Arithmogon Triangle Puzzle 3A: Solve the math puzzle by finding missing numbers in triangles.

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

🔍 Understanding the Rule:

In each triangle:
- The numbers in the circles (at the corners) are added together to make the number in the rectangle between them.
- So, for any side of the triangle:
- Circle + Circle = Rectangle
- We need to find the missing numbers (in circles or rectangles) based on this rule.

We'll go through each triangle one by one.

---

🔹 Triangle 1:

```
10
/ \
30 ?
/ \
20 50
\ /
70
```

We know:
- Top circle: 10
- Left circle: 20
- Right circle: 50
- Bottom rectangle: 70

Check:
Left side: 20 + ? = 30 → So top-left corner is 10? Wait — no. Let's clarify.

Actually, the rectangles are on the sides, and they represent the sum of the two adjacent circles.

So:
- Left side: 20 (bottom-left) + 10 (top) = 30 → ✔ Matches
- Right side: 50 (bottom-right) + 10 (top) = 60 → But rectangle says "?", so we can compute it:
→ 50 + 10 = 60 → So right rectangle = 60
- Bottom side: 20 + 50 = 70 → ✔ Matches given

✔ So missing number is the right rectangle: 60

Answer: Right rectangle = 60

---

🔹 Triangle 2:

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

We have:
- Top: 6
- Left: 14
- Right: 12
- Left rectangle: 20
- Bottom rectangle: ?
- Right rectangle: ?

Left side: 14 + 6 = 20 → ✔ Good

Right side: 12 + 6 = 18 → So right rectangle = 18

Bottom side: 14 + 12 = 26 → So bottom rectangle = 26

✔ Missing numbers:
- Right rectangle: 18
- Bottom rectangle: 26

---

🔹 Triangle 3:

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

Top: 60
Left: 40
Right: 50

We don't know the side rectangles yet.

But:
- Left side: 40 + ? = left rectangle → but we don’t know the top value yet? Wait — top is 60, which is sum of left and right circles?

Wait — no! The top circle is 60, and the side rectangles are sums of adjacent circles.

So:
- Left side: 40 (left circle) + 60 (top) = ? → So left rectangle = 100
- Right side: 50 (right circle) + 60 (top) = 110 → right rectangle = 110
- Bottom side: 40 + 50 = 90 → bottom rectangle = 90

But wait — that would mean:
- Left rectangle: 40 + 60 = 100
- Right rectangle: 50 + 60 = 110
- Bottom rectangle: 40 + 50 = 90

But nothing contradicts — all are consistent.

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

✔ Done.

---

🔹 Triangle 4:

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

Top: 7
Left: 15
Right: 14

We need to find:
- Left rectangle: 15 + 7 = 22
- Right rectangle: 14 + 7 = 21
- Bottom rectangle: 15 + 14 = 29

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

✔ All filled.

---

🔹 Triangle 5:

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

Top: 13
Left: 11
Right: 17

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

So:
- Left: 24
- Right: 30
- Bottom: 28

✔ Done.

---

🔹 Triangle 6:

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

Top: 12
Left: 9
Right: 15

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

So:
- Left: 21
- Right: 27
- Bottom: 24

✔ Done.

---

🔹 Triangle 7:

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

Top: 15
Left rectangle: 21
Right rectangle: ?
Bottom rectangle: ?
Left circle: ?
Right circle: 8

We know:
- Left side: left circle + top = 21 → left circle + 15 = 21 → left circle = 6
- Right side: right circle + top = ? → 8 + 15 = 23 → right rectangle = 23
- Bottom side: left circle + right circle = 6 + 8 = 14 → bottom rectangle = 14

So:
- Left circle: 6
- Right rectangle: 23
- Bottom rectangle: 14

✔ Done.

---

🔹 Triangle 8:

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

Top: 17
Right rectangle: 26
Left circle: 7
Right circle: ?
Left rectangle: ?
Bottom rectangle: ?

From right side:
- Right circle + top = 26 → ? + 17 = 26 → right circle = 9

Left side:
- Left circle + top = 7 + 17 = 24 → left rectangle = 24

Bottom side:
- 7 + 9 = 16 → bottom rectangle = 16

So:
- Left rectangle: 24
- Right circle: 9
- Bottom rectangle: 16

✔ Done.

---

🔹 Triangle 9:

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

Top: ?
Left rectangle: 21
Right rectangle: 19
Left circle: 13
Right circle: ?
Bottom rectangle: ?

We know:
- Left side: left circle + top = 21 → 13 + top = 21 → top = 8
- Right side: right circle + top = 19 → right circle + 8 = 19 → right circle = 11
- Bottom side: 13 + 11 = 24 → bottom rectangle = 24

So:
- Top circle: 8
- Right circle: 11
- Bottom rectangle: 24

✔ Done.

---

✔ Final Answers:

Let’s list them in order from top to bottom, left to right.

#### 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:
- Left circle: 6
- Right rectangle: 23
- Bottom rectangle: 14

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

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

---

📌 Summary of Missing Numbers:

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

---

✔ Check One Example:

Take Triangle 7:
- Left circle: 6
- Top: 15
- Left rectangle: 6 + 15 = 21 ✔
- Right circle: 8
- Right rectangle: 8 + 15 = 23 ✔
- Bottom: 6 + 8 = 14 ✔

All checks pass.

---

🟩 Final Answer:

You can now fill in the puzzle with these values.

Let me know if you'd like a visual version or printable solution!