Multiplication Square Puzzles worksheet for practicing multiplication skills.

Let's solve each of these Multiplication Square Puzzles step by step.

Each puzzle is a 2×2 grid where:
- The top-right number is the product of the top row.
- The bottom-right number is the product of the bottom row.
- The bottom-left number is the product of the left column.
- The bottom-right corner (the total) is the product of the entire 2×2 square, i.e., the product of all four numbers, or equivalently:
$$
\text{Total} = (\text{Top-left}) \times (\text{Top-right}) \times (\text{Bottom-left}) \times (\text{Bottom-right})
$$
But more practically, we can use:
$$
\text{Bottom-right} = (\text{Top-left}) \times (\text{Bottom-right}) \quad \text{(no, that’s not right)}
$$

Wait — let’s clarify with the example:

Example:

```
| 5 | 6 | → 30
|---|---|
| 3 | 8 | → 24
| 15| 48|
```

So:
- Top row: 5 × 6 = 30 ✔
- Bottom row: 3 × 8 = 24 ✔
- Left column: 5 × 3 = 15 ✔
- Right column: 6 × 8 = 48 ✔

So the rules are:
- Top-right = top-left × top-right cell
- Bottom-right = bottom-left × bottom-right cell
- Bottom-left = top-left × bottom-left cell
- Bottom-right (total) = top-right × bottom-right cell

Wait — actually, the values outside the square are:
- Right side: product of each row
- Bottom side: product of each column

So:
- Top-right number = product of top row: A × B
- Bottom-right number = product of bottom row: C × D
- Bottom-left number = product of left column: A × C
- Bottom-right corner (last one) = product of right column: B × D

And the bottom-right corner is also the product of the two bottom values? Wait — no.

Looking at the example:
- Top-right: 30 = 5 × 6
- Bottom-right: 24 = 3 × 8
- Bottom-left: 15 = 5 × 3
- Bottom-right corner: 48 = 6 × 8

So:
- The numbers on the right are: row products
- The numbers on the bottom are: column products

So:
- Let the grid be:
```
| A | B |
|---|---|
| C | D |
```
- Then:
- A × B = right-top value
- C × D = right-bottom value
- A × C = bottom-left value
- B × D = bottom-right value

So we have:
- A × B = R1
- C × D = R2
- A × C = C1
- B × D = C2

We need to find A, B, C, D such that these equations are satisfied.

---

We’ll solve each part using this logic.

---

a.

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

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

Let’s solve.

From A × B = 6 → possible pairs: (1,6), (2,3), (3,2), (6,1)

Try A = 2, B = 3 → then A × C = 14 → 2 × C = 14 → C = 7

Then C × D = 49 → 7 × D = 49 → D = 7

Then B × D = 3 × 7 = 21 ✔

Perfect!

So:
- A = 2
- B = 3
- C = 7
- D = 7

Check:
- Top row: 2 × 3 = 6 ✔
- Bottom row: 7 × 7 = 49 ✔
- Left col: 2 × 7 = 14 ✔
- Right col: 3 × 7 = 21 ✔

✔ So:
```
| 2 | 3 |
|---|---|
| 7 | 7 |
```

---

b.

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

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

A × B = 27 → possible: (1,27), (3,9), (9,3), (27,1)

Try A = 9, B = 3 → then A × C = 36 → 9 × C = 36 → C = 4

Then C × D = 32 → 4 × D = 32 → D = 8

Then B × D = 3 × 8 = 24 ✔

Perfect!

So:
- A = 9
- B = 3
- C = 4
- D = 8

Check:
- Top: 9×3=27 ✔
- Bottom: 4×8=32 ✔
- Left: 9×4=36 ✔
- Right: 3×8=24 ✔

✔ Grid:
```
| 9 | 3 |
|---|---|
| 4 | 8 |
```

---

c.

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

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

Factor 42: (1,42), (2,21), (3,14), (6,7), (7,6), etc.

Try A = 6, B = 7 → A × C = 54 → 6 × C = 54 → C = 9

Then C × D = 72 → 9 × D = 72 → D = 8

Then B × D = 7 × 8 = 56 ✔

Perfect!

So:
- A = 6
- B = 7
- C = 9
- D = 8

Check:
- Top: 6×7=42 ✔
- Bottom: 9×8=72 ✔
- Left: 6×9=54 ✔
- Right: 7×8=56 ✔

✔ Grid:
```
| 6 | 7 |
|---|---|
| 9 | 8 |
```

---

d.

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

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

Try A = 3, B = 4 → A × C = 18 → 3 × C = 18 → C = 6

C × D = 30 → 6 × D = 30 → D = 5

B × D = 4 × 5 = 20 ✔

Yes!

So:
- A = 3
- B = 4
- C = 6
- D = 5

Check:
- Top: 3×4=12 ✔
- Bottom: 6×5=30 ✔
- Left: 3×6=18 ✔
- Right: 4×5=20 ✔

✔ Grid:
```
| 3 | 4 |
|---|---|
| 6 | 5 |
```

---

e.

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

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

Start with A × C = 7 → possible: (1,7), (7,1)

Try A = 7, C = 1

Then A × B = 56 → 7 × B = 56 → B = 8

Then C × D = 9 → 1 × D = 9 → D = 9

Then B × D = 8 × 9 = 72 ✔

Perfect!

So:
- A = 7
- B = 8
- C = 1
- D = 9

Check:
- Top: 7×8=56 ✔
- Bottom: 1×9=9 ✔
- Left: 7×1=7 ✔
- Right: 8×9=72 ✔

✔ Grid:
```
| 7 | 8 |
|---|---|
| 1 | 9 |
```

---

f.

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

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

A × B = 9 → try A = 3, B = 3

Then A × C = 21 → 3 × C = 21 → C = 7

Then C × D = 35 → 7 × D = 35 → D = 5

Then B × D = 3 × 5 = 15 ✔

Perfect!

So:
- A = 3
- B = 3
- C = 7
- D = 5

Check:
- Top: 3×3=9 ✔
- Bottom: 7×5=35 ✔
- Left: 3×7=21 ✔
- Right: 3×5=15 ✔

✔ Grid:
```
| 3 | 3 |
|---|---|
| 7 | 5 |
```

---

g.

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

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

Try A = 4, B = 6 → A × C = 16 → 4 × C = 16 → C = 4

Then C × D = 36 → 4 × D = 36 → D = 9

Then B × D = 6 × 9 = 54 ✔

Perfect!

So:
- A = 4
- B = 6
- C = 4
- D = 9

Check:
- Top: 4×6=24 ✔
- Bottom: 4×9=36 ✔
- Left: 4×4=16 ✔
- Right: 6×9=54 ✔

✔ Grid:
```
| 4 | 6 |
|---|---|
| 4 | 9 |
```

---

h.

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

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

C × D = 2 → possible: (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 ✘ Not integer

Try C = 2, D = 1

Then A × C = 18 → A × 2 = 18 → A = 9

Then A × B = 63 → 9 × B = 63 → B = 7

Then B × D = 7 × 1 = 7 ✔

Perfect!

So:
- A = 9
- B = 7
- C = 2
- D = 1

Check:
- Top: 9×7=63 ✔
- Bottom: 2×1=2 ✔
- Left: 9×2=18 ✔
- Right: 7×1=7 ✔

✔ Grid:
```
| 9 | 7 |
|---|---|
| 2 | 1 |
```

---

i.

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

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

Try A = 6, B = 8 → A × C = 36 → 6 × C = 36 → C = 6

Then C × D = 18 → 6 × D = 18 → D = 3

Then B × D = 8 × 3 = 24 ✔

Perfect!

So:
- A = 6
- B = 8
- C = 6
- D = 3

Check:
- Top: 6×8=48 ✔
- Bottom: 6×3=18 ✔
- Left: 6×6=36 ✔
- Right: 8×3=24 ✔

✔ Grid:
```
| 6 | 8 |
|---|---|
| 6 | 3 |
```

---

j.

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

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

Notice: A × C = 40 and A × B = 40 → so B = C?

Try A = 5, B = 8 → A × B = 40 ✔

Then A × C = 40 → 5 × C = 40 → C = 8

Then C × D = 64 → 8 × D = 64 → D = 8

Then B × D = 8 × 8 = 64 ✔

Perfect!

So:
- A = 5
- B = 8
- C = 8
- D = 8

Check:
- Top: 5×8=40 ✔
- Bottom: 8×8=64 ✔
- Left: 5×8=40 ✔
- Right: 8×8=64 ✔

✔ Grid:
```
| 5 | 8 |
|---|---|
| 8 | 8 |
```

---

k.

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

So:
- 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

Then A × C = 42 → 7 × C = 42 → C = 6

Then C × D = 6 × 1 = 6 ✔

Perfect!

So:
- A = 7
- B = 1
- C = 6
- D = 1

Check:
- Top: 7×1=7 ✔
- Bottom: 6×1=6 ✔
- Left: 7×6=42 ✔
- Right: 1×1=1 ✔

✔ Grid:
```
| 7 | 1 |
|---|---|
| 6 | 1 |
```

---

## ✔ Final Answers:

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 |
```

All puzzles solved! Each satisfies the multiplication rules.