Mathematical Games Hosted by Ed Pegg Jr. [Episode 9: Multistate Mazes]

Sa. Welcome everyone. I'm Ed Peg Jr. And we're going to wait a few minutes. I'm going to be talking about multi state mazes or logic mazes. And welcome to my chat. This is another in a series of discussions of mathematical games. We're still waiting a little bit for, for people to, to line up their schedules, but I'm ready to go here and still watching people trickling in here. Let's see here. Make sure I've got everything else ready and things are looking good here. Okay, I'm Ed Pink Jr. And for 25 years Martin Gardner ran the mathematical games column for Scientific American. And this, this series that he did mathematical games was very popular and influenced many different mathematicians, including myself. I've been revisiting some of the items that Martin Gardner did for Scientific American and today I'm going to focus on one puzzle type that he introduced in 1962 by Robert Abbott, multi state mazes or logic mazes, which are now very popular. So let's talk. Okay, so here's the demonstrations project which I run. We've got 66 items related to mazes and a few of them happen to be logic mazes. And I'm going to shut up Alexa over here. All right, Alexa is unplugged. Sorry about that. We've got many different types of mazes at the demonstrations project and here are a few of them and feel free to go to demonstrations.wolfram.com if you'd like to see some of them. So first let's talk about the, the simplest types, a labyrinth and a normal maze. So a labyrinth typically has a single path. So here's a classic labyrinth and there's only one single path without branches from the entrance to the end. Here a maze is a collection of paths branched and loops. And this is one of the demonstrations that can generate mazes of different sizes. Mazes can be designed so that they show images of various types or they can have different things added to make them more complicated. But usually it's just a simple tree graph of some sort or a graph with some various loops, but there's various wriggles in it to make it more complicated. And it's possible to make a, a maze much more complicated if you add special rules. One thing with, with corn mazes that's been found usually there's not too much you can add, but they found if you add a bridge somewhere within a corn maze that, because it's now not planar, that tends to throw people off a lot because the, as human beings we're kind of expecting a planar set of rules when walking around. When you add that extra dimension with a bridge and you're going over and under a bridge, sometimes that can confuse people and make the maze more complicated. So you'll frequently see bridges within foreign mazes. So here's how things changed in 1962. The farmer goes to Market maze. And there's a set of rules here. You need to go through the town of Floyd's Knob and get through this grid of air, this grid of the town, which has various arrows indicating the traffic laws. And you basically. So let me try this here. So we go through here. We just go straight. Go straight. We take this curve down, up, take this curve over here. We'll take this loop. We can't go this way, so we'll just continue this through here. Maybe go straight and up. And I think we're back into a loop here. So there's various loops and traffic and traps where you basically, you wind up in a loop that you can't get out of. So that's one of the tricks within a maze like this. Within some logic mazes, you can be led into loops or various other traps that are inescapable. This was developed by Robert Abbott, who's kind of the pioneer for logic mazes. He did many different maze types for Games magazine and for mathematical games, which have now become sort of the classic types for logic mazes. He also ran the site logicmazes.com which is still active. You can visit it and see some of the other mazes that Robert Abbott made. When I did the show Numbers for a while, I was the advisor. We had one episode that was devoted to logic mazes, and we used some of the various concepts of Robert Abbott's mazes in that episode. And it's considered one of the best episodes of the show. So how do we solve this particular puzzle? Here's how I solved it. I started with a grid graph. And with that we can pick out their various edges of the graph and the vertices. And the arrows within this graph basically go from one end, one edge to another edge. So here's one way of. Of coming up with all the possible arrows. And since this is a grid graph with two additions, I basically added these extra points. So here's all the 192 possible arrows that could be in this town. And basically by hand, I selected which of the arrows are the allowed turns. And this, this list of yes. Items is the allowed turns. Within the. Within the maze, here's a possible path through. And we have the graphic for the maze. And we can make a Mathematica version with, with B spline curves. So here at the bottom, we have the original maze in color. And next to it is the mathematical representation of the maze where we see all the various directional arrows and whatnot. And if you check, you'll see that I've closely depicted the maze as given by Robert Abbott. So here's, here's a mathematical representation of the maze given as a series of, of arrows. So here's the maze again. And now that we've built it, we can come up with the graph behind it. And it looks pretty terrifying. Here's the graph here. Here's a layered digraph embedding. It's a really complicated graph. I haven't come up with a way of making it look any much nicer. It's not planar at all. So this is a way of doing it. And the numbers here, this is basically means you're going from vertex 13 to vertex 14. You have to keep track of your direction while going through this maze. While you're going through, you could go through this way, through a train, or you could be going this way. There's four different directions you could be going through. And for this one, there's basically six different directions you could be going through. You could be going this way, that way, up, down, left, right. It's. It's possible that you could visit one of these vertices up to, I guess this is the maximum you could visit this one up to eight times in the course of solving this maze. So let's take a look at the actual path through. And here it is. Here we start over here we go straight through 16, 17, 18, loop down through 19, go straight down. And then we take this bottom corner and we start going up. And here the path diverges. We're not sure which direction this green line is going to. However, up here, this is one way. So we know what we have to be going through the. This green loop this way. So from 10, 10, 15, 20, 25, 24, 23, 18, 13, 14, 15, 10. Now we're going the other way through five. We came from this direction, so we need to stay in the bottom here through 4, 3, 2, 7. So 17, 22, 21, 16, 11. And here we, we take the down loop 6, 1, 2, 3, 5, 8, 7. We come back up this way. 11, 16, 21, 22, 17, 12, 7. Then we branch this way. 8, 9, 10. And then through and out. So it's a very complicated maze. We end up going through stage seven in three different states. So this, this was kind of an amazing idea back in 1962. So Samwise and Frodo talked about getting lost in Lord of the Rings. And the money quote is. This looks strangely familiar. That's sort of what, when, when you're making a maze, that's what you want somebody to feel. You want them to come to a spot where they think they've been there before. Because that's kind of the instant way that a person starts feeling lost when they've come back to the same place. And it feels like they're going in circles. With these logic mazes, you get that a lot faster than with a typical maze because here you're returning to the same spots over and over again. It's just that your state has changed. Here's a simpler version of Farmer Goes to Market that Robert Abbott made for me about 20 years ago. Here we have the. A simpler version. And here's the state diagram of how to get through it with the various loops and whatnot added on. So for this One, you basically 4 East, 4 E. So basically you go from 4 up, we're here to 1, then to 2 through 6 through 5, 9, 10 east and then out. When you, when you make things this simple, you can't make. You can't build them as much complexity. But there are quite a few loops added on that you can get, you can get trapped in. If you don't know where you're going. Going through this maze, Here's another type of rule that you can add. You can force alternations. So without that rule, we can just go from start to finish. We can just ignore everything. But if you have to alternate going up and down, it's much trickier. So we go up, go down, both of those would be down again. So we have to go to C. We went up to C. So we go down the D, up to E, and so on. So we can solve this by making a chart of all the up moves that are possible. And it was while discussing something like this that Martin Gardner made the comment that up upside down is down if you rotate it, which I thought was always interesting. Here's what the graph looks like when you force alternations of ups and downs. And solving it is somewhat routine. Once you've got the map, you start at A and then you get to the finish after U. So basically up through here and then down through this way. But there's a. A possible sidetrack you can go to get lost Various Loops and, and dead ends that you can end up at. However, now that we've coded this up, it's possible to tweak the maze to make it possibly difficult. More difficult. So up here, imagine putting a card N under this region, what might happen, and we're going to be placing it under start A and G. So that makes a new list of ops. And then when we rebuild the graph, we get this more complicated object. With a normal maze, if you do a tweak, say add one wall, you're not going to make the maze all that much more complicated. But with a logic maze, making a small change can vastly change the maze. That's one of the reasons logic mazes are more interesting than a typical maze. The ability to tweak things to make things more complicated is much easier. And having a program to help with it can really help hone a maze down to something which is at the right level of control complexity to really be satisfying for solving. So here's another maze that Robert Abbott did. Meteor Storm was in Games magazine. And the idea is you have two different people trapped in a spaceship and they need to reach a place together in order to solve a problem. So here we have people on two crew members on room D and room F. D and F are the locations. And one of them can open up hallways of the color for the room that they are on. So since they're both on blue rooms, one of them can open up the blue corridors and one of them can then go through a blue corridor to another room. So for example, from this position, the person in D would stay in D while F goes to room I in the blue corridor. And then I could either go back to F, which wouldn't be all that productive, or I could stay there. The the brown corridor or tanned corridors now opened up. He can't go through those corridors since he basically has to stay in the room to keep them open. But now D could go to room B. And basically they move one at a time. Either one can move at any point, but they both need to reach room F. And what's. What's called this, I call it a covering maze. Basically one person is giving cover to the other person. And when the TV show Numbers was run, CBS had one of these mazes on the CBS site for promoting Numbers. So here's a solution for this, for this maze. So DF di bi BH and then it gets a lot more complicated. And finally down at the end, they both reach room F together. And we can tweak this maze a Lot. Basically we have seed values for coloring the rooms and corridors and Mathematica will just basically solve what is the hardest path and we can add a further tweak. Say we sort of like this one. Well, what happens if we just change one corridor? What happens? So this is one way of adding a tweak to the graph, to the, to the maze to see what happens. With a combination of large tweaks and small tweaks we can build balance out getting a complicated maze and we can change this up. We can change the underlying graph for how the rooms are connected and get a completely different maze. The bigger the underlying graph, the more complicated the the maze can be. And if we increase the number of colors, things can, can really get tricky. Let's see what we can get here. Let's change the, the graph again. So we can make a lot of different variations on this type of maze with this, with this covering maze demonstration. I'm going to be adding this one to the demonstration site soon. So you can, you can play around with this one if you'd like. And one with each of these. You can take a look at the solution graph for what the solution looks like. Here's a maze by Hiroshi Yamamoto. 1, 2, 3. You start on S, you can move one square followed by two squares, followed by three squares in the direction. And then you start over again. 1, 2, 3. And you basically have to get from S to F. So I've got code here to solve this. One thing to note is if you move 1, then 2, then 3, you'll wind up on a square with the same parity. So when I set up this maze, I also colored the squares in to show their parity. And here's where I set up the maze. The various paths. Basically you need to be able to reach A, from one square to another square within those three moves. First going 1, then 2, then 3. So can we get from say 1 to 6? We have to move 1 first and then we have to move say 2 in one of these two directions and then we have to move 3, but those will lead us away from 6. So you can't go directly from 1 to 6, but you can go from directly from 12 to 10. You do that by going towards 10, then 2 away from 10, and then 3 to 10. So 1, let's see, 12 to 10, we'd go 1, 2, 3 for 12 to 10. So since you have to act physically, be able to make those moves, it's, it's somewhat tricky to, to come up with the various paths Needed, but quite doable to do. And then this code here will find out how to connect these up. And here's what the graph looks like. We basically need to get from one to 11. So here's 11 here and one here. You basically need to figure out a way of getting in this graph from 1 to 11 and find path of that. So these are two of the shortest paths. So we get 1 to 12. That's just basically 1, 1, 2, 1, 2, 3. From 12 to 10, which we've done several times now. 1, 1, 2, 1st, 2, 3. From 10 to 3, 1, 2, 3. From 3 to 7. 1, 2, 3. And then from 7 to 11. 1, 2, 3. And we solve the maze. So that uses this root through the graph. One thing to note is that each edge in this graph is three different moves. 1, 2, and 3. So this, this maze was quite popular. So Eric Friedman added onto it, you need to move 1, 2, 3, 4, and 5. So here again, you start on S and you have to get to F by using a series of moves of length 1, 2, 3, 4 and 5 in in straight lines. And we basically solve this one the same way we set up the maze. And here's a graphical version of it. This is a, a short path which we eventually get to. And the graph is much more complicated since there's not a parity issue with this maze. 1, 2, 3, 4, and 5. You'll land on a square with a different parity. So it's possible to visit every, every square in the maze within one cycle. And here's what the graph looks like. It's somewhat more complicated than the other one we use. Find path again. And with those paths, we want to start with getting from 1 to 16. Let's see. 1. 1, 2. 1, 2, 3. 1, 2, 3, 4. 1, 2, 3, four, five. That brings us from 1 to 16. Then from 16, they're 37. 1. 1, 2. One, two, three. One, two, three, four. One, two, three, Four, five. From 37 to 36. One, one. No, that's not going to work. One, one, two. That doesn't work either. One. One, two, one, two, three. I'm not sure how to get from 37 to 36 quickly. I'm sure it's possible, though. The code said it's possible. Let's see. 1, 1. Let's see 1, 1, 2. 1, 2, 3. 1, 2, 3, 4. 1, 2, 3, 4, 5. Yeah, I need to look at that. But I'm sure it's possible one of these will work. So once you have the, the Path you can, you can finish the maze and you could also add squares to the maze and just redo the code and it will show you whether you've made the, the maze more interesting or not. Another type of developed by Robert Avenue initially was the tilt maze. Basically you have a object that you need to get to a certain location and you can tilt it left, right, up or down, but the object will move all the way in one direction. So if you tilted left, it would go to here. These maces are subject to being solved backwards, so they're not quite as interesting. Also, you'll visit a state multiple times. For example, you can slide through this square in four different directions. But the positions where you stop, it's basically a static stop. You don't need to worry too much about direction at that point. So basically if you look at this as a series of stops, it's a simpler type of maze and we can sort of figure out that get to here, we need to stop there, which means stopping here, stopping there, going through here, either stopping here or here. Let's try stopping here. That would require a stop there, probably a stop here, stop there, looping around, a stop there, stop there. And so from that we basically had the solution. You'd basically begin here, that way, stop, stop, stop, stop, stop, stop, stop, stop, stop, stop. Through here, stop, stop, stop, stop, stop. And then down. And that solves the problem. I need to come up with code for solving these. The original rolling block maze was made by a friend of Robert Abbott, Richard Tucker. You start with a one by one by two cuboid on start. And these walls here are positions where you cannot topple the cuboid onto. So from the start here, if you imagine this cuboid standing on, in, on start, it has to basically topple this way. And then you'd have a one by one by two slab over these two squares. And from there you could say roll it up to these, this position here. So the cuboid has basically five different positions. It can be or three different positions. It can be on end, it can be horizontal or it can be vertical. And you wind up going through various places in dismayed multiple times while you're rolling around. So it's a multi state maze or a logic maze. Another Robert Abbott invention was the no left turn maze. We need to get from a starting point to a goal without making left turns. You can go straight or you can turn right. So here you could go there and then there, but then you'd be trapped because you can't make a left turn. So here's the first place where you have a real choice. And you could turn here, then you have to go down this way. And here you basically have a choice between turning right or going straight. But that would basically put you back in a loop. And here you could get back to the starting point and so on. So this is another type of of maze that I need to come develop code for solving. But it should be very similar to the Farmer Goes to Market maze. Here's a type of maze I don't see very often. Here are the two states, or whether you're on one side of a, a, a path or the other side of a path. So this guy here needs to reach from here to here, but there's two sides to the path. So he could say, go this way, loop around and come to this point here, loop around and back. And he's on the other side of the same path that he started on. I haven't seen very many mazes like this one. This one was by Dave Phillips. It was an Omni magazine a while ago. I'm not sure how I would code this one up, but I'd love to have Mathematica able to develop more mazes like this one. It would be very interesting to work with splines to come up with something like this. Here's an extremely complicated logic maze. You basically need to get this treasure from the Kronos Corporation. You enter the maze on a Monday and you need to leave the maze with this treasure on a Monday. There's various red locked doors labeled A to H. So for example, here's the button that opens door H here. So if you reach this button, you can press it and that will open door H. However, there are also time loops built into the maze that will move you either forwards or backwards in time as you go through them. So for example, if you're currently in a Monday and you go through this doorway here, you're suddenly in Wednesday. And then if you come through here, you're back in Monday again. So the states in this maze are whether you've pressed a button for a door and what day of the week you happen to be in. But it's slightly more complicated for that than that, because for each of these buttons you have to keep track of which day of the week you press the button. Because if, say, your here you're in a Tuesday and gate E isn't opened until Thursday, you'd basically have to wait a couple days to go through it, and then it would be Thursday when you pass through this door. Basically, you need to keep track of all the states for each button and for the day you're in and whether or not the treasure has been collected. So there's. I think there's five to the eight different states you need to keep track of while going through this maze. It's not quite so bad, but there's a lot of complexity built into this and I haven't been able to solve this one yet. It's a very hard maze. And this is another maze that I need to try to come up with code for solving. Here's another type where each of these cuboids has two mirrors and two open faces. You need to use the clues given by the figures in order to figure out where the mirrors are. And after that, you need to come up with a path that goes through the two empty frames in each cuboid. And so there's a lot of logic that goes into solving this and quite a few optical illusions built into it. Here's another of the type seen in Meteor Storm. You have two pointers, both starting at start I. The. The. The goal is to get to the goal area simultaneously, and the pen moves horizontally or vertically as many squares as a current number shows. Both pins move at the same time, but one pin must move horizontally, horizontally and the other vertically. And this happens to be a very tricky maze that I haven't coded up yet. But I think it takes 40 moves in order to get from start to goal, which is surprisingly hard based on the size of the maze. The Theseus and the Minotaur maze is another type developed by Robert Abbott. You start here as the Black Dart, as. As Theseus, and you move one square at a time. The Minotaur here will move two squares directly towards you first, going left or right, if that's possible, and then up or down if that will move it closer to you. However, it will only move towards you and never away from you. So from the start, Theseus can basically move anywhere he wants, so long as he doesn't go under this line. Once he moves down, that will free up the Minotaur to start moving. So if Theseusa moved to was on this square and made one move towards here, the Minotaur would move down first to get closer and then this way the Minotaur always moves two spaces to to one space by Theseus. So for this one, you need to keep track of the state that the Minotaur is in. And since the Minotaur can move faster, you basically need to move Minotaur into traps so that Theseus can Move around safely. This is a, a very interesting maze. And if you're familiar with I think the. There's another company that did. I think their name was Popcaps or something like that, where you had to shoot at bubbles and whatnot. They developed a whole bunch of games and they pretty much stole this idea for one of their games. It was, I think it was something based on mummies, Mummy's tomb or something like that. And Robert Abbott basically sent them a note and the game company kindly decided to give Robert Abbott a thousand dollars as thanks for helping develop this concept for them, which was very nice of that game company. Here's one that I made. Basically with every move you need to change your angle. Basically you can going through this, you can go straight, you can take a, a soft turn, or you can take a sharp turn. And going through this, here's the path from leaving and getting back to the, the starting position. This change of angle is. You need basically keep track of your state while going through the maze. And if you check the path, you'll see that the, the angle changes with every move. Other WAY MAZE so you have a, a set of arrows. There's. There's two coins or something on the arrows. And basically you need to. The one person needs to move in the direction that the other person is on. So I call this the other ma way Other way maze. And you have two people starting at the center. One of them has to move to here. And then the goal is to get them both back to the center again following the rules of the maze. And here's how to set it up. We have the maze described here is a set of arrows. Here are the different directions for the arrows. We can set up the grid and build up the math for it, build up the graph for it. And then we can find the shortest path leading to a solvable state. And we can basically step through the solution here. From here, green follows the red arrow, red follows the green arrow. Green follows red, red follows green. And again and again. So we keep going through this. The two people wander around on the maze following the arrows. And they might wonder whether they're getting anywhere near the goal because it's a rather tricky maze and here they've basically solved it actually. Have they? Oh, looks, looks like I need to rethink this maze a little bit. This doesn't actually solve it. Basically the other person needs to get to there. Let me see if I can, I can come up with a solution for this one real fast. So this would be. Four, three. Yes. Let's see if this is actually solvable. These are not only hard to do but, but tricky to, to fall. So it looks like this one is broken. Maybe. Oh, maybe it is, maybe it isn't. Let's see. Ah, here we go. So move similar, but the ending is slightly different. Let's see what the ending is like here. Yes, this is the next to last. Then we have this and then the red can follow green and you've solved the maze. And that is a length path of length 55. That's a good reason to have things coded because these are really, really tricky to solve by hand. It's, it's easy to go wrong. So this is a solution for, for that maze. Here's another maze. When Roger Penrose was a teenager, he rather liked the Robert Abbott maze in the 1962 Scientific American. So he developed the railway maze when he was a teenager. Roger Penrose. And here's one of the mazes here. Oops, doesn't seem to want to work. Let's try that one again. Of course, now it doesn't want to work, but the Penrose Railways maze. On the demonstration site you can see sandwagon and antennas, a version of Penrose railway mazes. And you're basically forced into loops to, to solve the maze very similar to what Robert Abbott did. Here's another type of maze called the fractal maze. Basically you have multiple copies of this particular shape. And once you go into this inner part, you are in a copy of the outer part and you can basically nest these pieces as you go in deeper and deeper. So we can nest these pieces like this and we need to get from 1 to 8. So with 1, we go down one level, we come back out, we go back in again, we loop around up here and we go down several levels and then we can get back out again. So here in the fractal maze, you basically need to keep track of what level of depth you're in as you're going through the maze. If you look up fractal maze on Google, you'll see various sites that have more than one copy that you need to, to use. And there can be like eight holes in the, in the fractal maze. And each one is a complicated circuit diagram you would need to go through. And sometimes you need to go through four different levels through this maze. And it's very complicated to keep track of what depth you're at and where in the maze you are as you're going through it. Here's a mechanical version where basically you need to solve three mazes at the same time. And there are metal discs going through to limit your options. As you're moving the sliders and different mazes around the different levels of the maze can move, the disc can move. And as you remove objects through the larger openings, you get more freedom to move things around. Typically this requires 200 moves to solve. Here's a earlier version by Oscar Van Deventer, the Oscar Cube. I highly recommend his YouTube site, Oscar Puzzle where he has about a thousand different mechanical muzzles made. He's the world's foremost maker of mechanical puzzles. And in this one you basically have three different mazes, simple tree mazes on the three sides, but they combine to make a harder three dimensional maze. Here's a different type of maze I developed. You basically have two people starting at the bottom here, facing up. You, one of them travels in the direction the other one is facing and then they, as they land on a number, they have to make a series of, of turns that many places. So if they landed on a four, they would basically do a U turn and both of them need to get back to the starting position and at least one of them needs to be pointing up. This is a very, very complicated maze based on its size. I coded this a while ago, but I couldn't find where my code was. A counter step maze. Another by Oscar Van Deventer. You, you have a red dot moving and the blue dot moves in the absolute opposite direction. They can only move if both can move and you basically have to join them together at the center. This, this is a very nice set of puzzles@clickmazes.com and there's many different maze types@clickmazes.com Many different logic puzzles. Another developed by Andrew Gilbert, who owns Clickmazes.com is the plank puzzles. This was made as the river crossing maze. And you basically start here and you can move these three planks along which can join up tree stumps within a river. So the various state is states are where you are and where you've placed the planks. So you could basically start here and then you could move, move to here and then you could move this plank to go straight up. You can only move a plank if you're touching it. And then once you've got that plank there, you could say move the plank over to here instead or you can move this plank to go down. So this, these, this is a very hard type of puzzle to solve. It's been proven to be hard and there's a wide variety of these Planck puzzles. Available either online or you can get the physical version from Think Fun. Color sequence mazes are also fun where you basically have to follow a sequence. They're given color sequence like red, blue, yellow, and here's one of the puzzle types. Robert Abbott, Adrian Fisher, Andrew Gilbert and Bradley have all looked at puzzles of these types. At community wolfram.com you can see Brad Cleese write up on color sequence mazes. There's also the Hanayama Cast series which all of these are mazes, usually two piece mazes where you need to maneuver the pieces into many different positions in order to solve it. For example, Oscar's disk puzzle. This is a Hanayama Cast disk here. This is basically a 7 by 7 toroidal maze where you need to keep track of which of the openings you've gotten into at each step. And it works very much like a maze on a torus. This one here called Medallion is actually a four dimensional maze. There's, there's mazes on this side and the other side and the, the two halves of this medallion will split apart to let the, the pegs move. And some of these are just ridiculously difficult like Radix up here. I have no idea how this puzzle was designed, but it's basically a 3D maze. That looks absolutely beautiful. It's not at all obvious that it's a maze, but it works. Rotor here is another very nice puzzle which is also an extremely clever maze. This is considered the best series of puzzles ever made in the world. The Cast series by Hanayama Maze Log is another site that has many different multi state mazes. So here you need to alternate between matching shapes and matching colors. And another rule that I should have written here is you can't do U turns. This no U turns rule is a very good way of adding a logic item to the maze because even though you've got plenty of freedom while going through something, there's one direction that's restricted. You can't go back in the direction you just came from. And that means you need to keep track of your direction while going through the maze. And therefore it's a logic maze. There's multiple states you can be in. So central solving method, turn it into a graph and then solve the graph. That's what I had prepared and I'll be glad to answer questions. I see Daniel had one, an algorithm that puts dead end signs in mazes. That would be easy to do. I think I have seen a program that does that for normal mazes. It would be much trickier to do that in a logic maze because you often have to revisit a state or position in multiple places, and it's a dead end depending on how you've entered it, so it wouldn't always work. Does anybody else have any questions that they can ask here? Oh, many thanks for attending my talk. I hope you've learned a bit about mazes and graphs and all that fun stuff, and feel free to write up any suggestions you'd like. All right. Many thanks.