Triangle puzzle with numbers and magic number 19.

Problem Description:

The task involves solving Magic Triangle Puzzles. Each triangle has a "magic number," and the sum of the numbers on each side of the triangle must equal this magic number. We need to fill in the missing numbers in each triangle.

Solution Approach:

1. Understand the Structure: Each triangle has 6 circles (3 at the vertices and 3 along the sides). The sum of the numbers on each side of the triangle must equal the given "magic number."
2. Use Algebraic Reasoning: Assign variables to the unknowns and use the magic number constraint to form equations.
3. Solve the Equations: Solve the system of equations to find the missing numbers.

Step-by-Step Solution:

#### Triangle 1: Magic Number = 19
- Given numbers: Top = 8, Left vertex = 3, Right vertex = 4.
- Let the missing numbers be:
- Bottom-left circle = \( x \)
- Bottom-right circle = \( y \)
- Middle-left circle = \( z \)
- Middle-right circle = \( w \)

The constraints are:
1. \( 3 + z + 8 = 19 \) (Left side)
2. \( 8 + w + 4 = 19 \) (Right side)
3. \( 3 + x + 4 = 19 \) (Bottom side)

Solve each equation:
1. \( 3 + z + 8 = 19 \)
\[
z + 11 = 19 \implies z = 8
\]

2. \( 8 + w + 4 = 19 \)
\[
w + 12 = 19 \implies w = 7
\]

3. \( 3 + x + 4 = 19 \)
\[
x + 7 = 19 \implies x = 12
\]

So, the completed triangle is:
- Top: 8
- Left vertex: 3
- Right vertex: 4
- Bottom-left: 12
- Bottom-right: Not needed (already solved by symmetry)
- Middle-left: 8
- Middle-right: 7

#### Triangle 2: Magic Number = 20
- Given numbers: Top = 1, Left vertex = 8, Right vertex = 5.
- Let the missing numbers be:
- Bottom-left circle = \( x \)
- Bottom-right circle = \( y \)
- Middle-left circle = \( z \)
- Middle-right circle = \( w \)

The constraints are:
1. \( 8 + z + 1 = 20 \) (Left side)
2. \( 1 + w + 5 = 20 \) (Right side)
3. \( 8 + x + 5 = 20 \) (Bottom side)

Solve each equation:
1. \( 8 + z + 1 = 20 \)
\[
z + 9 = 20 \implies z = 11
\]

2. \( 1 + w + 5 = 20 \)
\[
w + 6 = 20 \implies w = 14
\]

3. \( 8 + x + 5 = 20 \)
\[
x + 13 = 20 \implies x = 7
\]

So, the completed triangle is:
- Top: 1
- Left vertex: 8
- Right vertex: 5
- Bottom-left: 7
- Bottom-right: Not needed (already solved by symmetry)
- Middle-left: 11
- Middle-right: 14

#### Triangle 3: Magic Number = 21
- Given numbers: Top = 3, Left vertex = 3, Right vertex = 6.
- Let the missing numbers be:
- Bottom-left circle = \( x \)
- Bottom-right circle = \( y \)
- Middle-left circle = \( z \)
- Middle-right circle = \( w \)

The constraints are:
1. \( 3 + z + 3 = 21 \) (Left side)
2. \( 3 + w + 6 = 21 \) (Right side)
3. \( 3 + x + 6 = 21 \) (Bottom side)

Solve each equation:
1. \( 3 + z + 3 = 21 \)
\[
z + 6 = 21 \implies z = 15
\]

2. \( 3 + w + 6 = 21 \)
\[
w + 9 = 21 \implies w = 12
\]

3. \( 3 + x + 6 = 21 \)
\[
x + 9 = 21 \implies x = 12
\]

So, the completed triangle is:
- Top: 3
- Left vertex: 3
- Right vertex: 6
- Bottom-left: 12
- Bottom-right: Not needed (already solved by symmetry)
- Middle-left: 15
- Middle-right: 12

#### Triangle 4: Magic Number = 23
- Given numbers: Top = 7, Left vertex = 5, Right vertex = 9.
- Let the missing numbers be:
- Bottom-left circle = \( x \)
- Bottom-right circle = \( y \)
- Middle-left circle = \( z \)
- Middle-right circle = \( w \)

The constraints are:
1. \( 5 + z + 7 = 23 \) (Left side)
2. \( 7 + w + 9 = 23 \) (Right side)
3. \( 5 + x + 9 = 23 \) (Bottom side)

Solve each equation:
1. \( 5 + z + 7 = 23 \)
\[
z + 12 = 23 \implies z = 11
\]

2. \( 7 + w + 9 = 23 \)
\[
w + 16 = 23 \implies w = 7
\]

3. \( 5 + x + 9 = 23 \)
\[
x + 14 = 23 \implies x = 9
\]

So, the completed triangle is:
- Top: 7
- Left vertex: 5
- Right vertex: 9
- Bottom-left: 9
- Bottom-right: Not needed (already solved by symmetry)
- Middle-left: 11
- Middle-right: 7

Final Answer:

\[
\boxed{
\begin{array}{c}
\text{Triangle 1: } \{8, 3, 4, 12, 8, 7\} \\
\text{Triangle 2: } \{1, 8, 5, 7, 11, 14\} \\
\text{Triangle 3: } \{3, 3, 6, 12, 15, 12\} \\
\text{Triangle 4: } \{7, 5, 9, 9, 11, 7\}
\end{array}
}
\]