Mazes and Other Route-Finding Puzzles

To solve a route-finding puzzle, one must navigate a traveler piece (sometimes one's whole body - or maybe just a fingertip...) through a pre-established network of routes, from some starting point to some goal point, perhaps obeying some rules or constraints along the way. Usually the challenge is to find any route, but sometimes one must find the shortest route among many. The route-finding class is a subclass of Sequential Movement type puzzles, since where you can go depends on where you've been - i.e. current state options depend on prior state actions.

The difficulty of route-finding puzzles lies in the confusing complexity of the network, which will usually present many choices and dead-ends, and in some cases loops. This is different from a route-building puzzle, where the network is not pre-established (or at least does not appear to be), but must be constructed from some kind of units (often tiles depicting route segments), according to some rules (often edge-matching constraints), as play progresses.

Think about a maze (Wikipedia entry), and the legend of Theseus and the Minotaur might come to mind. Confined within a structure on the island of Crete at Knossos, the Minotaur - half-man and half-beast, and evidently very pissed off, would claim a human sacrifice each year, until the hero Theseus arrived to slay him. However, the structure in which the Minotaur was confined is usually called a labyrinth, not a maze. The distinction lies in the characteristic that a maze contains pathways with intersections - choices which can lead to dead-ends or run-arounds - while a labyrinth contains essentially only a single path - and is therefore not much of a puzzle, though the path may be very convoluted on itself. This being the case, it would be a mystery why Theseus required the help of Ariadne, in the form of a thread spooled from the entrance, to find his way out. Let it suffice to say that there remains an absence of universal agreement on the terminology.

Another famous maze appeared in the first computer adventure game, Advent, and spawned the immortal phrase, "You are in a maze of twisty little passages, all alike." This phrase was used as the description of all the locations within the maze, and one would quickly become lost or go in circles. To map the maze and get through it without relying on sheer luck, the solver had to realize they could drop various items at different locations in order to distinguish them. At another point, Advent employed a variation on the original theme - each location's description was a unique variation on the phrase "You are in a maze of twisty little passages, all different." For example, "You are in a twisty maze of little passages, all different." Once one realizes the descriptions are all actually different, solving the maze becomes easy! Unfortunately, mazes became over-used in adventure games, often poorly done and coming to be seen as a hackneyed device for adding time-wasting filler.

The Cretan labyrinth design has appeared down through the centuries in many decorative and architectural motifs - for example, on coins and carved in stone - and seems to recur in many cultures. See examples at Mirko Elviro's site. The schematic can be fairly easily constructed - see instructions at Maths aMazes.

The maze is certainly a contender for the title of World's Most Ancient Puzzle.

When discussing and analyzing route-finding puzzles, it is useful to employ notions and terminology from graph theory. A graph, or network, can be described by defining a set of nodes (also known as vertices) and paths (also known as edges or arcs) each of which connects two nodes. (If an edge connects more than two nodes, you're talking about a hypergraph, which we won't discuss here.) The paths do not intersect, otherwise that would define a new node. The degree of a node is the number of edges connected to it, and can be odd or even depending on the sum. Sometimes paths are one-way - then the graph is a directed graph or digraph.

A graph is connected when all the vertices can be reached from each other along the defined paths (i.e. the maze is all of one contiguous piece). The order of a graph is the number of its vertices, and the size of the graph is the number of its paths.

Navigating one's auto among the world's roadways can be (and often unintentionally is!) a kind of route-finding puzzle - I have enjoyed a road rally where a list of puzzling clues defined the route. Perhaps the most recognizable route-finding puzzle is the traditional pathway maze built of walls, either life-sized or in miniature.

It may be hard at first to see how a traditional maze of walls and pathways can be represented as a graph, but it is easily done - imagine each junction, as well as the starting point and goal(s) as nodes, and then draw all the paths that connect them. Sometimes the outside of the maze must be included as a node, too.

Mazes and Labyrinths for People - Panel, Hedge, and Maize Mazes

The life-sized variety may technically be mechanical puzzles (Slocum classifies them as 5.7 Mazes and Labyrinths for People), but it is difficult to "collect" them, except in the sense that a bird-watcher collects birds - you visit them in the wild.

Mazes built at a scale intended for people to walk through have been made using various material for the walls, including hedges, panels, and maize. Sometimes the maze is merely a pattern on the ground, perhaps marked out in stone.

Perhaps the most famous hedge maze is The Maze at Hampton Court Palace in England.

There is an indoor panel maze in Montreal, Canada, at Hangar 16 (caution - link opens a full-sized browser window), which I visited and enjoyed.

Here are a few additional venues on the web:

Interest in mazes and labyrinths continues unabated today. Adrian Fisher is perhaps today's master maze maker. Adrian has created many large-scale commissioned mazes, including hedge, corn, and mirror mazes. Find a list of locations of Adrian's maize mazes here.

There are many other web resources devoted to mazes and labyrinths - here are some I have found to be especially interesting or informative:

Hand-Held Route-Finding Puzzles

Aside from large-scale structures meant to be walked through, mazes also appear in many portable mechanical puzzles, which are our primary concern here. Starting with the very simple concentric rings in the original Pigs in Clover design, there have been countless rolling-ball-in-maze puzzles issued. The Pigs in Clover design is a trivial maze - but the passages and openings are cleverly contrived so that the challenge of that puzzle is primarily one of dexterity - tilting a ball through one passage will often, frustratingly, tilt another ball through a different passage in the wrong direction.

I will try to sort out dexterity puzzles and keep them in the Dexterity section, but the distinction can sometimes be a fine one. You will also see a blurring of this class with the Tanglement class, since the traveler piece can be thought of as being tangled in the network, from which it must be freed. I will also exclude pencil-and-paper printed mazes, though they have become quite complex in their own right, some even requiring 3-D glasses!

The Hordern-Dalgety classification has several sub-classes of route-finding puzzle:

One key difference between the walk-in maze and the hand-held puzzle is that in the latter you usually have a bird's-eye view of the entire maze, whereas in the larger puzzle the network unfolds from a first-person, limited viewpoint. Having a view of the whole maze at once would tend to reduce the challenge, but designers have come up with many tactics to make their mazes more interesting and/or difficult...

Some of these modifications can be and have been used in parallel.

Traditional Mazes

This section includes basic mazes - they might have multiple levels, wrap around a cube, be "reprogrammable" to make different challenges, pit you against a timer, or release a switch when solved, or even be expressed in the form of a logic puzzle, but the maze is "all there" in front of you and nothing about it changes as you go.


Backflip

Bilz Box

A-Maze-Ing
by Jajaco, from 1963

Color Cube Maze - DaMert

Boston Subway
made by George Miller

Moire Maze - designed by Scott Elliott. Scott writes about his Moire Maze on his blog.

Fire Escape by SmartGames
(also sold as "Tower of Logic Inferno")
A series of 48 two-sided cards pose route-finding problems. Fit a card into the tower. Ladders, fires, and a victim on each card are visible through the tower. Navigate the fireman from the starting position to the victim, using the ladders and the wrap-around terraces on the tower, and avoiding the fires. The fireman is equipped with one or more colored fire extinguishers which can each be used once to pass through a fire of the corresponding color. (Playing with this last feature admittedly moves this to the Complex category.)

Minute Maze - Mag-Nif

An acrylic 3-D maze
This reminds me of "Next Floor" by Oskar van Deventer - see it at Oskar's website (scroll down).
 
Ethereal Maze - designed by Steve Winter
See Steve's Shapeways shop
a gift from Brett - Thanks!
 

 

This is the XMATRIX Quadrus puzzle,
developed in 2009 by artist and designer Jeremy Goode and issued by www.xmatrix.co.uk
You can see Goode's European patent GB2472581(A) online.
The Quadrus retails for £20 - Jeremy kindly sent me a copy to try. Thanks, Jeremy!

Quadrus is a large (140 x 140 x 30mm) and attractive traditional rolling-ball multilevel maze in a gold-tinted transparent acrylic case, nicely packaged in a cardboard slipcase tray that shows off the ambigrammatic XMATRIX logo. Quadrus is also available in a blue tint, and has a smaller cubic sister puzzle called Cubus.

In Quadrus, the maze network is defined by three layers of internal latticework structures and interstices - one lattice on each large face, and a third suspended between them, with an empty thickness between pairs of adjacent layers - giving an overall thickness of 5 layers between the "floor" and "ceiling" faces. The walls in a layer are 4mm wide, and the pathways between walls are 8mm wide. Walls and pathways in a layer are arranged on a 34 x 34 virtual grid of 4mm x 4mm squares. The ball occupies a 2x2x2 space within the lattice.

The maze contains a central 12x12 square compartment - a white panel with a stylized 'X' cutout separates the compartment into a gold-framed side and a silver-framed side. Each side of the compartment has a single entrance into the maze. To solve Quadrus, one must navigate the ball from the central gold-framed compartment to the central silver-framed compartment on the opposite side, by tilting the puzzle and guiding the ball through the maze. I have found that the occasional ill-planned tilt can send the ball somewhat further than one intends, adding a dexterity dilemma to the already-considerable routefinding challenge.

This style of maze is similar to the Boston Subway puzzle designed by Oskar van Deventer for the 2006 IPP Exchange. (See Boston Subway at Oskar's website.) Boston Subway is a much smaller puzzle, but also comprises a "sandwich" of 5 layers of transparent acrylic, through which one navigates an internal metal ball from point A to B and back. Unlike Quadrus, Boston Subway requires the solver to use an included magnetic wand to move the ball through the maze. As interesting as Oskar's Boston Subway puzzle is, I find the Quadrus more convenient to hold and manipulate, and it is far easier to see and keep track of where the ball is. While less portable, it is more engaging to the casual puzzler.

For the 2009 IPP, Oskar also designed Next Floor. (See Next Floor at Oskar's website.) Produced from laser-cut MDF, this maze is formed from 5 grooved layers with four interstices. A version of Next Floor is marketed by Bits and Pieces, but unfortunately folks reported that the layers can be loose and the ball can squeeze through unintended paths or even fall out, spoiling the fun. No such issues plague the robust, high-quality Quadrus puzzle.

I found an older seven-layer version (in an eBay auction). Unfortunately it included no documentation and I am unware of its provenance.

You can read some reviews of the XMATRIX puzzles here, here, here, and here.

My own impressions? When solving a maze, I do not typically sit with the puzzle tilting it to and fro. I manipulate and study the puzzle enough to systematically create an accurate representation of the network "on paper" and then exhaustively map it out. The physical object itself becomes somewhat unimportant and is usually stored away. With the XMATRIX Quadrus, however, two things are true: It is just as much a pleasure to hold and play with the physical object as it is to intellectually solve the maze, and this is one maze puzzle that will be left out for others to enjoy rather than being put away and forgotten.

Complex Mazes

In these mazes, the designers have incorporated various devices to confound the would-be solver. Pieces of the maze might be hidden, or might be movable.


In this "Maze Ball Game,"
the segments move and
re-configure the maze.

The horizontal slices
rotate in Raintree's
Tower of London,
reconfiguring the maze.

The Medallion

Synapse
Each leg can rotate on its long axis. The ball must be navigated internally from leg to leg to the exit.

Missing Marble

Amazin' Marble Die

Exit - issued by Reiss in 1974
Place a steel ball into the wooden block, which contains maze passages to be navigated blind. What is odd about the photo on the back?

Odd Ball

Skill-It
Milton Bradley 1966
The frying pan contains a maze of half-height walls. A permanently attached but rotating transparent lid has another maze of half-height walls. Maneuver the ball and the lid to get the ball from the center to the handle. (U.S. Quarter in pic for size.)

Inside Cube Orange

No Peekie
Ideal 1971
The maze has a lid showing the solution. Close the lid and navigate the ball. Harder than it would seem, as the dead-ends complicate things and you become unsure of just where the ball is. (U.S. Quarter in pic for size.)

KO Labyrinth
By the Lepato Co. Ltd.
Combines a maze with a 3x3x3 twisty sphere.

Amaze
Here, you can move some of the walls, but only from certain positions and only in certain directions.

Perpetual Motion, by Bill Darrah
A hidden maze. Maneuver the top off. Purchased from Bill at IPP28.
Made by George Miller.

Life's Maze
A switched maze, by Kirill Grebnev.

Octamaze, by Pavel Curtis.
Read about Octamaze at Pavel's blog. Pavel has several pages giving progressive hints.

The No Dexterity maze, designed by Oskar van Deventer and made by Tom Lensch.
Set the maze on a table with the single opening in the center of one side face up. Drop in the ball. Now get the ball out back through the same hole. For each move, you may only rotate the maze flat onto a new face without picking it up or shaking it. You must plan your sequence of moves carefully to force the ball to drop from cell to cell using only gravity.

I found The Black Jack Labyrinth by Constantin at Eureka.


22 Card Maze, designed, made, and exchanged by Mike Snyder at IPP32
First, build the maze by arranging the 22 tiles so that all cut-outs are filled by another tile. Then, navigate the maze by always proceeding from a card "on top" to one it is "covering."

Paradox Box
Designed by Ivan Moskovich, issued by Fat Brain Toy Co.
Drop the steel ball bearing into a hole in the corner of the "top" of the cube, then navigate it through the blind maze to an exit hole in the center of the "bottom" face. Various clues on the outside of the box must be decoded to help - arrows on the clear face panels (up, down, left, right), a 5x5 grid of 10 colors on each face (white, black, purple, red, orange, yellow, light blue, dark blue, light green, dark green), and plus/minus/infinity signs at the grid crossings.

Acrobat designed and made by Diniar Namdarian, exchanged at IPP32 by Goetz Schwandtner

Vikings Brainstorm - designed by Raf Peeters, issued by Smart Games
A 3x3 grid of overlapping circles is populated with a set of nine "water" pieces of two types. These water pieces interlock, rotate, and interfere with each other in interesting ways, and at certain locations there will be boat-shaped gaps. The puzzle includes four Viking boats and each challenge calls for you to move one or more boats from given starting positions to specific goal positions, navigating through the board by figuring out how to exploit the rotating water pieces and gaps. An unusual and very satisfying style of movement!

Sleeve-on-Cylinder Mazes

One type of maze puzzle entails a sleeve riding on a cylinder. Either the sleeve or the cylinder contains a maze of grooves, and the other component contains a peg which moves in the grooves.


Screw Loose
One of the "Lost Puzzles of My Childhood."
Issued in 1970 by Lakeside Industries, a division of Leisure Dynamics Inc. The packaging says "patent pending" but I could find no online record of the application.

Dool-O-Rinth set
This is a recent series of puzzles called "Dool 'O' Rinth" (aka Crazy Maze) made by CorToys. There are 6 puzzles in the series - in order from easiest to hardest: yellow, orange, green, blue, red, and black. (According to the vendor from whom I purchased it, the red sleeve on an orange spool is a special edition. Since the sleeve contains the maze, this is a red.)
Recently re-branded as "Groove Tube."

Two sleeve-on-cylinder type mazes (Blindlabyrint) designed in 1983 by Lauri Kaira.
Only 2000 copies of each were made; most were sold back in 1984.
1A is a single-track labyrinth; 1C is a branching maze.
Purchased from Finnish company Oy Sloyd Ab.

This is Mental Block Puzzle #6 Double Semi-Maze, by R. D. Rose. It is crafted from aluminum, and consists of an outer sleeve with various paths (some not visible) and two half-cylinders riding inside. One of the half-cylinders has a mark on its edge corresponding to a similar mark on the sleeve's edge. The objective is to move the inner cylinder's mark through a full 360 degree circuit. Neither inner piece is meant to come out of the sleeve.

Tube Maze
a sleeve-on-cylinder maze, 3D printed on his Makerbot by Jon Taylor.
You might be able to find one for sale via his eBay account 19snowden89.

The Roto-Maze issued by Little Harbor Corp. of Lewiston, Maine.
Mine is labeled as a "Chevron" pattern, of Intermediate difficulty. The last image is a copy of my hand-drawn solution.
Jerry Slocum's collection contains a copy with a rectilinear pattern, and a copy with a curly pattern.

Revomaze

Revomaze, run by Chris Pitt, offers a series of "sleeve-on-cylinder-type" maze puzzles, of high quality and sophistication.


At auction I won a special edition Blue (BU00006SE), and a Bronze. I've also acquired an Extreme Silver (V2) - as of this writing, very few people in the world (less than 100) have solved one!

Revomaze has issued several products - the main series of puzzles are called the Extreme V1 series. They include the Blue, Green, Bronze, Silver, and Gold (in increasing level of difficulty) and the Titanium which was only offered in a collector's edition set. Shown below for reference (I don't have this).

The Extreme V1 series is made from aluminum and nickel-plated brass.

From the manufacturer, here is a video of the series. Here are some notes on and links to reviews of the series members:
  • Blue - static progression
    Allard , Brian , Neil , Kevin , A video that includes reassembly
  • Green - more complex static progression, longer bridges, twist - hearing click no longer necessarily means reset!
    Allard , Brian , Neil , Kevin
  • Bronze - dynamic progression maze - moving maze parts - swimming pool, jacuzzi traps, "the euro"
    According to several folks who have tried them all, the Bronze may be the best of the series.
    Allard , Brian , Neil , Kevin , Oli
  • Silver - v1, v2 - dynamic progression - gravity pins, the dead end, the canal, the rabbit run, the motorway, the swamp, "not a canyon"
    Allard , Brian , Neil , Kevin , Oli
  • Gold - algorithm
    Neil

Revomaze has also issued an Extreme V2 series that have smooth bodies - including the blue, green, and bronze. The mazes are identical to their V1 counterparts.

There have also been several Limited Edition (LE) puzzles issued:

  • Titanium - only in collectors set - static
    Will
  • Orange issue 2, designed by competition winner Mark James - dynamic progression - castle theme - village, forest, guards, battlements, moat, turrets
    Allard
  • Lime - "Dino's Revenge" (inspired by Enzo Ferrari’s fabled Dino V6) - pistons, cylinders, gears - "stamatic"
    Puzzle Place entry
  • Black - ltd ed of 20 - contains Revomaze company name in maze
    Allard , Neil
  • Red - only 99 sold - rainbow is a red herring
    Allard
  • Purple (6000GBP!) - offered only as a prize
    Revomaze website
  • Handmade v1, Handmade v2 - ltd ed of 20
    Puzzle Place entry

There are also inexpensive versions issued in plastic, called the Obsession series. Allard blogs about them. Gabriel blogs about them, too. In the original Obsession series, unlike the Extreme versions where the core can come out after the puzzle is solved, the cores cannot be removed, so you cannot see the maze. However, I am told that the new Obsession series does allow the core to be removed.

  • Level 1 Blue
  • Level 2 Green
  • Level 3 (Pro) Black - same as Level 1, times 6
  • No longer available: Green, Black v1 (REVOMAZE), Black v2 (MERCK), Red (flawed)
  • Titanium was planned but never issued

Check out the R2 Saturn.

While it is great to be able to see the maze once the puzzle is solved, actually being able to see the maze as you're trying to solve it kind of defeats the purpose of this type of puzzle - the intent is to force you to feel your way through and build up a mental image of what's going on inside the maze. Nevertheless, lots of folks have wanted a way to show off and demonstrate the clever Revomaze concepts once they have solved theirs. Enter the clear sleeve...


Clear Revomaze Sleeve - purchased from, and designed and made by Neil Hutchison.
Neil discusses the genesis of the clear sleeve in his 30 Dec. 2012 blog post, with a follow-up on 5 Feb. 2013. Allard reviewed his on his own blog. The clear sleeve has also been discussed on the Revomaze forum.

Shuttle Mazes

Here are some Shuttle maze puzzles, where the maze is navigated by some element other than a rolling ball.


"16 to 1" from Bits and Pieces, and Hanayama's version called Laby
U.S. Patent 598855 - Carter 1898
This is a 2-sided maze, and the shuttle requires you to solve both sides simultaneously.

Vermont Castings
Like the 16-to-1 puzzle, this is a 2-sided maze.
It was issued circa 1981 by Vermont Castings, of Randolph Vermont, as a premium accompanying stove purchases.
It seems like the objective is to move the shuttle on and then visit the four areas marked I, II, III, and IV.

Oskar's Cube
In plastic and in metal
Here, the shuttle forces you to solve three mazes simultaneously!

Culax
Culax is an enhancement to Oskar's Cube - now the shuttle can be rotated within the network.

George Miller's Moby Maze is a maze on the surface of a Moebius Strip - it's got only one side, but it behaves like a 2-sided maze!

Brain-Chek - two-state faces, and three-state traffic lights.
Here, the shuttle interacts with the network as it moves, and changes the state of the nodes. You must find a route such that all the nodes achieve the desired state.

Bookworm - Doug Engel

Fishyhookery - Doug Engel

Rolling Block Puzzles

Here, the shuttle changes state as it moves around the network. Achieving the objective entails finding a sequence of moves of the shuttle and a route it can take so that it is in the desired state when it arrives at its goal.


Color Cubes

Cmetricks Cubicle

Hedgehog Escape, designed by Oskar van Deventer and Wei-Hwa Huang, and issued from Popular Playthings.

Creeping Block - exchange from Dirk Weber
Thanks, Dirk!

Say Cheese - Popular Playthings
A rolling-block route-finding puzzle, designed by Eric Harshbarger.

Creeping Block 3D, designed by Dries de Clercq, made and exchanged by Dirk Weber at IPP32

Plank Puzzles

River Crossing (and River Crossing 2) - Thinkfun

"River Crossing" is a commercial version of a type of puzzle known as "plank" puzzles. Plank puzzles were invented by UK maze enthusiast Andrea Gilbert. Visit Andrea's website Clickmazes.com.

You can visit the River Crossing Home Page at Puzzles.com. Here is a link to play plank puzzles on-line at Clickmazes.com. You can also play an online version at Thinkfun.com.

Step Mazes

This is a vintage puzzle called "Pike's Peak or Bust." It was patented by Judson M. Fuller in 1894 (518061) and made by Parker Brothers circa 1895. Move the metal traveler from peg to peg from the base to the top of Pike's Peak. Featured in Slocum and Botermans' "Puzzles Old & New" on page 136.

Here is my solution:


The Yankee Puzzle - a vintage route-finding puzzle patented in 1896 by W. G. Adams (554565), described in Slocum and Botermans New Book of Puzzles on page 110:

"Yankee Puzzle
Patent Allowed
Take the disk off by moving from one pin to another,
USING NO FORCE.
Then replace it so it will cover the circle.
Adams & Forbes, sole owners & Mfrs.
Philadelphia, Penna."


Here are some creative step maze puzzles made by George Miller, designed by Oskar van Deventer:


Sunflower

This is "Bronco" by Oskar van Deventer. Move the Bronco out of the starting gate.

Another Oscar van Deventer design made by George Miller - Free Willy.
 
The Rotten Apple
Yet another Oscar van Deventer design made by George Miller.


The "Sunflower" design was picked up by Hanayama, who created an entry called O'Gear in their wonderful "Cast" puzzle series.

This is my solution to the Hanayama Cast O'Gear puzzle...

The puzzle consists of two pieces - a hollow cube and a "key." The key piece has five tabs, and on one side there appear "dots" imprinted on the tabs. There is one tab having a hole through it. Hold the key so that the tab with the hole is on top and the side with the dots is facing you. Number the tabs starting with assigning 1 to the tab clockwise of the tab with the hole. This tab number 1 has a notch in one edge. Proceed clockwise, ending by assigning 5 to the tab with the hole. Tab 4 will also have a notch in it.

The cube has six faces and a crossed hole in each face. One hole contains an extension which allows tab 1 to be freely inserted into and withdrawn from the cube. This is the exit hole. Once a tab is inserted into the cube, the key can be "rolled" in various directions around the cube, transitioning from face to face without being released via the clever geometry of the key and holes. On each face, the key can also be twisted to re-orient the direction in which the dots side of the key is facing.

The cube face opposing the exit face contains 2 small holes at diagonally opposing corners, which permit the key to "perch" on this face. This is the start hole. Another face contains a small triangular symbol. Hold the cube so that the face with the exit hole is upwards and the face with the triangular symbol is facing you. Label the face which is upwards Up (U) and the face which is downwards Down (D). Label the face towards you South (S), the face away from you North (N), the face to the right East (E), and the face to the left West (W).

Using these conventions it is now possible to uniquely label every possible state of the puzzle using three characters: first, the letter of the face in which the key is currently embedded. Second, the number of the tab embedded in the cube. Third, the direction in which the dots side of the key is facing. The number of total possible states is 120: 6 cube faces X 5 tabs X 4 key facings possible for each cube face. These 120 states can be depicted as nodes in a graph. The exit node is U1N. The "perching" state can be reached from either D5S or D5E.

Each of these 120 nodes can be connected by at least one and at most three edges to other nodes: a single edge representing the act of twisting the key while remaining on the same face and tab and thus changing the direction in which the key is facing; an edge representing rolling the key clockwise (relative to looking at its dot face); and an edge representing rolling the key counterclockwise. Not every face of the cube permits all possible actions - you will note that some of the cube's edges are rounded while others are sharp. The sharp edges prohibit the key from rolling across them.

Here is a path from Start to Exit:

D5E - D5S - E1S - E1U - S5U - W4U - N3U - N3E - D2E - D2S - E3S - E3U - S2U - W1U - N5U - N5E - U1E - U1N


Other step mazes:


This is my nicest route-finding puzzle -
it is called The Wanderer.
Tom Lensch made it.

Hanayama picked up the Wanderer, too, and calls it Cuby.

Oskar's Disks

Free the Key

Locomotion

Big Wheel

19th Hole - Pentangle


Here is my analysis of Hanayama's Cuby puzzle. Don't read too far if you want to try this great puzzle yourself!

The Cuby is another wonderful design from the diabolical mind of Oskar van Deventer. This puzzle consists of a traveler (originally known as the Wanderer) shaped like a quarter of a watermelon, and a hollow cubic cage.

Each face of the cube has a square central opening, and around each opening's perimeter there may be up to eight notches, two on each side of the four sides of the opening. Some of the notches are absent - I have indicated them in yellow and numbered them. One of the notches, shown in black, is wider than the others. One of the faces of the cube has two decorative "eyes" at one corner - this allows you to easily orient the cube.

The traveler has a square cross-section, but two faces are curved in such a way that it can be rotated from face to face within the cubic cage. On one flat face, the traveler has an oblong peg - each end of the peg comes to a point and there is a half-circle bulge in the center of the peg.

Initially, the traveler is trapped inside the cubic cage, with each of its two ends sticking out the openings in two opposing faces of the cube. (If you've got it through two adjacent faces, it's in the middle of a move - fix it.)

Because of its clever shape, you'll find that you can maneuver the traveler by retracting one end fully into one face of the cube, then exploiting its curvature to move the other end into a perpendicular face. Some moves will be prevented by the absence of a notch in the perimeter of a face's central opening - the notches are required to accomodate the peg on the traveler.

Only the wide notch will allow the bulge in the peg to pass through - so the traveler can only be freed from the cubic cage by navigating it to a specific orientation within the cage.

The state of the puzzle can be fully described by giving the current orientation of the traveler within the cube. To follow my arbitrary naming convention, hold the cube with the face having the eyes towards you, with the eyes in the upper-right-hand corner. Label the top face UP (U), the bottom face DOWN (D), the eyes face SOUTH (S), the opposite face NORTH (N), the right face EAST (E) and the left face WEST (W).

One can now specify the orientation of the traveler by giving two letters - first, the letter of the face through which whose opening you can see the peg, and second the letter of the face (perpendicular to the first) toward which the peg's bulge is pointing. The peg can face towards 6 faces, and at each such position the bulge can be pointing towards 4 perpendicular faces so there are 6x4=24 possible states. Each of the 24 states is shown in the diagram below as a large circle containing the two-letter code for that state.

The goal state is SE and the starting position is SW - the peg is initially visible through the opening in the face with the eyes, and the peg looks kind of like an enigmatic smile.

 

A move consists of two parts - first, retract the traveler fully into one of the two faces through which an end is poking, then rotate the other end into one of two perpendicular faces. So from each state, there are 2x2=4 possible moves - however, because of the missing notches, some will be prohibited since the peg will be blocked in some way. In my notation, a move is also labeled using two letters - first, the face into which you retract the traveler, then the face toward which you move the opposite end. The moves (and their inverses) are given on the arcs of the graph.

The graph has five "zones" - the red portion is kind of a "railroad" from state US to the solution. There is one dead end, shown in purple. There are three loops, in orange, green, and blue. Here is my eight-step solution sequence - states are shown in parens and moves between them:

(SW) DE (SU) ES (EU) ND (ES) UE (US) WU (WS) UN (WD) SW (SD) WU (SE)

There are three ways in which a missing notch can block a move - (1) it can prevent the first, lateral part of a move by blocking the peg; (2) it can prevent the second part of a move by blocking the peg at the destination face; and (3) it can prevent the rotation by blocking the peg at the pivot edge.

In the diagram, for each node I have also shown in the yellow rectangles how a given absent notch blocks a move. The notation gives the move, an X, the notch number, and L, D, or E depending on the blocking method as given above. An L-type (lateral) block really blocks two moves.

I have shown one of the blocked moves, at the dead end, in magenta. In my copy of the puzzle, it seems like the traveler is impeded from even retracting into the U face. I am not sure if this is intentional, or merely a manufacturing defect.

Some thoughts and unanswered questions to ponder:

Ring-in-Plate Mazes

Ring-in-Plate puzzles are a variety of step maze.

   

I finally found an example of the vintage Maze Puzzle, patented in 1892 by Joel W. Thorne (483820) (483820 on Google Patents)
This is a ring-in-plate type maze - mine is missing the ring. Interestingly, the patent specifies an extra "decoy" hole, and a bull motif not used by the actual puzzle. The gap in the ring is supposed to be only just wide enough to fit the plate, not the raised maze walls. Also, note the elongated holes, which will allow the ring position to be adjusted for certain moves. Both those features are unusual for a ring-in-plate maze.

 
The Spider Puzzle
made in Occupied Japan

Here are some examples in my collection - some of these appear in the Tanglements section, too.

(The above appear in the Tanglements section, too. The puzzles shown below do not.)

At left is Kohner's Toothache puzzle, from 1971. Maneuver the C ring from the upper left to the lower right. It does not exit the board. This was a member of a series of Kohner puzzles, which also included Heartache (1971) (see my section on Sliding Piece Puzzles), and Belly-Ache (which I do not have). You may also remember Headache, which was a Pop-O-Matic game, not a puzzle.


Below is my solution to Hanayama's Cast Plate puzzle. Refer to the image of the plate to identify the hole numbers and follow the route.

Multi-Plate Mazes

In this type of maze, multiple plates and/or travelers must be moved in coordination.


Saunders' puzzle
aka the National Puzzle
U.S. patent 766118 - Samuel L. Saunders - 1904
The traveler piece comprises two small brass buttons connected by a short axle - the axle rides in the channels cut in the circular plates. The two plates are connected in the center by a hollow rivet and can be rotated with respect to each other. Both plates are identical - the one in back is flipped. The objective is to navigate the traveler from the central hole towards the outside where it comes free, then back in, via the channels formed when the two plates are moved into various alignments.

I've mapped out a solution to the Saunders Puzzle:

The channels in the puzzle plates are organized in concentric rings, and can be divided into radial pathways and circular channels. In the diagram, I have assigned identifiers to the concentric rings, starting with the channel at center labeled C1, and alternating radial and circular rings outwards to the outside (free) position I label C6.

There are six circular channel rings, C1 through C6, and five radial pathway rings, R1 (innermost) through R5. I've colored the radial rings red and the circular rings cyan, and numbered the individual segments in each ring clockwise, starting on the upper right.

The identifier for a circular channel segment is Cij where "i" is the ring and "j" is the segment. Similarly, the identifier for a radial path segment is Rij where "i" is the ring and "j" is the segment. We can label the two sides of the puzzle (the two plates) A and B. Since the two plates are identical, we can use the same notation on each side and prefix with the symbol for the side.

The position of the traveler anywhere in the puzzle can be given by noting the identifiers on the A and B sides of the circular channel segments where the axle resides, e.g. ACij*BCxy. The traveler can be moved from ring to ring only when a radial pathway in the A layer lines up with a radial pathway in the B layer. My notation for a pathway is ARij*BRxy.

In the plate, there is only one path - easily seen - from the center out (or vice versa). The solution entails navigating the traveler along this path in both plates simultaneously. There are numerous dead ends to be encountered if you deviate from this, and it is easy to screw up by making an incorrect rotation.

Below is the solution path using my notation - channel locations alternate with radial paths. From each location, you have to rotate the plates to create the necessary path to proceed to the next location. The key to staying on track is to note that in every case, for either ACij*BCxy or ARij*BRxy, x=i and y=j. In other words, to re-iterate, each movement and rotation must be done such that the traveler is visiting the same location in both plates simultaneously.

AC11*BC11, AR11*BR11, AC21*BC21, AR21*BR21, AC31*BC31, AR32*BR32,
AC42*BC42, AR42*BR42, AC54*BC54, AR43*BR43, AC43*BC43, AR33*BR33,
AC32*BC32, AR34*BR34, AC44*BC44, AR44*BR44, AC56*BC56, AR45*BR45,
AC45*BC45, AR35*BR35, AC33*BC33, AR36*BR36, AC46*BC46, AR41*BR41,
AC51*BC51, AR51*BR51, AC61*BC61

Here are further examples of multi-plate mazes.


Mysterians
Made by and purchased from George Miller
Designed by Oskar van Deventer and presented by Nick Baxter at IPP23 in Chicago.
Named after the '60s rock group the ???Mysterians, who had one hit 96 Tears.
The three plates each contain one question-mark-shaped channel. The acrylic barbell traveler comes free after 124 correct moves.

4D
Designed by Oskar van Deventer
Made by and purchased from George Miller

Hanayama's L'Oeuf
Designed by Oskar van Deventer

Holey Plates and Holey Plates II
Markus Goetz
See U.S. patent 932147 - Levine 1909.
 
Maze Medal - designed by Oskar van Deventer
Comparison to an earlier version made by George Miller
 

     

Cross and Crown
U.S. patent 1071874 - Louis S. Burbank - 1913
The first two photos show the cross-side and pin-side with all four pins at their innermost state - this is the starting state of the puzzle. The goal state is to move all four pins to the outside, so that the cross (with the pins) can be freed from the crown. According to the patent, this will require "not less than 681 moves." The second two photos show the cross-side and pin-side with the pins in some intermediate stage.

Prolific puzzle-designer Jean-Claude Constantin issued a puzzle called Kugellager, along with several derivatives, very similar in concept to the Cross and Crown. Puzzle collector Goetz Schwandtner discusses this class of puzzles on his website, and has written a nice article about the Kugellager family.

I brought this puzzle with me to IPP32, knowing I would see Goetz there, so that he could have a try at solving it. And solve it he did! See the solved photo below (thanks, Neil)...

 
Ling Meiro / Panel & Ling
An interesting ring-in-plate maze from Japan.
Three maze cards - blue-green (level 1), yellow (level 2), and pink (level 3),
each fit into a shuttle that contains a sliding 4x4-hole frame and the ring.
You must navigate the ring, within the frame, from each card's start position to its goal position.

Unicursal Puzzles

In graph theory, an Eulerian Path is a tour which visits each edge exactly once. The tour is allowed to cross itself - i.e. vertices may be visited more than once. An Eulerian Cycle is a tour that starts and ends at the same vertex - it's a closed path. This terminology follows the famous mathematician Leonard Euler who investigated the Seven Bridges of Koenigsberg problem in 1736.

Here is a key result: a graph will have an Eulerian Circuit only if it has no odd nodes, and will have an Eulerian Path only if it has exactly two odd nodes.

Such a graph is also called unicursal - one can draw the required circuit from start to finish without taking one's pencil from the paper, and without retracing any edge, though crossing an already-drawn segment is usually allowed (although technically you're not supposed to cross any segment more than once). So, a unicursal puzzle calls for you to find an Eulerian Circuit (sometimes just an Eulerian Path) of a graph. Sometimes they're called single-stroke figures. If you can draw the graph right and correctly determine the degree of each node, you can easily tell if you can solve a unicursal puzzle.

Unicursal problems are discussed in Ball and Coxeter's Mathematical Recreations and Essays (11th Ed. 6th printing 1973) in Chapter IX starting on page 242.

You can see some unicursal drawing problems at The Unicursal Marathon. There are also several unicursal puzzles at mathpuzzle.ca.

Hoffmann gives some Single-Stroke Figure problems in Chapter X, No. IX.

A Hamiltonian Path is kind of the complement of an Eulerian Path - it visits every vertex exactly once, but may repeat edges. A Hamiltonian Circuit starts and ends at the same vertex (which, therefore, you're allowed to visit twice). In general, determining whether a Hamiltonian Path exists for a given graph, and what it is, is an NP-complete (i.e. very hard) problem. See the Traveling Salesman Problem (TSP) for an example.

In 1857, Sir William Rowan Hamilton invented the Icosian Game, in which one must find a Hamiltonian Circuit of the vertices of a dodecahedron. Also called the Hamiltonian Game, it is discussed in Ball and Coxeter on page 262. It was produced as a puzzle, but there are only four known surviving examples - see a picture of an original at Dalgety's Puzzle Museum site.

The Knight's Tour puzzle is also a type of unicursal route-finding puzzle, requiring the discovery of a Hamiltonian Path around the 64 squares of a chessboard, following the edges that connect them defined by legal knight's moves. Dan Thomasson gives some example puzzles on his site.


Here is an example of a unicursal puzzle. It's called Chain 16 and was issued by the "Are Jay Game Co., Inc." of Cleveland Ohio. David Singmaster calls this pattern the "brick pattern" and cites several references to it in the puzzle literature. One challenge associated with the brick pattern is to draw it in only 3 strokes. This is impossible, but drawing it in 4 strokes is possible. Another challenge is to draw a path crossing every wall once - this is the challenge posed by the Chain 16 puzzle.

Chain 16 is simply a wooden block printed with a figure of the bricks pattern, and a long thin brass chain. The "Object of the Puzzle" is as follows:

"There are 16 'walls' with an opening in each. Using the chain, can you lay out one continuous line going through each opening? You are not allowed to go through the same opening more than once or cross over the chain."

I've drawn a graph overlaying the puzzle, and defined six nodes A through F, and sixteen paths 1 through 16, corresponding to the openings. I've also shown the degree of each node.

So, can you solve it?

Trick solutions entail having the path go through a vertex, or within a wall. Another trick, when the puzzle is printed on paper, entails folding the paper. I don't think they're valid. The brick pattern puzzle is mentioned in Gardner's First Scientific American Book of Mathematical Puzzles and Games, in Chapter 12. It is also discussed in Dudeney's Amusements in Mathematics, as problem #239, where the object is to draw the pattern in three strokes. Dudeney includes it again in 536 Puzzles & Curious Problems, as #414 "Crossing the Lines," where the challenge of drawing the path crossing each wall once only is discussed.


Here is another unicursal puzzle. Catch of the Day, from Bits and Pieces, consists of a board illustrated with several fish. There are nailheads protruding from the board at several locations, and a long line with a hook on the end affixed to the center. The objective is to run the line around the nailheads and return it to the center, such that each nail is touched only once, and two fish end up enclosed in each loop of the line. The solution is included.

This is similar to the vintage Fisherman's Puzzle described in Slocum and Botermans' The Book of Ingenious and Diabolical Puzzles on pages 106-107, U.S. patent 552167 - Alphonse W. Ziegler 1895 - and manufactured by Jarvis & Company. Also see U.S. patent 658083 - Favour 1900.

In their New Book of Puzzles on pages 108-109, Slocum and Botermans describe another series of similar "stringing" problems played on a nail-studded square grid, called the Oklahoma Puzzle.


Rectangle String Route, designed by Tod Muroi, made by Here to There Puzzles, exchanged by Saul Bobroff at IPP32
Given a large tile with 5 horizontal and 5 vertical evenly-spaced grooves cut into each face and wrapping around the edges, route the single string through all the channels without crossing itself. The route must be a mirror image on the opposite side of the grid.

Thread the Maze Golf Tangle
Invented by Shane Murphy
The history of the Thread the Maze puzzle is online at toy-and-game-inventor-success.com.