Arithmogon Triangle Puzzle 3A: Solve addition puzzles by finding missing numbers in triangles.

I will solve this Arithmogon Triangle Puzzle by carefully analyzing the rules and applying them to each triangle. The rule is that "The numbers in the circles added together makes the number in the linking rectangle." This means for each side of the triangle, the sum of the two circles at its ends equals the number in the rectangle on that side.

Let me break down the solution step by step for each triangle.

First Row

Triangle 1 (Top Left):
- Top circle: 10
- Bottom-left circle: 20
- Bottom-right circle: 50
- Left rectangle: 30 (between top and bottom-left circles)
- Bottom rectangle: 70 (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

We need to find the missing number in the right rectangle.
According to the rule: Top circle + Bottom-right circle = Right rectangle
So, 10 + 50 = 60

Triangle 2 (Top Middle):
- Top circle: 6
- Bottom-left circle: 14
- Bottom-right circle: 12
- Left rectangle: 20 (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

First, find the bottom rectangle:
Bottom-left circle + Bottom-right circle = Bottom rectangle
14 + 12 = 26

Next, find the right rectangle:
Top circle + Bottom-right circle = Right rectangle
6 + 12 = 18

Triangle 3 (Top Right):
- Top circle: 60
- Bottom-left circle: 40
- Bottom-right circle: 50
- Left rectangle: ? (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

Find the left rectangle:
Top circle + Bottom-left circle = Left rectangle
60 + 40 = 100

Find the bottom rectangle:
Bottom-left circle + Bottom-right circle = Bottom rectangle
40 + 50 = 90

Find the right rectangle:
Top circle + Bottom-right circle = Right rectangle
60 + 50 = 110

Second Row

Triangle 4 (Middle Left):
- Top circle: 7
- Bottom-left circle: 15
- Bottom-right circle: 14
- Left rectangle: ? (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

Find the left rectangle:
7 + 15 = 22

Find the bottom rectangle:
15 + 14 = 29

Find the right rectangle:
7 + 14 = 21

Triangle 5 (Middle Middle):
- Top circle: 13
- Bottom-left circle: 11
- Bottom-right circle: 17
- Left rectangle: ? (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

Find the left rectangle:
13 + 11 = 24

Find the bottom rectangle:
11 + 17 = 28

Find the right rectangle:
13 + 17 = 30

Triangle 6 (Middle Right):
- Top circle: 12
- Bottom-left circle: 9
- Bottom-right circle: 15
- Left rectangle: ? (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

Find the left rectangle:
12 + 9 = 21

Find the bottom rectangle:
9 + 15 = 24

Find the right rectangle:
12 + 15 = 27

Third Row

Triangle 7 (Bottom Left):
- Top circle: 15
- Bottom-left circle: ?
- Bottom-right circle: 8
- Left rectangle: 21 (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: ? (between top and bottom-right circles)

First, find the bottom-left circle:
Top circle + Bottom-left circle = Left rectangle
15 + ? = 21
? = 21 - 15 = 6

Now, find the bottom rectangle:
Bottom-left circle + Bottom-right circle = Bottom rectangle
6 + 8 = 14

Find the right rectangle:
Top circle + Bottom-right circle = Right rectangle
15 + 8 = 23

Triangle 8 (Bottom Middle):
- Top circle: 17
- Bottom-left circle: 7
- Bottom-right circle: ?
- Left rectangle: ? (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: 26 (between top and bottom-right circles)

First, find the bottom-right circle:
Top circle + Bottom-right circle = Right rectangle
17 + ? = 26
? = 26 - 17 = 9

Now, find the left rectangle:
Top circle + Bottom-left circle = Left rectangle
17 + 7 = 24

Find the bottom rectangle:
Bottom-left circle + Bottom-right circle = Bottom rectangle
7 + 9 = 16

Triangle 9 (Bottom Right):
- Top circle: ?
- Bottom-left circle: 13
- Bottom-right circle: ?
- Left rectangle: 21 (between top and bottom-left circles)
- Bottom rectangle: ? (between bottom-left and bottom-right circles)
- Right rectangle: 19 (between top and bottom-right circles)

First, find the top circle using the left rectangle:
Top circle + Bottom-left circle = Left rectangle
? + 13 = 21
? = 21 - 13 = 8

Now, find the bottom-right circle using the right rectangle:
Top circle + Bottom-right circle = Right rectangle
8 + ? = 19
? = 19 - 8 = 11

Finally, find the bottom rectangle:
Bottom-left circle + Bottom-right circle = Bottom rectangle
13 + 11 = 24

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Final Answer

Here are all the missing numbers filled in:

First Row:
- Top Left Triangle: Right rectangle = 60
- Top Middle Triangle: Bottom rectangle = 26, Right rectangle = 18
- Top Right Triangle: Left rectangle = 100, Bottom rectangle = 90, Right rectangle = 110

Second Row:
- Middle Left Triangle: Left rectangle = 22, Bottom rectangle = 29, Right rectangle = 21
- Middle Middle Triangle: Left rectangle = 24, Bottom rectangle = 28, Right rectangle = 30
- Middle Right Triangle: Left rectangle = 21, Bottom rectangle = 24, Right rectangle = 27

Third Row:
- Bottom Left Triangle: Bottom-left circle = 6, Bottom rectangle = 14, Right rectangle = 23
- Bottom Middle Triangle: Left rectangle = 24, Bottom rectangle = 16, Bottom-right circle = 9
- Bottom Right Triangle: Top circle = 8, Bottom-right circle = 11, Bottom rectangle = 24

All puzzles are now solved!