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 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.
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.
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.
When most people think of
mazes (Wikipedia entry),
the legend of Theseus and the Minotaur probably comes 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 fact 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 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.
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.
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.
Each year, some farms invite guests to get lost in their maize mazes, and parks or palaces entertain visitors with
elegant hedge mazes.
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:
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, thought they have become quite complex in their own right,
some even requiring 3-D glasses!
There are several types of route-finding puzzle - species include:
|
 Backflip |
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|
 Bilz Box |
|
 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 |
 Missing Marble |
 Amazin' Marble Die |
 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.) |
 A-Maze-Ing by Jajaco, from 1963 |
|
 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.) |
 Odd Ball |
|
 KO Labyrinth By the Lepato Co. Ltd. Combines a maze with a 3x3x3 twisty sphere. |
 Color Cube Maze - DaMert |
 Boston Subway made by George Miller |
 Amaze Here, you can move some of the walls, but only from certain positions and only in certain directions. |
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. |
|
River Crossing - 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 
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 "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
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.
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.
