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.

The goal of Subway Shuffle is to move the red subway car to destination, encircled in red. There are other subway cars blocking your way (in the example above right, there are four blue cars, one yellow and one green). To free your path to the destination, the other cars will have to be moved away. However, they only can move along their own lines (in the same color as the cars). By the way: there is also a free version of Subway Shuffle with only a few levels to try it out.

Although the graphs are really small, the puzzles get incredibly complex, up to the point where it is no fun anymore for a human being. However, the simpler levels are really great puzzles!

The idea to make sliding block puzzles on a graph actually predates the 15-puzzle and all other sliding block puzzles. Botermans & Slocum (1986: 122) describe the “Siege of London Puzzle” which uses counters that have to moved along the lines of a graph to reach a particular end configuration. The earliest known mention of this puzzle is from 1803! Another early sliding puzzle on a graph is the Moulin Rouge puzzle from 1890, developed by M. Fleury (shown to the left). The goal of Moulin Rouge is to rearrange the letters to from the word ‘moulin’.

The real breakthrough of Subway Shuffle, which makes it way more interesting (and complex) is that not all pieces move along the same graph. This seems to be an original idea of Bob Hearn. There is already for a long time a collection of similar puzzles available online, the Meandering Monk Maze by James Stephens (whose Picnic puzzle already came up in this post). However, he explicitly credits Bob Hearn as the inspiration for these puzzles, and specifically his “Martin Gardner coin puzzle”.

A next step now of course would be to introduce larger “block” again, for example in the form of cars that are connected by a fixed bar in between them, allowing them only to slide in such a way that the distance between the cars remains the same. But, as always, the possibilities of new puzzle designs is infinite – it only takes a dedicated developer to also design actual puzzles, and find a nice balance between complexity, appearance, and gameplay.

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