Addition Math Puzzles #5 Worksheets | 99Worksheets - Free Printable

Let's solve the Arithmogon Triangle Puzzle 3A step by step.

🔍 Understanding the Rule:

Each triangle has:
- Three circles (at the corners),
- Three rectangles (on the sides),
- The number in each rectangle is the sum of the two numbers in the circles it connects.

So, for any side:
> Rectangle = Circle₁ + Circle₂

We’ll use this rule to find missing numbers.

---

Let’s go through each triangle one by one.

---

🔹 Triangle 1

```
10
/ \
30 ?
/ \
20 50
```

We know:
- Left side: 20 + ? = 30 → So top-left circle is 10, but we already have 10 at the top. Let's check consistency.
- Top: 10 is given.
- Bottom-left: 20
- Bottom-right: 50
- Left rectangle: 30 → 20 + ? = 30 → So the top-left corner must be 10 → ✔ matches top value.

Now, right rectangle: ? = top + bottom-right = 10 + 50 = 60

Middle rectangle (bottom): 70 = left circle + right circle = 20 + 50 = 70 → ✔ correct.

So, missing:
- Right side rectangle: 60
- Top-right circle: ??? Wait — no, the top is 10, and the top-right circle is part of the top and right side.

Wait — actually, the top circle is shared between the left and right sides.

Let me label properly:

Let’s define:
- A = top circle = 10
- B = bottom-left circle = 20
- C = bottom-right circle = 50

Then:
- Left rectangle = A + B = 10 + 20 = 30 → ✔
- Right rectangle = A + C = 10 + 50 = 60
- Bottom rectangle = B + C = 20 + 50 = 70 → ✔

✔ So, missing number is the right rectangle: 60

Answer: 60

---

🔹 Triangle 2

```
6
/ \
20 ?
/ \
14 12
```

Top = 6
Bottom-left = 14
Bottom-right = 12

Left rectangle = 20 = top + bottom-left = 6 + 14 = 20 → ✔

Right rectangle = top + bottom-right = 6 + 12 = 18

Bottom rectangle = 14 + 12 = 26

So missing:
- Right rectangle: 18
- Bottom rectangle: 26

But the bottom rectangle is blank → so fill that with 26

Answer: Right rectangle = 18, Bottom rectangle = 26

---

🔹 Triangle 3

```
60
/ \
? ?
/ \
40 50
```

Top = 60
Bottom-left = 40
Bottom-right = 50

Left rectangle = top + bottom-left = 60 + 40 = 100
Right rectangle = top + bottom-right = 60 + 50 = 110
Bottom rectangle = 40 + 50 = 90

So:
- Left rectangle: 100
- Right rectangle: 110
- Bottom rectangle: 90

Answer: Left = 100, Right = 110, Bottom = 90

---

🔹 Triangle 4

```
7
/ \
? ?
/ \
15 14
```

Top = 7
Bottom-left = 15
Bottom-right = 14

Left rectangle = top + bottom-left = 7 + 15 = 22
Right rectangle = top + bottom-right = 7 + 14 = 21
Bottom rectangle = 15 + 14 = 29

So:
- Left rectangle: 22
- Right rectangle: 21
- Bottom rectangle: 29

Answer: 22, 21, 29

---

🔹 Triangle 5

```
13
/ \
? ?
/ \
11 17
```

Top = 13
Bottom-left = 11
Bottom-right = 17

Left rectangle = 13 + 11 = 24
Right rectangle = 13 + 17 = 30
Bottom rectangle = 11 + 17 = 28

Answer: 24, 30, 28

---

🔹 Triangle 6

```
12
/ \
? ?
/ \
9 15
```

Top = 12
Bottom-left = 9
Bottom-right = 15

Left rectangle = 12 + 9 = 21
Right rectangle = 12 + 15 = 27
Bottom rectangle = 9 + 15 = 24

Answer: 21, 27, 24

---

🔹 Triangle 7

```
15
/ \
21 ?
/ \
? 8
```

Top = 15
Left rectangle = 21 = top + bottom-left → 15 + ? = 21 → bottom-left = 6

Right rectangle = top + bottom-right = 15 + 8 = 23

Bottom rectangle = bottom-left + bottom-right = 6 + 8 = 14

So:
- Bottom-left circle = 6
- Right rectangle = 23
- Bottom rectangle = 14

Answer: Bottom-left = 6, Right rectangle = 23, Bottom = 14

---

🔹 Triangle 8

```
17
/ \
? 26
/ \
7 ?
```

Top = 17
Right rectangle = 26 = top + bottom-right → 17 + ? = 26 → bottom-right = 9

Left rectangle = top + bottom-left = 17 + 7 = 24

Bottom rectangle = 7 + 9 = 16

So:
- Left rectangle = 24
- Bottom-right circle = 9
- Bottom rectangle = 16

Answer: Left = 24, Bottom-right = 9, Bottom = 16

---

🔹 Triangle 9

```
?
/ \
21 19
/ \
13 ?
```

Left rectangle = 21 = top + bottom-left → top + 13 = 21 → top = 8

Right rectangle = 19 = top + bottom-right → 8 + ? = 19 → bottom-right = 11

Bottom rectangle = 13 + 11 = 24

So:
- Top circle = 8
- Bottom-right circle = 11
- Bottom rectangle = 24

Answer: Top = 8, Bottom-right = 11, Bottom = 24

---

✔ Final Answers:

Let’s compile all answers in order:

#### Triangle 1:
- Right rectangle: 60

#### Triangle 2:
- Right rectangle: 18
- Bottom rectangle: 26

#### Triangle 3:
- Left rectangle: 100
- Right rectangle: 110
- Bottom rectangle: 90

#### Triangle 4:
- Left rectangle: 22
- Right rectangle: 21
- Bottom rectangle: 29

#### Triangle 5:
- Left rectangle: 24
- Right rectangle: 30
- Bottom rectangle: 28

#### Triangle 6:
- Left rectangle: 21
- Right rectangle: 27
- Bottom rectangle: 24

#### Triangle 7:
- Bottom-left circle: 6
- Right rectangle: 23
- Bottom rectangle: 14

#### Triangle 8:
- Left rectangle: 24
- Bottom-right circle: 9
- Bottom rectangle: 16

#### Triangle 9:
- Top circle: 8
- Bottom-right circle: 11
- Bottom rectangle: 24

---

📝 Summary Table:

| Triangle | Missing Numbers |
|--------|------------------|
| 1 | Right rect: 60 |
| 2 | Right rect: 18, Bottom rect: 26 |
| 3 | Left: 100, Right: 110, Bottom: 90 |
| 4 | Left: 22, Right: 21, Bottom: 29 |
| 5 | Left: 24, Right: 30, Bottom: 28 |
| 6 | Left: 21, Right: 27, Bottom: 24 |
| 7 | Bottom-left: 6, Right rect: 23, Bottom: 14 |
| 8 | Left rect: 24, Bottom-right: 9, Bottom: 16 |
| 9 | Top: 8, Bottom-right: 11, Bottom: 24 |

---

✔ All solved using the rule: Rectangle = Sum of two adjacent circles

Remember to double-check! 😊