How to escape any maze
Tonight's idea
You are lost in a maze. Here is a rule that needs no map. Put your right hand on the wall as you walk in. Keep it there. Follow the wall round every corner and into every dead end.
Why? Look at the outside wall of maze A. It has two gaps: the way in and the way out. The piece of wall on your right runs all the way round from one gap to the other. Inner walls joined to it are only detours. Slide your hand along and you must reach the way out.
The rule is sure, not short. You may walk into every dead end on the way.
The rule fails when the thing you want is in the middle, inside walls that touch no other wall. That block of wall is an island. Your hand never crosses to an island.
Watch one: maze A
Follow the orange line with your finger. A dead end is a square with walls on three sides. The route turns into two dead ends and has to come back out of each.
Count the squares it steps on, counting a square again each time it comes back. That makes 27. The shortest route uses only 15 squares.
So the rule did not find the best way. But it never needed a guess.
Your turn
- In maze A, use your left hand. Draw the route in pencil. (a) Does it reach the way out? (b) How many dead ends does it go into?
- In maze B, walk in with your right hand on the wall. Draw the route. Where do you finish? Do you reach the star?
- Now start at the star in maze B. Put your right hand on the wall of the star's room and walk. Do you ever get out of the maze? Say what happens.
- Draw one new piece of wall in maze B so that the hand-on-wall rule does find the star. Explain why your wall works.
Stretch: left and right
Maze A has six dead ends. The right-hand route goes into two of them. The left-hand route goes into the other four. No dead end is visited by both routes. Why not?
Answers
- (a) Yes. The wall on your left also runs from the way in to the way out. (b) Four. The route goes up and round the top of the maze. It is longer than the right-hand route.
- You walk along every passage outside the star's room, and you come back out of the way in. You pass the doorway of the room, but your hand is on the other side of the passage. So you never go in. You do not reach the star.
- No. You leave the room, walk once round the outside of it, and come back in through the same doorway. Then it all happens again. The room's walls are an island, and your hand cannot leave them.
- Join the island to another wall. Any wall across the passage that runs round the star's room will do. For example, draw a wall from the bottom left corner of the room straight down to the wall below it. Now every wall is joined into one piece. Your hand slides along all of it, including the inside of the room.
Stretch. Draw the shortest route. It cuts the maze into two sides. The right hand goes into every side passage on the right of that route, and none on the left. The left hand does the opposite. Each dead end lies on one side only, so exactly one of the two routes visits it.