# Guide the dots to land on the portals at the same time

Question:

First, I'll define the top left dot of the empty middle square as dot 1, and the bottom right dot of the empty middle square as dot 2. Left being <, right being >, up being ^, and down being v, what set of these moves can guide the two dots to land on the portals at the same time?

Rules:

(0) The two side-by-side dots are the portals.

(1) You are the yellow dot (bottom left of entire grid).

(2) You can move (one cell) in all directions except diagonally.

(3) When you move, dot1 moves (one cell) in the opposite direction.

(3) When you move, dot2 moves (one cell) in the same direction as you.

(4) Both dots move when you move (in their respective directions).

(5) If dot1 and dot2 are facing each other, and you move, they stay in the same spot, and you keep your move.

(6) If dot1 and dot2 try to move onto the same spot, they stay in the same spot, and you keep your move.

(7) If you and dot1 try to move onto the same spot, you die, and the game resets.

(8) No one can move off the grid or on the empty cell.

(9) If you and dot2 are next to each other, and you want to move on the spot dot2 is on, dot2 will move off it in that direction and you will land on it.

(10) If (9) happens with dot2 facing the edge of the grid or the empty cell, then no one moves.

(11) If dot1 or dot2 would move off the grid or onto the empty cell because of your move, they remain in place and you (and not necessarily the other dot) keep your move.

(12) If you try to move off the grid or onto the empty cell, no one moves.

Remember each time you move dot1 and dot2 move, excepting the cases where it's not allowed.

• Is this too complicated? – user39732 Aug 11 '17 at 22:42
• It's very much like the Goriya seen here. – user39732 Aug 11 '17 at 22:49
• I'm quite sure this rule set is comprehensive for all scenarios, however, I could be wrong. – user39732 Aug 11 '17 at 22:53
• Can they move onto the portals during their time as long as at some point they move onto them together, and if so do they go the other one? – boboquack Aug 11 '17 at 22:56
• If you are you changing the empty cell(s)' arrangement, consider putting that as a different question then, since it would fundamentally change the answer – boboquack Aug 12 '17 at 1:17

You can:

Not do this

How:

Note that in the initial configuration, both dots are at a 180° degree rotation of each other around the centre square.

Now:

You cannot interact per se with the dots, all you do is give them instructions, one of which is the 180° rotation of the other (since it is the opposite direction)

So then:

Each dot must remain at a 180° rotation of the other around the centre square, because nothing can break the symmetry

And finally:

Since the two portals are not at 180° rotations of each other around the centre square, it is impossible to get both dots on a portal at the same time.

In fact (as @WilliamNathanael pointed out):

The absence centre square doesn't even matter - neither of the other dots can move onto it since then both would move onto it, which is explicitly forbidden by the rules.

• @uwnojpjm Yes, because if one is hitting the edge/the central cell the other must be too because they are at a 180° rotation of each other – boboquack Aug 12 '17 at 0:45
• So, the vacant center has no importance, I believe? – William Nathanael Aug 12 '17 at 1:37
• @WilliamNathanael Exactly – boboquack Aug 12 '17 at 1:38