Table-game problem that a bit looks like Knight's Tour problem

Question

Given a table containing some instruction in each cell ($L$ - $U$ - $R$ - $D$ = Left - Up - Right - Down). The coefficient of the instruction means the number of steps (for example, $2L$ means you should move two squares to your left). The task is to find which cell you should start from if you are to visit all cells and finish at the blank square. Also, find the value (instruction) of the cell with $"?"$. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline 2D & ? & 2D & 3R & 1L & 2R & 2D & 1D & 1D \\ \hline 1U & 1R & 2R & 1D & 1U & 2L & 5L & 2L & 8L\\ \hline & 2U & 3R & 4R & 4R & 2U & 5L & 3L & 2U\\ \hline \end{array} $$

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Update: Solution

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Well, at first glance, this reminds me of Knight's tour problem. However, I don't really know if the current problem restricts the number of visits to each cell, as well as "Knight's tour" does.

I have done some further research on the "Knight's tour" problem on $3 \times 9$ grid and found this page representing the following picture:

Despite these two problems have something in common, I can't reduce the current problem to KT (since the trajectory of steps is not (always) the same).

Any help is appreciated.

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Attaching the picture of the problem anyway:

Answer 1

Start from the blank square as the end because you can't leave it. Look for a square that can go there. You have to come from the top left corner. Then to get there you need to come from the middle left. Keep going backwards. You should get to a square you can't get to unless it is from the question mark square, which tells you what to put in there. Then go until you have gone through all the squares and you know where to start.

Answer 2

Here is the order of the steps to be taken ($i$ in the table stands for the $(27-i+1)^{\text{th}}$ step).

As @RossMillikan mentioned in his answer, we should start from the end (it means we assume the movements start from the blank cell marked with $1$ in the table below): $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline 2 & \color{blue}{18^{\boxed{1R}}} & 17 & 22 & 23 & 14 & 21 & 13 & 5 \\ \hline 3 & 26 & 25 & 10 & 24 & 11 & \color{green}{27_\text{start}} & 12 & 4 \\ \hline \color{red}{1_\text{finish}} & 19 & 16 & 9 & 7 & 15 & 20 & 8 & 6\\ \hline \end{array} $$ Hence, we observe that $ \ \ \boxed{?}=1R$ (marked as $18$ in the table). And the starting cell seems to be the cell on the $2^{\text{nd}}$ row and $7^\text{th}$ column