D=9 → D=9
B×D=72 → B×9=72 → B=8
A×B=7×8=56 ✔

✔ So:
Top row: 7, 8
Bottom row: 1, 9

> Answer e:
```
7 8
1 9
```

---

## 🔹 f.
```
? ? |9
? ? |35
-----
21 15
```

Need:
- A×B=9
- C×D=35
- A×C=21
- B×D=15

Try A=3 → A×C=21 → C=7
C×D=35 → 7×D=35 → D=5
B×D=15 → B×5=15 → B=3
A×B=3×3=9 ✔

✔ So:
Top row: 3, 3
Bottom row: 7, 5

> Answer f:
```
3 3
7 5
```

---

## 🔹 g.
```
? ? |24
? ? |36
-----
16 54
```

Need:
- A×B=24
- C×D=36
- A×C=16
- B×D=54

Try A=4 → A×C=16 → C=4
C×D=36 → 4×D=36 → D=9
B×D=54 → B×9=54 → B=6
A×B=4×6=24 ✔

✔ So:
Top row: 4, 6
Bottom row: 4, 9

> Answer g:
```
4 6
4 9
```

---

## 🔹 h.
```
? ? |63
? ? |2
-----
18 7
```

Need:
- A×B=63
- C×D=2
- A×C=18
- B×D=7

Note: 7 is prime → B×D=7 → B=7, D=1 or vice versa

Try B=7 → B×D=7 → D=1
C×D=2 → C×1=2 → C=2
A×C=18 → A×2=18 → A=9
A×B=9×7=63 ✔

✔ So:
Top row: 9, 7
Bottom row: 2, 1

> Answer h:
```
9 7
2 1
```

---

## 🔹 i.
```
? ? |48
? ? |18
-----
36 24
```

Need:
- A×B=48
- C×D=18
- A×C=36
- B×D=24

Try A=6 → A×C=36 → C=6
C×D=18 → 6×D=18 → D=3
B×D=24 → B×3=24 → B=8
A×B=6×8=48 ✔

✔ So:
Top row: 6, 8
Bottom row: 6, 3

> Answer i:
```
6 8
6 3
```

---

## 🔹 j.
```
? ? |40
? ? |64
-----
40 64
```

Need:
- A×B=40
- C×D=64
- A×C=40
- B×D=64

Interesting — left and right products are same as top and bottom.

Try A=5 → A×C=40 → C=8
C×D=64 → 8×D=64 → D=8
B×D=64 → B×8=64 → B=8
A×B=5×8=40 ✔

✔ So:
Top row: 5, 8
Bottom row: 8, 8

> Answer j:
```
5 8
8 8
```

---

## 🔹 k.
```
? ? |7
? ? |6
-----
42 1
```

Need:
- A×B=7
- C×D=6
- A×C=42
- B×D=1

B×D=1 → only possibility: B=1, D=1

Then A×B=7 → A×1=7 → A=7
A×C=42 → 7×C=42 → C=6
C×D=6×1=6 ✔

✔ So:
Top row: 7, 1
Bottom row: 6, 1

> Answer k:
```
7 1
6 1
```

---

## ✔ Final Answers Summary:

a.
```
2 3
7 7
```

b.
```
9 3
4 8
```

c.
```
6 7
9 8
```

d.
```
3 4
6 5
```

e.
```
7 8
1 9
```

f.
```
3 3
7 5
```

g.
```
4 6
4 9
```

h.
```
9 7
2 1
```

i.
```
6 8
6 3
```

j.
```
5 8
8 8
```

k.
```
7 1
6 1
```

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You can now fill in the worksheet with these answers! 🎉