Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards?
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To determine the number of paths from point \( A \) to point \( B \) on the given grid, we need to consider the constraints: the paths must go along gridlines only, not pass through any point twice, and never go upwards.

1. **Understanding the Grid and Constraints**:
   - The grid is a 6x6 grid.
   - Movement is restricted to rightward or downward directions only.
   - No point can be visited more than once.

2. **Counting the Paths**:
   - Each path from \( A \) to \( B \) must consist of exactly 6 rightward moves and 6 downward moves.
   - The total number of such paths without any restrictions can be calculated using combinations. Specifically, we need to choose 6 positions out of 12 for the rightward moves (or equivalently for the downward moves).

3. **Calculating the Total Number of Paths**:
   \[
   \text{Total number of paths} = \binom{12}{6} = \frac{12!}{6! \cdot 6!}
   \]

4. **Simplifying the Calculation**:
   \[
   \binom{12}{6} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}
   \]
   \[
   = \frac{665280}{720} = 924
   \]

5. **Considering the Constraints**:
   - The initial solution provided seems to have misunderstood the problem constraints. The paths must be counted considering the grid structure and the non-repetition of points, but the combinatorial approach already ensures that no point is visited more than once and no upward movement is allowed.

Therefore, the correct number of paths from \( A \) to \( B \) is given by the combinatorial calculation.

The final answer is \( \boxed{924} \).