# Twilight Zelda Guardian Puzzle

I'm playing a game (see here), and in it is a puzzle, which I've made a TIKZ graph seen below for illustration:

$\hspace{2.5cm}$

There is one player. The player ($\mathscr{L}$) moves from node to node limited by the directed arrows. As the player moves, $\mathscr{G}_{\Huge\cdot}$ chooses to move in the opposite direction while $\mathscr{G}_{\circ}$ moves in the same direction as the player, all three step simultaneously. If the player makes a move that would cause the guardians to jump off the playing board they remain in place, and the player keeps his/her move. Also, if the player makes a move that would cause him/her to jump off the playing board, then he/she remains in place. The object of the game is to guide the guardians to the circle-cross ($\bigoplus$) nodes so that they individually land on each in whatever order and whatever way simultaneously (see linked video for clarity). A 12-step solution to solve the puzzle has been uncovered, however, I am seeking the shortest possible step solution to this puzzle.

## Special rules:

Game ends if above move is performed.

Guardians remain in place and player keeps his/her move.

Game ends if above move is performed.

$(\mathscr{L}\rightarrow\mathscr{G_\circ}\rightarrow\circ)$ OR $(\mathscr{G_\circ}\rightarrow\mathscr{L}\rightarrow\circ)$ conga allowed.

• A computer brute force search is probably appropriate to verify whether a $\le 11$ move solution exists, considering there is only $4^{11} = 4194304$ possibilities. May 11, 2015 at 23:14
• A* search probably.
– Liz
May 11, 2015 at 23:21
• Wait, according to your diagram, the empty circle follows Link's moves, while your description says the solid follows Link. Which is it? May 12, 2015 at 1:33
• Is there such a thing as a cross-site duplicate? May 12, 2015 at 3:04
• If Link would make a move that would take him off the board, do the guardians move to follow what would have happened, or not? May 12, 2015 at 10:25

The puzzle as shown in the video is different from the one you have presented here. In the video, the top guardian moves in the opposite direction and the bottom guardian moves in the same direction, while as written here it is the opposite.

A brute force search turned up no solutions shorter than 12 steps for the puzzle with the top guardian moving opposite the player.

There are three pairs of solutions in 12 steps, but one requires the player and a guardian to pass through each other as they switch spots, and one requires the guardians to pass through each other. The solution that does not require either of these is LDUUURRDDDLU (and its reflection). The other two are LUURURDDDDLU and LDRRULUUUURD (and their reflections).

If the top guardian moves in the same direction as the player, there is a 9 step solution (LDLDDDDRU), which is unique except for reflection.

I'll have to pull out my Wii tonight -long time since I've said that- to test this out and verify that I am keeping all the rules in check and not getting myself stomped by stone, but my early scribblings amounted to what I think will work in 11 steps.

DULLUULDDRU

• It's not clear to me whether this works. On the eighth step, Link is one space above the same-direction guardian, which can't move any further down. What happens if Link tries to move down? If Link and the same-direction guardian don't move, and the opposite guardian does move, then I think this is a valid solution. If Link moves and the same-direction guardian doesn't, then Link is squished. Here is a recreation. May 12, 2015 at 19:04
• @Kevin That's where I'm hung up on it, too. During my first attempt, I realized I was ending up with 12 steps (as Homer said "anything that takes 12 steps isn't worth doing"), so I looked to cut wherever I could. It's hard to visualize on paper or by pointing at a computer screen, but your recreation was spot on. I'll have to use that resource in the future. May 12, 2015 at 21:53