Math Puzzles Printable for Learning | Activity Shelter - Free Printable

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 last cell) is the product of the diagonal or can be derived from the other products.

But actually, looking at the example:

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

So:
- Top right: product of top row (5 × 6 = 30)
- Bottom right: product of bottom row (3 × 8 = 24)
- Bottom left: product of left column (5 × 3 = 15)
- Bottom right corner: product of right column (6 × 8 = 48)

Wait — but in the example, the bottom-right corner is 48, which is the product of the right column (6 × 8), and the bottom-right number is 24, which is the product of the bottom row (3 × 8).

So the layout is:

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

So the four numbers around the square are:
- Right-top: A×B
- Right-bottom: C×D
- Bottom-left: A×C
- Bottom-right: B×D

We need to find A, B, C, D such that:
- A×B = given top-right
- C×D = given bottom-right
- A×C = given bottom-left
- B×D = given bottom-right (wait — no, that’s already used)

Actually, from the example:
- Top-right: 30 = 5×6 → A×B = 30
- Bottom-right: 24 = 3×8 → C×D = 24
- Bottom-left: 15 = 5×3 → A×C = 15
- Bottom-right corner: 48 = 6×8 → B×D = 48

So yes, we have:
- A×B = top-right
- C×D = bottom-right
- A×C = bottom-left
- B×D = bottom-right (no — wait, it's labeled as "bottom-right" but that's actually the right-bottom value)

Wait — let's clarify the labels.

In the example:

```
5 6 → 30
3 8 → 24
15 48
```

So:
- Top-right: 30 = 5×6
- Bottom-right: 24 = 3×8
- Bottom-left: 15 = 5×3
- Bottom-right corner: 48 = 6×8

So the four outer numbers are:
- Top-right: A×B
- Bottom-right: C×D
- Bottom-left: A×C
- Bottom-right (corner): B×D

But note: Bottom-right (the label under the second column) is B×D, and bottom-right (under the first column) is A×C

So:
- Top-right: A×B
- Bottom-right (right column): B×D
- Bottom-left (left column): A×C
- Bottom-right (bottom row): C×D

Wait — this is confusing. Let's label the positions clearly.

Let’s define the grid:

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

So:
- Top-right: A×B
- Bottom-right (row): C×D
- Bottom-left (column): A×C
- Bottom-right (column): B×D

So the four numbers outside are:
- Top-right: A×B
- Bottom-right (right side): B×D
- Bottom-left (bottom side): A×C
- Bottom-right (bottom side): C×D

Wait — no! In the example:

The numbers are:
- Top-right: 30 = A×B = 5×6
- Bottom-right (on the right side): 24 = C×D = 3×8
- Bottom-left (on the bottom): 15 = A×C = 5×3
- Bottom-right (on the bottom): 48 = B×D = 6×8

So:
- A×B = 30
- C×D = 24
- A×C = 15
- B×D = 48

Now, we can solve for A, B, C, D using these equations.

Let’s go through each problem.

---

a.

```
? ? → 6
? ? → 49
14 21
```

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

We need to find integers A, B, C, D.

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

From A×C = 14 → A divides 14 → A ∈ {1,2,7,14}

So A must be common factor of 6 and 14 → gcd(6,14)=2 → so A=1 or 2

Try A=2:
- Then B = 6/A = 3
- C = 14/A = 7
- Then D = 49/C = 49/7 = 7
- Then B×D = 3×7 = 21 ✔ matches

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

Check:
- A×B = 2×3 = 6 ✔
- C×D = 7×7 = 49 ✔
- A×C = 2×7 = 14 ✔
- B×D = 3×7 = 21 ✔

✔ So answer:
```
[2] [3]
[7] [7]
```

---

b.

```
? ? → 27
? ? → 32
36 24
```

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

A divides both 27 and 36 → gcd(27,36)=9 → possible A: 1,3,9

Try A=9:
- B = 27/9 = 3
- C = 36/9 = 4
- D = 32/4 = 8
- Check B×D = 3×8 = 24 ✔

Perfect!

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

Grid:
```
[9] [3]
[4] [8]
```

Check:
- A×B = 9×3 = 27 ✔
- C×D = 4×8 = 32 ✔
- A×C = 9×4 = 36 ✔
- B×D = 3×8 = 24 ✔

✔

---

c.

```
? ? → 42
? ? → 72
54 56
```

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

A divides 42 and 54 → gcd(42,54)=6 → A ∈ {1,2,3,6}

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

Perfect!

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

Grid:
```
[6] [7]
[9] [8]
```

Check:
- 6×7 = 42 ✔
- 9×8 = 72 ✔
- 6×9 = 54 ✔
- 7×8 = 56 ✔

✔

---

d.

```
? ? → 12
? ? → 30
18 20
```

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

A divides 12 and 18 → gcd(12,18)=6 → A ∈ {1,2,3,6}

Try A=6:
- B = 12/6 = 2
- C = 18/6 = 3
- D = 30/3 = 10
- B×D = 2×10 = 20 ✔

Good!

So:
- A=6, B=2, C=3, D=10

Grid:
```
[6] [2]
[3] [10]
```

Check:
- 6×2 = 12 ✔
- 3×10 = 30 ✔
- 6×3 = 18 ✔
- 2×10 = 20 ✔

✔

---

e.

```
? ? → 56
? ? → 9
7 72
```

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

A divides 56 and 7 → so A divides 7 → A=1 or 7

Try A=7:
- B = 56/7 = 8
- C = 7/7 = 1
- D = 9/1 = 9
- B×D = 8×9 = 72 ✔

Perfect!

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

Grid:
```
[7] [8]
[1] [9]
```

Check:
- 7×8 = 56 ✔
- 1×9 = 9 ✔
- 7×1 = 7 ✔
- 8×9 = 72 ✔

✔

---

f.

```
? ? → 9
? ? → 35
21 15
```

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

A divides 9 and 21 → gcd(9,21)=3 → A ∈ {1,3}

Try A=3:
- B = 9/3 = 3
- C = 21/3 = 7
- D = 35/7 = 5
- B×D = 3×5 = 15 ✔

Perfect!

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

Grid:
```
[3] [3]
[7] [5]
```

Check:
- 3×3 = 9 ✔
- 7×5 = 35 ✔
- 3×7 = 21 ✔
- 3×5 = 15 ✔

✔

---

g.

```
? ? → 24
? ? → 36
16 54
```

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

A divides 24 and 16 → gcd(24,16)=8 → A ∈ {1,2,4,8}

Try A=4:
- B = 24/4 = 6
- C = 16/4 = 4
- D = 36/4 = 9
- B×D = 6×9 = 54 ✔

Perfect!

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

Grid:
```
[4] [6]
[4] [9]
```

Check:
- 4×6 = 24 ✔
- 4×9 = 36 ✔
- 4×4 = 16 ✔
- 6×9 = 54 ✔

✔

---

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:
- A×C = 18 → A×1 = 18 → A=18
- A×B = 63 → 18×B = 63 → B = 63/18 = 3.5 ✘ not integer

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

Perfect!

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

Grid:
```
[9] [7]
[2] [1]
```

Check:
- 9×7 = 63 ✔
- 2×1 = 2 ✔
- 9×2 = 18 ✔
- 7×1 = 7 ✔

✔

---

i.

```
? ? → 48
? ? → 18
36 24
```

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

A divides 48 and 36 → gcd(48,36)=12 → A ∈ {1,2,3,4,6,12}

Try A=6:
- B = 48/6 = 8
- C = 36/6 = 6
- D = 18/6 = 3
- B×D = 8×3 = 24 ✔

Perfect!

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

Grid:
```
[6] [8]
[6] [3]
```

Check:
- 6×8 = 48 ✔
- 6×3 = 18 ✔
- 6×6 = 36 ✔
- 8×3 = 24 ✔

✔

---

j.

```
? ? → 40
? ? → 64
40 64
```

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

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

Also: C×D = 64, B×D = 64 → since B=C, consistent.

So B = C

Then from A×B = 40 and A×C = 40 → same.

And C×D = 64 → B×D = 64

So we can pick any A,B such that A×B = 40, and then C=B, D=64/B

But also, C×D = B×D = 64 → D = 64/B

But C×D = B×(64/B) = 64 ✔ always

So we just need A×B = 40 and B divides 64

So B divides 64 and B divides 40 → gcd(40,64)=8 → B ∈ {1,2,4,8}

Try B=8:
- A = 40/8 = 5
- C = B = 8
- D = 64/8 = 8

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

Check:
- A×B = 5×8 = 40 ✔
- C×D = 8×8 = 64 ✔
- A×C = 5×8 = 40 ✔
- B×D = 8×8 = 64 ✔

Perfect!

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

A×C = 42 → 7×C = 42 → C=6

C×D = 6×1 = 6 ✔

Perfect!

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

Grid:
```
[7] [1]
[6] [1]
```

Check:
- 7×1 = 7 ✔
- 6×1 = 6 ✔
- 7×6 = 42 ✔
- 1×1 = 1 ✔

✔

---

## ✔ Final Answers:

Here are all the completed grids:

a.

```
[2] [3]
[7] [7]
```

b.

```
[9] [3]
[4] [8]
```

c.

```
[6] [7]
[9] [8]
```

d.

```
[6] [2]
[3] [10]
```

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!