The Dodeca Match puzzle consists of 6 fat rhombic (with acute angle 60 degree) tiles, 6 thin rhombic (with acute angle 30 degree) tiles and 3 square tiles. The object of the puzzles is to put them to form regular dodecagon (regular 12-gon) with matching patterns on them.
Note: Let n be an integer greater than 1. In general, regular 2n-gon (the regular polygon having even number of vertices 2 * n) can be filled with rhombic tiles with their edge length all the same as the edge length of 2n-gon and their acute angle varies multipe of 180/n degree. Thus, the number of variations of rhombic shape is n/2 for even n and (n-1)/2 for odd n, where for even n, one of the them is square (foursquare); its acute angle is 180/n * n/2 = 90 degree. To fill the 2n-gon, it requires n rhombic tiles for each variation of shapes except square tiles which requires n/2 for even n.
The object is to put the tiles shown in the above picture together to form dodecagon (12-gon) with matching patterns. Tiles are not reversible. The tiles collects all possible patterns such that: a tile is divided into two connected regions of different colors with the border line passing through midpoints of 2 edges.
It seems impossible to make every border lines form loops or single loop.
This puzzle is an alternate design of the previous puzzle, but logic is different. The object is to put tiles together to form dodecagon (12-gon) with matching line patterns and their thickness as shown in the picture above. The logic is equivalent to Dodeca Match (Dapple) if colors unmatched at every edges of its tiles. It seems also impossible to make every lines form loops or single loop.
The object of this puzzle is to form a dodecagon with matching colors at every edges of tiles. The tiles are reversible, and the patterns on both faces are mirror images each other for each tile. The patterns on the faces collects all possible patterns on the rhombus or square tiles (12 or 6 kinds of patterns, respectively) such that 4 edges of the shape painted with 4 different colors. This puzzle is fairly difficult to solve.
If one match colors pairwise such as matching green with blue and red with transparent, as shown in the second picture, another puzzle will be made. This puzzle is much harder than the previous one.
This puzzle is an alternate design of the above puzzle, where one can play it with various constraints. The tiles are reversible, and the patterns on both faces are mirror images each other for each tile.
In Dodeca Match (Dapple), the border line passes through midpoint of 2 edges. Another puzzle will be made by changing the crossing points to be at vertices. Similar to the previous Dodeca Match, the object of this puzzle is to put the tiles shown in the above picture together to form dodecagon (12-gon) with matching colors. The tiles are not reversible. This puzzle seems to be much easier than the other Dodeca Match puzzles, but additional conditions will make it interesting:
A solution exists for the first conditon. For the others, solutions not discovered yet. Another puzzle will be made by changing the matching rule so that colors unmatched at every edges of tiles. (In this case, it is impossible to make all the outer edges of dodecagon to be uniform color.)
The object of this puzzle is to put the tiles shown in the above picture together to form dodecagon (12-gon) with forming the pattern at every vertices to be hexagon. Tiles are not reversible. The patterns collects all possible placement of regular hexagons at each vertex of a rhombus/square such that: (1) All 4 hexagons are congruent. (2) The center of each hexagon exactly meets one of the vertices. (3) The angle between one of diagonal lines of hexagon and one of edges of a tile is 15 dgree. (4) The pattern on a tile can not be congruent with rotation of the tile itself.
Another design will be made by replacing the criterion (3) such that (3') The angle to be multiple of 30 degree. The logic is equivalent, because a solution of one puzzle can be transformed to a solution of the other one by rotating each of hexagons on the whole pattern by 15 degree.
(Solutions not found yet.)
Similar to Dodeca Match (hex), the object of this puzzle is to put 15 tiles shown in the above picture together to form dodecagon with forming the pattern at every vertices to be squares. Unlike Dodeca Match (hex), tiles are reversible, and the patterns on both faces are mirror images each other for each tile. The patterns of faces collects of all possible placement of squares at each vertex of a rhombus/square such that: (1) All squares are congruent. (2) The center of each square exactly meets one of the vertices. (3) The angle between one of diagonal lines of square and one of edges of a tile is multiple of 30 dgree. (4) for each tile, squares on it are oriented to the same direction except one. (Note: There are 45 possible patterns if criterions (4) is removed.)
Like Dodeca Match(4 colors), the object of this puzzle is to form a dodecagon with matching colors at every edges of tiles.
The tiles are reversible, but unlike the others, the patterns on both faces are not mirror images each other for each tile, instead:
(a) 2 arcs drawn for each face (i.e. 4 arcs on a tile) with colors all different (in the picture above, one of 4 colors is 'transparent').
(b) Each arc passes through midpoint of an edge with its center on a vertex of the tile.
(c) 2 arcs on the same face do not overlap at edges.
(d) The arcs on a tile forms a single loop (passing through both faces of the tile).
The patterns collects all possible drawings under this configuration (12 or 6 kinds of patterns for rhombus or square tiles, respectively), which are shown in the first and second picture above. In the pictures, the pattern at the same position indicates the pattern of both faces of the same tile.
3 additional conditions can be applied:
Solutions exists for each conditions. There is a solution that satisfies 2 of 3 conditions.