A primer on maze puzzles

Mazes are a favorite kind of puzzles for many. Actually, I will argue below that all sequential movement puzzles are actually concealed mazes. However, one important distinction to be made first is that between a maze and a labyrinth. A labyrinth (like the famous one shown to the left here) is only a single twisted path without junctions. There is no possibility to get lost here, and consequently it cannot really considered to be a puzzle. Only when junctures and dead ends are introduced, the puzzling aspect becomes worthwhile, and then the puzzle is called a maze.

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Chip’s Challenge: a Sokoban Laboratory

For those of you who like the idea of Sokoban, but find it a bit abstract, Chip’s Challenge is probably a good alternative. The original game was designed in 1989 by Chuck Sommerville for the Atari Lynx (in a sense an iPhone predecessor). In Chip’s Challenge, you guide around Chip McCallahan through a series of increasingly difficult puzzles. Many of the puzzles have Sokoban parts in them, but there are many other additions, like ice for sliding, conveyor belts, floors that only be passed once, and bouncing balls that ask for precise timing.

There are various games in the app store that take more or less direct cues from Chip’s Challenge, although I have no idea whether there are any direct links. None of the games even mention Chip’s Challenge. Only one of the games (Loopy Laboratory) cites Chip’s Challenge so directly that it is clearly inspired by it.

Sokoban: not just sliding, but pushing blocks

In his classic definition of Sliding Block puzzles, Edward Horden explicitly noted that “there must be no requirement that they be pushed or pulled by other pieces.” So, let’s change the topic from sliding blocks (as discussed in the last few posts) to pushing blocks.

The most well-known instantiation of a pushing block puzzle is Sokoban (Japanese for ‘warehouse keeper’), created in 1980 by Hiroyuki Imabayashi. In this puzzle, various blocks must be pushed to their destination in a twisted warehouse. The blocks cannot be pulled, so when a block is pushed into a corner it is stuck, and eventually the puzzle will have to be restarted.

Rubik’s Cube: great apps, but wasted efforts…

Ernö Rubik’s Cube is possible one of the most well-known puzzles of recent years. Almost exactly a century after the 15-puzzle craze of 1880, Rubik’s Cube set off its own craze from 1980 onwards. It’s a fantastically great puzzle (it actually was the puzzle that got me into the hobby of collection, solving and designing puzzles), and now it has come to the iPhone as well. However, it might just as well have passed this opportunity. The iPhone is not the right place for a Rubik’s Cube.

Re-Shuffling Rush Hour

A sliding block puzzle is normally played inside a tray, in which the block can move around orthogonally. How about other possibilities of moving block around? Enter Subway Shuffle: Rush Hour on a graph. Or, as the developer Bob Hearn explains it:

Imagine playing Rush Hour, but make all the cars 1×1 instead of 1×2 or 1×3. But still some can only move vertically, others horizontally. Replace the cars by tokens, red for horizontal, blue for vertical, and put the tokens on nodes of a graph with colored edges: red tokens can only move on red edges, blue on blue. If you use a grid graph, with horizontal edges red and vertical edges blue, then you have 1×1 Rush Hour. If you relax the grid graph constraint, and also allow more colors, then you get Subway Shuffle. These relaxations allow for lots of interesting possibilities in puzzle configuration. It’s also possible to have extremely complicated puzzles in a very small space: I’ve generated some puzzles on small graphs that require over 1,000 moves.

Know How to extend Rush Hour

Things move fast in the app store. I just wrote this summary on Rush Hour puzzles, and then today a new variation becomes available: Know How [iTunes Link]. I have not bought the game (at 3€ I find it too expensive to try it out; hopefully a lite version will become available soon). However, from the description and the other information on the web it is clearly a Rush Hour variant. The block can only slide in the direction of the elongation, and the treasure box has to be moved out to a destination outside the playing field (at the bottom in the screen shot to the right).

Rush Hour

One approach to make sliding block puzzles more complicated is by restricting the movements of the blocks. The most prominent example of this strategy is Rush Hour. In this puzzle, all sliding blocks are rectangles of size nx1. These blocks can only be moved in the direction of the elongation. The original trick to make this restriction transparent to the user is to show the block as cars (hence the name “Rush Hour”). It is then immediately obvious that the car can only be moved forwards or backwards, and not sideways.

Rush hour was invented by Nobuyuki “Nob” Yoshigahara and marketed as a physical puzzle by ThinkFun (formerly Binary Arts). There are at least four implementations of Rush Hour available in the app store (links redirect to iTunes):

Toroidal sliding block puzzles

A new variation on the sliding block theme was recently introduced in the app store: sliding blocks on a torus! Of course, the marketing of the puzzles do not call it like that, but it actually is a nice further development, which uses the possibilities of the touch screen to produce a puzzle variation that is not easily reproduced in a physical puzzle.

So, what is a toroidal sliding block puzzle? Well, the basic idea is that the left and the right sides of the puzzle are connected, and likewise the top and the bottom sides. When you slide something off the top, it will reappear on the bottom, and likewise for the other sides. Mathematically speaking, this means that the pieces are sliding on a torus.

Pennant-like sliding block puzzles

After the 1880’s craze of the 15-puzzle (see the previous post) developments lay still for a few decades. The next development was a sliding block puzzle adding larger blocks of sizes 1×2 and 2×2, and enlarging the tray from 4×4 to 4×5. The first such puzzle was patented in 1907 by L. W. Hardy (U.S. Patent 1,017,752, patent granted 1912) and copyrighted on the name ‘Pennant puzzle’ in 1909. The goal of this puzzle is to move the 2×2 piece to a predetermined position on the bottom of the tray.

This puzzle is one of the most widely commercialized puzzles in the world, existing in dozen of variants. Likewise, there are at least seven implementations of this puzzle available in the app store (links redirect to iTunes Store):

A primer on sliding block puzzles

Given the gorgeous touch screen it has, the iPhone is destined to be used for sliding block puzzles. Edward Hordern, the grandmaster of sliding block puzzles defined these kind of puzzles as follows:

“A sliding block puzzle consists of a group of pieces of any shape(s) enclosed within a confined area, in which the purpose is to rearrange the pieces either into a certain order or to get a particular piece to a specified position. This is accomplished by sliding the pieces or “blocks” —hence the name sliding block puzzles — usually one at a time into areas not occupied by other pieces. The lifting of pieces is never allowed — nor must they hop or jump over other pieces.”

Edward Hordern’s classic 1986 volume on these puzzles ‘Sliding Piece Puzzles’ is unfortunately long out of print, but a selection of the wide array of puzzles that have been develop in this realm can be found on Nick Baxter’s Sliding block page on PuzzleWorld and Rob Stegmann’s Sliding Block page.