Multistate mazes

A nice multistate maze by Hiroshi Yamamoto.  Start at S.  Take 1 step in a direction, then two steps, then three steps.  Repeat taking 1, 2, and 3 steps to finish at F.  You may not turn a corner or turn back while taking a step.  Tricky.  See answer.

Erich Friedman was one of many impressed by the Hiroshi's maze.  Here is his homage to it ... the 1 2 3 4 5 maze!  Start at the S.  Move one space north, south, east or west.  Follow that by moving two spaces in one of the 4 main compass directions.  Follow that by moving three spaces in one of the 4 main compass directions. Follow that by moving four spaces in one of the 4 main compass directions. Follow that by moving five spaces in one of the 4 main compass directions.  Repeat this sequence of moving 1 2 3 4 and 5 spaces as many times as you like, until you finish at F.  See answer.


The 1 2 3 4 5 maze by Erich Friedman

Matthew Daly -- "Enclosed is a variant of the maze whose solution is (for the most part) unique (I believe).  This maze contains six circular paths: the Even path, the Fibonacci path, the Prime path, the Square path, the Triangular path, and the 2-digit path.  Start at the red circle.  As with your maze, increase your tally with every blue circle you encounter.  The only way to change directions is from one path to another, and only if your tally at a junction meets the conditions of BOTH of the paths.  The object is to travel on at least a portion of each path."


Matthew Daly's Dot Maze
It's a nice one.  Joseph DeVincentis solved it.  If one of the dots in the 2-digit circle is moved, how many more transfers are possible?

On the Prime Minister Maze (below), one thought I had was to use red stones (-1) and green stones (+1) in such a way that each complete loop had an equal or greater number of red stones.  This would prevent a person from cycling around a loop until they had a large enough prime.  Instead, a person would need to continually look for path sequences that would build up the total.  If you drop to zero, you have to restart.  If you make a nice, loopy prime maze of this type (you can use your own loops and circles), please send it to me.
I was cleaning out some old files, and found a new maze idea.  It's not a very good maze, but it's pretty.  It might have an infinite number of solutions.  I call it the Prime Minister maze.  You start with zero point, at S.  As you walk on the paths, keep track of how many circles you've walked over.  If you come to junction and your running total is prime, you may change direction.  Otherwise, you must continue going forward.  The object is to finish at F with a prime number.  Is it possible to make an interesting maze of this type with a unique solution?

The Prime Minister Maze.  Larger image.

A multilevel rolling column maze by Erich Friedman is here.  This is a rolling cube maze on four levels.  Timothy Firman and Joseph DeVincentis sent me solutions.

The Rolling Rhomboid by Wei-Hwa Huang
Wei-Hwa Huang sent me a rolling rhomboid maze.  "You might find the rules for this particular maze interesting.  Start the die at the upper left, on the two colored cells.  You'll notice that there are two ways to put the die on those two cells; the die can "lean" towards the lower left, or "lean" towards the upper right. The goal -- start the die in one of those positions, and, by rolling, get it to the other position!

The die cannot land on black cells, although it may "hang" over them. Think of the black cells as having immovable tetrahedra on them.  The below is a folding diagram.  Can you solve it?  If not, you can see a hint, solvers, and a solution on my Solutions page.

In 1999, the ACM Programming Contest used one of Robert Abbott's rolling dice mazes as a programming challenge - without attribution.  Steve Stadnicki spotted it, informed me, I informed Robert Abbott, who wrote to ACM, who apologized and allowed him to put Abbott's Revenge in this years ACM Programming Contest.  Here it is -- just click on 'Problem Set.'  Robert has developed a new variety of maze which is quite difficult.  Donald Knuth has solved it.  Robert Abbott is working on a write-up on his site.

At Robert Abbott's site, a Javascript Number Maze is available.  It was solved by Nick Boone, Chris Lusby Taylor, Francis Heaney, and Joseph DeVincentis.

Another Rolling U maze by Erich Friedman is here.  It requires 49 moves, and was solved by Robert Abbott and Luc Kumps.

The following Rolling U Pentomino maze is by Erich Friedman.  It was solved by Joseph DeVincentis and Robert Abbott.


Erich Friedman has created a new concept in rolling block mazes, below.  Only the red blocks may move.  The 2x2x1 slab can only stay on level 0, the floor.  The 2x1x1 brick must stay a distance of 1 above the floor.  The 1x1x1 cube must stay a distance of 2 above the floor.  Rolling the red objects around, get the 1x1x1 cube above the F.  Solution.

If you are unfamiliar with rolling block mazes, please visit Robert Abbott's Things That Roll.  Ishihama Yoshiaki has written an applet version of the Rolling Arrow Cube maze.  John Bailey has put together a page for Manson's Maze book.

The original double cube maze, created by Richard Tucker, is now at Robert Abbott's multistate maze site.  Here is a new double cube maze by Juha Saukkloa of Finland.  Tape two cubes together, and put it end up on the square marked S (for Start).  Roll the double cube around until it lands end up on the square marked F (for Finish).  At no time may the doubled cube land on red brick.  Is a harder 8x8 double cube maze possible?


Rolling Cube Maze (by Ed Pegg Jr.)  Print out the above diagram.  Cut out the cross, and fold it into a cube.  Next, place the red arrow face up on the indicated square.  From there, you may roll the cube to a neighboring square if the top face matches the direction of the neighboring square.  For example, at the start, the top face points up, and both neighboring faces point up, so you may roll the cube to either square.

The above set of puzzles I originally attributed to Reiner Muller, from his book Dominoes.  Since then, I discovered an earlier printing of the same puzzle in the 1981 issues of Top Puzzles Magazine.  They require 41 and 35 moves respectively.
1.  Move the 0-0 domino (1-1 in second puzzle) to the upper right-hand corner.  To do so, slide dominoes into the free space.
2.  The pip numbers on the last domino moved may not border the same pip number on an adjacent square.
3.  At the end, the free space must be in the lower left-hand corner.

Chris Lusby Taylor used Zillions to help him find new rolling die mazes.  He made a very nice one, which is now at Robert Abbott's site.  You can download see his Zillions program here.  A separate version for Arrow Cube mazes is here.  The following is a maze I built with the latter.  You need to move the cube to the lower right square.  The bottom of the cube points down.  The space underneath points up.  The cube is the same as the one in the arrow cube maze above.

Object:  Reach the square marked F (for Finish).  This is a very interesting little maze, I hope you'll try it.

RED, GREEN, YELLOW.  The above maze is from SuperMazes, from Robert Abbott's Multistate Maze Site.  Traveling with the repeating sequence R G Y R G Y, get through the maze. This is a multistate maze, which means you'll be visiting several areas multiple times.

Both of Robert Abbott's maze books, Mad Mazes and SuperMazes, are excellent.  These are the best books for mazes of this type, and I highly recommend both of them.  My arrow maze above was inspired by these books.

Adrian Fisher Maze Design is an excellent place to visit for normal-type mazes.  Adrian makes the largest mazes in the world, and has an excellent list of where you can find walk-through mazes in your area.  He has a few multistate mazes at his site as well.

If you have an interesting multistate maze, I'd like to see it!

Here's a Card Maze.  By alternating Down and Up, go from Start to Finish.  The answer is in my Solutions area.

Art by Robert Abbott. New traffic laws by Ed Pegg Jr.

   PRIME CLIMB.  Your running total must always be a prime number (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 ...) while you're in the town.  No U-turns are allowed.  Can you get through town with a ending total of 97?  Example:  Starting, you have 5.  North is bad from there (5+4=9, nonprime), but east (5+2=7) and south (5+6=11) are both fine. Here are the solvers and an answer going to 61.  Juha Saukkola has solved the 97 problem.