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The problem asks to find the total number of paths from point \( A \) to point \( B \) in the given figure, where only downward motion is allowed.

The shape consists of six small equilateral triangles, and you can only move along the edges downward, either diagonally or vertically.

### Step-by-step analysis:

1. **Understand the grid structure**: The figure resembles a hexagon divided into six equilateral triangles. Moving from \( A \) to \( B \) means you're only allowed to move downwards either to the left or right, following the triangular grid structure.

2. **Possible moves**: 
   - From \( A \), you can move to three adjacent vertices in the middle row.
   - From these middle points, you can then move to three different points in the bottom row, where \( B \) is located.

3. **Paths from top to bottom**:
   We can start counting the paths systematically from top to bottom, treating each vertex as a "junction" where decisions are made:
   - From \( A \), there are 3 paths to the middle row.
   - From each of the 3 middle row vertices, there are 2 downward paths leading to the bottom row, so a total of \( 3 \times 2 = 6 \) paths to \( B \).

Thus, the total number of paths from \( A \) to \( B \) is **6**.

Would you like further details or have any questions about this?

### Related questions:
1. How would the solution change if both upward and downward motion were allowed?
2. What would the total number of paths be if the figure had more levels of triangles?
3. How many total possible vertices are in the triangular grid, excluding the starting and ending points?
4. How could this problem be generalized to a larger triangular grid?
5. What if lateral (horizontal) movement was also allowed?

### Tip:
In problems involving paths or grids, using combinatorics or dynamic programming techniques can be very helpful in counting possible movements.

In the plane figure, only downward motion (movement leaving you relatively lower than where you were) is allowed. Find the total number of paths from A to B.

Discover how to calculate the total number of paths from point A to point B in a hexagonal grid with only downward motion allowed. This problem uses combinatorics to explore path counting and is suitable for students in Grades 10-12.

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