Consider the field with infinitely many boxes, "S" means start, "D" destination, and I already found a way to move a $1\times 2\times 4$-cuboid (as you can see on the right at this picture) from the start position to destination trough tilting this cuboid. My question:
If I choose an arbitrary startposition S and destination D, is it always possible to tilt such an $1\times 2\times 4$-cuboid from S to D?
Regards
Edit:Ok first question is now clear, thanks to achille hui. What is if the field has only 10x15 units? How to find out a startposition and endposition such that there is no possibility to move the cuboid from start to end? I could try every possibility but it is an overkill. the starting point is always standing up and the end standing up again and the field now has 10x15 units. Maybe there are some problems in the corners. Could you give me a strategy/hint?
Yes, it is always possible.
The key is one can move the initial "standing up / horizontally aligned" position for one unit with only 6 flips in both horizontal and vertical directions (See elementary moves at end).
Using them, one can move the cuboid anywhere one want while keeping it "standing up / horizontally aligned". If the end configuration is vertically aligned like what was shown in the question, one can align it with 3 extra steps.
Elementary moves
Label the 4 sides of the base of the cuboid by E(ast), N(orth), W(est) and S(outh). We can index the set of possible operations on the cuboid using strings formed from 4 alphabets: $E, N, W, S$. For example, one can use the string "ENW" as a shorthand for following chain of operations
As shown in the two pictures below, the strings "ESWWNE / SWNNES" correspond to a unit move in horizontal and vertical direction respectively.
(ESWWNE - a horizonal move to the left for one unit)
(SWNNES - a vertical moves to the top for one unit )