The Persistence Of Memory

Question

You find yourself on the floor, slowly waking up and unable to remember how you got here. As you slowly come to your senses, you realize that another weird guy has trapped you in another unfamiliar room with what you're sure is another set of bizarre rules that you have to follow.

A previously-unseen speaker starts to produce a voice. "Hᴇʟʟᴏ, ᴀɴᴅ ᴡᴇʟᴄᴏᴍᴇ ᴛᴏ ᴍʏ ʟᴀʙʏʀ--"

You cut him off angrily. "Yes, I know. You've trapped me in a labyrinth, and I have to get out while following some strange rule that you've come up with. That's great. Just tell me the rule and give me the map. And let's make it quick - I have a report that I have to do by tomorrow."

"Cᴏᴍᴇ ᴏɴ, ᴀʀᴇ ʏᴏᴜ sᴜʀᴇ ʏᴏᴜ ᴅᴏɴ'ᴛ ᴡᴀɴᴛ ᴛᴏ ʜᴇᴀʀ ᴛʜɪs? I sᴘᴇɴᴛ ᴀʟʟ ᴡᴇᴇᴋ ᴍᴀᴋɪɴɢ ɪᴛ sᴏᴜɴᴅ ᴏᴍɪɴᴏᴜs ᴀɴᴅ ᴄᴏᴏʟ."

"Look, this is the fourth maze I've been trapped in since yesterday. I've already had a maze that contained itself, one with a ton of letters written on the tiles, and one where I had to follow directions written on dice. Just tell me the rules so I can get this over with."

"Uɢʜ, ғɪɴᴇ. Hᴇʀᴇ ʏᴏᴜ ɢᴏ."

The Memory Labyrinth Rules

You are currently just behind the door in the bottom right.

You must walk through each of the other rooms before returning to the first room. Steps must be to horizontally or vertically adjacent tiles - no diagonal movement.

You may not step on the same tile twice.

In the middle room, you must step on every tile. In the left room, you must step on every tilein the S-shaped region.

Every time you step between two tiles, a line will be drawn between those two tiles' centers. This means that the line will trace out your path along the tiles, moving like a rook in chess.

The most important rule: When your path is completed, every section of the same color and shape must look identical. (If you enter two blue rectangles from the bottom left going up, your paths through them must be exactly the same. You can trace them out in reverse, but the lines left behind must be identical.)

"But wait, where's the exit?"

"Iᴛ's ʜɪᴅᴅᴇɴ ɪɴ ᴏɴᴇ ᴏғ ᴛʜᴇ ᴡᴀʟʟs. I'ᴍ sᴜʀᴇ ʏᴏᴜ ᴄᴀɴ ғɪɢᴜʀᴇ ɪᴛ ᴏᴜᴛ."

The speaker shuts off and you notice that a map is sitting just below the door. You pick it up and unfold it...

(For the color-blind, I have marked all groups of 1×1 squares with symbols.)

(And yes, this is a [logic-puzzle]: logical deduction will be very useful here.) — Deusovi, Jul 16 '16 at 04:19
"look identical" and "can trace them out in reverse" - so without regard to symmetry? (We can't, for example, enter one blue rectangle at bottom-left, and go FL; and enter another from top-right and go FR?) — Jonathan Allan, Jul 16 '16 at 06:32
(Good sign - so busy looking at it upvoted 26 minutes later ;)) — Jonathan Allan, Jul 16 '16 at 06:59
@BmyGuest my guess is not cheating per-se (no no-computers tag), but if you want to "match the conditions given"... — Jonathan Allan, Jul 16 '16 at 07:10
@BMyGuest: No, no "cheating" - move one step at a time. And it's correct that orientation matters for all shapes - if one pattern starts facing north and turning east, all the others of that type must start facing north and turning east (or end facing west and turning south, if you come from the other direction) — Deusovi, Jul 16 '16 at 08:01
@JonathanAllan: Orientation does matter for all shapes - you can't have one 2x3 rectangle have an upside down version of another's (unless the pattern is rotationally symmetric already). — Deusovi, Jul 16 '16 at 08:06
@Deusovi I was thinking of making a program/tool that could be used to create and solve puzzles of this kind. Would that be ok? Also, I would love to integrate those other puzzles you mentioned. I couldn't find them but they sound interesting. — user64742, Jul 17 '16 at 05:23
@TheGreatDuck: Those other puzzles are the ones here. Also, puzzles like this are meant to be solved manually. Many of the shapes are specially built for you to make certain logical deductions rather then brute-forcing it. I can't stop you from writing a program, but I would recommend against it, since it would ruin enjoyment for both the creator and the solver. — Deusovi, Jul 17 '16 at 05:30
@Deusovi I meant write a program that you can personally solve it with. It would just be a player you could move whilst having the paths autotrack themselves. Does that make sense? I won't automate solving. I'll just make a minigame of the puzzle. — user64742, Jul 17 '16 at 07:02
@TheGreatDuck: Ah, I see - I say go ahead! It'd be very useful for both constructing and solving any puzzles involving movement on a grid. — Deusovi, Jul 17 '16 at 07:31
@Deusovi: that's what I figured and it might be nice to feature the puzzles in the competition. I will probable post it in meta when I am done. — user64742, Jul 17 '16 at 07:43
+1 for an interesting puzzle, and especially for taking steps to accommodate people with limited color vision. But next time I suggest that you avoid marking tiles with ▘ (▅), as that leads to confusing terminology like “X square,” “diamond square,” and “square square.” Consider color gradients, like this. — Peregrine Rook, Jul 26 '16 at 01:40
@PeregrineRook: Alright! I'm not very familiar with image editing tools, so I just used the pencil tool.. Next time I make a puzzle like this (and there probably will be a next time), I'll try to make distinguishing marks easier to understand. — Deusovi, Jul 26 '16 at 01:43

score 33 · Accepted Answer · edited Jul 26 '16 at 01:14

33

Solution:

1. I started with the left room. The key for this room is

The single-tile square marked with "X" (pale orange/light brown) being placed both at the lower door (i.e., against the bottom wall) and against the left wall is very restrictive.
Adding in the only way through it we need to go into or out of the pink C in a couple of ways: vertically at the bottom left and therefore also vertically at the top middle. After that, this seems to be the only way.

Like so

2. Next note that

the only way through the top door of the bottom right room is up from the middle of the pink C.
We can then extend lines the only way possible and find the turquoise L's very restricted.
Also, the green 1by2 rectangle is against the top wall of the top room, so we cannot enter one from above and there is such a situation in the bottom room to avoid, which completes another part of the turquoise L.

Like so

3. Next note that

A 2by2 square is against the left wall in the middle room and the 1by1 square (marked with a black square ▘ – colour?? beige??) in the bottom room then restricts the turquoise L further, which, in turn, requires that we pass through the top of the beige tile.
Adding that to the top room then restricts the dark-grey $\zeta$, then the light-grey T and hence in turn the 1by2 green rectangle.
Adding the light-grey T line to the bottom room then restricts the orange (diamond) single-tile square.

Like so:

4. Then

The only options for the orange (diamond) single-tile are to exit up or to the right, but up is not possible since that implies going into the 2by3 blue rectangle in the bottom room from the bottom of the bottom-right tile, which further implies, from the middle room, that we must enter the 2by2 from the top of its top-right tile, but this is not possible in the bottom room.
Adding the implied paths then leaves only one option for a) the bottom left most white tiles in the middle room, and b) the white tiles near the door from the bottom room into the middle room.

So we have

5. Now

in the top room, the light-grey T is against the bottom wall, so we cannot enter it from the bottom of its bottom tile, so from the left-most dark-grey $\zeta$’s bottom tile we can go left or right. However, going left implies we would go left into the top tile of the light-grey Ts, but the top-left light-grey T has that path blocked by a green rectangle.
Furthermore, we cannot go up into the right tile of the T since it would imply going up from the middle of the top-left and top-right $\zeta$s, but the third $\zeta$ would then be passing up into the bottom of a T. Since the path to the right from the right tile of the T is blocked near the centre of the room, the only option remaining for the right tile of the T is left.

We need to have a path out of the beige 1by1 square. It cannot be to the right, as one is against the right wall, and it cannot be to the left, as that would imply we go into the bottom of the two right most tiles of $\zeta$s (at the right most one) which in turn implies we come out of the top of the light-grey Ts to the left (into the left most $\zeta\:\!$) which in turn implies we go into the topmost beige 1by1 from the right. Hence we must go into the beige tiles from below.

The only remaining way into the light-grey T is from the top of its top tile, as we cannot cross the bottom (it would lead to a forced loop at the bottom right of the top room), we cannot enter from the left of the top (it would imply going right out of the top beige 1by1), and going in at the left of the middle tile implies crossing the bottom (due to a then otherwise over-connected white tile below the top beige cell). This completes the top room (unique up to the path through the top-right most nine white tiles, the top-left most four white tiles, and the four white tiles just beneath the beige 1by1 on the right wall).

The 2by2 square is restricted to having to traverse its bottom right tile, which could be done in two ways; however, we cannot go into that tile from below as it won't work with any arrangement of blue now possible in the middle room. Entering the tile from the right is the other way, and due to the white tiles we must walk across in the middle room it is clear the blue must be two parallel horizontal lines, which further restricts the red rectangle and finishes the middle.

Like so

6. Almost finally...

The two pairs of paths near the right wall (of the lower-right room) must be joined (the top blue one cannot be to the exit or the middle two would join and create a loop).
Then the start of our journey must go to the bottom orange (diamond) 1by1; and it must go to the beige (square) 1by1 (since otherwise there'd be a loop between the middle turquoise and the beige).

Like so

7. Finally

We could complete our route in two ways (turning left [up] or right [down] after traveling from the left room into the bottom room) without breaking any rules. Let us assume there is a door as stated, and not a tunnel; then we should take a left rather than a right, giving the solution at the top of the answer.

8. For completeness

the "tunnel" route would be

edited Jul 26 '16 at 01:14

Peregrine Rook

4,880
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26
40

answered Jul 16 '16 at 08:08

Jonathan Allan

21,150
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109

Nicely done! That is indeed the unique solution. (You didn't have to step on every tile in that room, though - the only room where you do have to do that is the middle one.) – Deusovi Jul 16 '16 at 08:10
This is an important statement, because it may influence the pink large fields in the other rooms... – BmyGuest Jul 16 '16 at 08:13
@BmyGuest: I can assure you that it does not. Jonathan's solution is the only one there is in that room (and it's also the correct one). – Deusovi Jul 16 '16 at 08:14
1

@BmyGuest Note that I filled in the X, the pink C and the white tile which would be fixed in the lower right room. – Jonathan Allan Jul 16 '16 at 08:16
@Deusovi BTW, I love the 'stepwise logic' your maze requires. Building puzzles which 'work' that way is not easy, but they are the most rewarding to work on! – BmyGuest Jul 16 '16 at 08:16
@BmyGuest: Thank you! I'm planning to make a lot more puzzles similar to this - this was my first attempt at this genre, so it may be a bit easy. – Deusovi Jul 16 '16 at 08:18
@BmyGuest, Jonathan: THERE IS A MISTAKE IN THE TOP ROOM. I'm fixing it now - I'll reupload shortly. – Deusovi Jul 16 '16 at 08:24
I've edited your images to the updated version of the maze. Sorry about the mistake! – Deusovi Jul 16 '16 at 08:31
Hah, thanks, just did too, went to upload and saw edit :) – Jonathan Allan Jul 16 '16 at 08:33
1

I think it will be useful to mark which tile boundaries cannot be stepped across. For example, for the 2x2 tile, you can't cross the left of the top-left tile because there is a wall next to one of them in the middle room. Also, the left of the red 1x1 tile (marked with a diamond) can't be crossed because the path that passes left of the tile. And when those uncrossable boundaries are adjacent to other regions, you can mark them in as well (the two here affect the grey T-shaped region and the green 2x1 region). – Volatility Jul 16 '16 at 08:44
@Volatility - could be, am just keeping the info in my head :) – Jonathan Allan Jul 16 '16 at 08:45
how do you reason that the beige 1x1 must have a path on it? could it not be untouched? – Jasen Jul 16 '16 at 11:48
1

@Jasen - a 2by2 is against the left wall in the middle room, so, due to the effect on the bottom room's 2by2, we cannot pass from the white cell above the beige 1by1 into the turquoise L, the only route left is down into the beige 1by1. (but you made me think about it!!) – Jonathan Allan Jul 16 '16 at 12:07
1

Gah, I knew I would've made a mistake somewhere. The intended solution had the square just to the right of the exit traversed as part of the second time through; at the very beginning, you would go through the orange tile, then go up to the green. You'd only pass through the yellow and blue that you have near the start at the very end, and that gives a unique solution for the exit door. Oh well - nicely done! :D – Deusovi Jul 16 '16 at 13:48
1

First time glitch only to be expected. Good puzzle!! – Jonathan Allan Jul 16 '16 at 13:54
1

@Deusovi I can't tell you how glad I am, that you've made a mistake! (Because then I'm not the only one. I hate it when it happens, though. Great puzzle! Deserves more upvotes.... maybe add a silly number-sequence-riddle-to-guess to increase popularity ?!? g) – BmyGuest Jul 16 '16 at 17:17
So which one is the solution? The top "Final solution" is different from the bottom "complete" solution because from the left room you can either go up or down. – justhalf Jul 19 '16 at 06:22
@justhalf - the top one is the expected final solution, unfortunately there was a slight glitch in the Matrix when Deusovi created the puzzle. I chose the top one (turning left and ending bottom-left) because there could not be "doors" in the section reached the other way. There was a change made early on during the puzzle's life to where the top green in the top room was to fix a problem, but (I believe) this meant the path in the bottom room changed to make the very start different and hence lead to the two solutions (I also believe the aim was to have a single tile where the door was). – Jonathan Allan Jul 19 '16 at 06:41
@JonathanAllan, I see. Well, then I think we can add one more single orange (diamond) tile on top of the gray tile in the first room to make it a single solution as well as pointing to specific wall section (the exit could be a stairs down =D) – justhalf Jul 19 '16 at 06:58
@JonathanAllan Also, I'm not convinced that your second step in deducing the path for the turquoise L-shaped tile is the only solution (at that point, at least, I haven't traced the implications). Can it not be that from the middle tile of the magenta tile, it goes down to the L-shaped tile, then goes left instead of right? It might not be possible later on, but at this point I don't see why it should be the way it is now. – justhalf Jul 19 '16 at 07:05
1

@justhalf adding the single orange diamond you suggest would work, but it would make deducing the path through the T's far, far easier - that took me quite some time. – Jonathan Allan Jul 19 '16 at 07:14
@justhalf - I can't quite see the reasoning I had - I posted as I worked through it so it should have fallen out somehow, maybe I'm too tired. – Jonathan Allan Jul 19 '16 at 07:39

The Persistence Of Memory

The Memory Labyrinth Rules

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