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The individual pieces can be easily arranged in many different ways that partially satisfy the rule - but overall there is usually only one total arrangement (or a relatively small number among all possible arrangements) that will completely meet the goal.
Finding a solution within a huge range of possibilities can be very hard, and this class is often attacked using computers - but beware, the class of edge-matching puzzles is NP-Complete .
It is evident in Haubrich's Compendium that these shapes comprise the majority of tile shapes used in existing edgematching puzzles. Grandpa's head must always be upright on every card, so the cards cannot be rotated.
Rectangles have also been used, as have octagons (allowing empty areas). Jacques says this is the first example of a corner dismatching puzzle. The Besco Soap Puzzle - from the Beaver Soap Company of Dayton Ohio.
Tile-laying puzzles (in German, "Legespiel") of both the edge-matching and polyform varieties have been explored and produced by Kate Jones at Kadon, and you can read a lot of interesting material at the Kadon site - see Edgematching Colors and Shapes, and More About Edgematching. You can see some original designs by Yukio Hirose here.
Take a look at George Hart's article "A Color-Matching Dissection of the Rhombic Enneacontahedron." Also see Peter Esser's page. Jacques suggests the following seven categories: Jacques also defines a classification scheme by which one can identify the puzzles abstractly and find isomorphisms (i.e. Jacques' conclusions, based on computer analysis, regarding the best approach to solving these puzzles agree with my empirical findings - in general, fix the tile in the middle and work around it.
According to Jacques, this is the first known example of a Heads/Tails puzzle (the two numbers which must add to ten comprising the head and tail).
Success is usually determined by careful inspection, checking to see that the given rule has been everywhere satisfied.
You might see some puzzles in this section that remind you of jigsaw puzzles - but the key difference here is that the edge features are compatible with several other potential mates and there are usually several other pieces that match any given piece.
A good tile to choose for this middle position is the tile with the most possible matches.
As discussed in my polyforms section, the only regular polygons which can be used to completely tile the plane are the equilateral triangle, the square, and the hexagon. It is described on page 36 of Slocum and Botermans' "Puzzles Old & New." The goal is to form a 3x3 grid such that at the points where the quarter-circles on the corners of the tiles meet (either four or two), there are always different colors on each of the meeting quarter-circles.
edges or corners) match (or complement, or dismatch) those at corresponding points on abutting pieces" or "the heights (or numeric values) of aligned pieces total a specific constant" or "defined sets of faces have all distinct (or equal) features." Sometimes, as in the case of matchstick puzzles or the "Eight Queens" puzzle, the individual pieces are indistinguishable, but their arrangement in a particular pattern is paramount.