The object of the puzzles here is to put regular triangular tiles together to form certain shapes with matching patterns on them.
The object of this puzzle is to put 24 tiles shown in the picture above together to form regular hexagon of edge length 2 units with matching pattern of colors at all inner edges (in the picture, the tile at the right most lower edge unmatched). The tiles are not reversible. The pattren of tiles differs each other and varies all possible paintings such that 3 vertices and 3 edges of a triangle can have one of 2 colors (black or azure).
In the above puzzle, if tiles are reversible and the patterns on both faces are mirror images each other for each tile, the number of variation of tiles is 20, since 8 of 24 patterns are not symmetric. Then, one can imagin that they can cover an regular icosahedron which can be another challenging puzzle. (Note that solutions may not exist. I have not tried it.) The puzzle is similar to 'DODECA-PUZZLE' designed by NOB Yoshigahara which utilizes regular dodecahedoron. Some of Tri-Match puzzle below can also be converted into 'Icosa Puzzle'.
In the previous Tri-Match, the colors must be matched. If one change the rule such that colors must be unmatched at all edges, another puzzle can be made. The picture below is a design which materialize this model.
The patterns collects all the possible paintings of 6 smaller regular triangles on a regular triangle, where 3 of them centered on vertices and the others centered on midpoints of edges, with their directions varies whether the same as the tile or half turn of it. In the design above, the length of edges of painted triangles are half of that of the tile. The matching rule is to form triangles (or partial triangles at the edges of the hexagon) at every verteices and edges of tiles. In addition, the following conditions may make it challenging.
The solutions satisfying both of the first and second conditions exists.
The following picture illustrates a prototype design of this puzzle, which you can also play the previous puzzle by altering the matching rule such that only letter X or I must appear at edges and letter * or transparent must appear at vertices instead of letter Y.
Like the previous Tri-Match, the object is to put 24 triangular tiles shown in the picture together to form regular hexagon with matching pattern of colored lines at all inner edges. The tiles are not reversible. The patterns on the tiles collects all possible drawing in a manner such that:
(1) 3 lines on a tile; drawn from an edge to another edge; crossing at 2/5 or 3/5 point on an edge.
(2) Lines on a tile can not overlap at edges of the tile.
(The conditions (1),(2) implies that for each vertex of a tile, there exists a line whose touching edges shrae the vertex.)
(3) Lines are colored chosen from 2 colors (in the picture, black and red).
It is fairly easy to solve, because some of tiles are equivalent if only colors are considered in matching. The following conditions will make it interesting:
(Nov. 12, 2004)
The underlying logic of the first condition is equivalent with Tri-Match (two tone), by the following correspondence of tiles: Let 'La' denote a line whose touching edges shrae a vertex 'Va'. For each tile of Tri-Match (two tone), one can obtain tiles of this puzzle by transforming [color of vertex Va to color of line La] and [color of the edge shared by 2 vertices Va and Vb to whether La and Lb cross or not]. With this transformation, a solution of Tri-Match (two tone) corresponds to a solution of this puzzle of the first condition and vice versa.
This puzzle is similar to the previous Tri-Match (rings) but much hard to solve. The patterns are obtained from the previous puzzle by removing a line or 2 lines from patterns of the previous puzzle, where lines of the same tile must have different colors in case 2 lines exist in the pattern. Tiles are not reversible. Following conditions can be applied:
Note: In the first condition, it seems impossible to make the lines into a loop. For black colored loop, there exists 4 tiles which contain no black lines. Thus, the loop must pass through 20 tiles of 24, but seems impossible to make such a loop in the hexagon of edge length 2.
Unlike the above Tri-Match puzzles, the tiles are reversible and the patterns on both faces are mirror images each other for each tile. The 48 pattrens on the tiles varies all possible paintings on a triangle such that fans (part of a unit disc) of 3 different colors chosen from 4 colors are painted on each vertex in arbitrary order of overlapping. The object of this puzzle is to put these tiles together to form a regular hexagon with matching colors and shape of discs at all inner edges. It is not so hard without additional conditions:
This is a variation of Tri-Match (discs). The difference is the pattern pairing on faces of tiles: The tiles are made as if clipped out from real 3 stacked discs. Instead of mirroring, the top most disc on a face becomes the bottom most disc on the back face for each tile. As you can see in the picture, mirroed pattern resides other tile for any pattern. The addtional conditions of Tri-Match (discs) can also be applied.
(Currently, unsolved) Put 24 tiles shown in the picture above together to form a regular hexagon with matching pattern of lines at opposite sides for each edge AND lines at opposite sides for each vertex. The tiles are not reversible.
There exists a similar puzzle which collects all possible paintings on a regular triangle such that each of edges can have a color chosen from 4 colors, which can be seen in the cover of the book 'Play Puzzle' (HEIBON-SHA, 1981, the author TAKAGI, Shigeo). It is a part of collection of puzzles designed by Percy Alexander MacMahon (1921) called MacMahon's Color Tiles.
In this puzzle, the pattern consists of 4 types of lines (one of line types is transparent) vertical to each edges of tiles, insetad of coloring edges. The difference of logic between two puzzles is that 'matching lines at opposite sides for each vertex' vs 'making colors of all outer edges of the hexagon to be the same'. This puzzle is much harder than the cited one. Can you solve it?
The title has been changed from 'Tri-Match (vertical lines)' .
This puzzle is a variation of Tri-Match (bisectors), but not equivalent. The number of line types is only 2, but lines are shifted to left or right in the tile. Lines shifted to the right matches to lines of the same type shifted to the left. Of course, all possible patterns are collected. The solution exists.
This is also a variation of Tri-Match (bisectors #1). It is a mix of Tri-Match (bisectors #1) and Tri-Match (bisectors #2). The number of line types is 3 (one of it is transparent in the picture), and the only one type of lines (narrow lines) are shifted to left or right. This puzzle can be equivalent to the puzzle 'TRIFOLIA' by Kate Jones (which you can see from this link at gamepuzzles.com ), if the condition of 'matching lines at opposite sides for each vertex' is replaced to 'making lines at all outer edges of the hexagon to be the transparent'. Solutions exist.
Put 24 tiles shown in the picture above together to form a regular hexagon with matching pattern of lines and making flyovers at all inner edges. The tiles are not reversible. The patterns collects all of possible drawings in such a way that connecting the mid points of the edges of the regular triangle by 3 lines of 2 types with making overlapping order for each crossing of 2 lines at the edges. There exists one to one correspondence between tiles of Tri-Match (two tone) and tiles of this puzzle ([color of an edge <-> overlapping order at an edge] and [color of a vertex <-> type of line]), however, the logic is different.
This puzzle is a variation of Tri-Match (flyovers), but the logic is not equivalent. There is only one type of lines, but lines have a distinction of 'left and right' side. Thus, they can be drawn on tiles in 2-way. The solution exists. It's very hard.
Put 16 tiles shown in the picture above together to form a regular triangle of edge length 4 units with matching pattern of lines and making flyovers at all inner edges. The tiles consists of the tiles of Tri-Match (flyovers), but removing 8 tiles on which 3 lines wraps over circularly. This puzzle is compratively easier because of flexibility of tile placement at corners. However, it migtht become extremely hard if one of the following conditions are applied:
Similar to Tri-Match (flyovers mini), the tiles consists of the tiles of Tri-Match (flyovers #2), but removing 8 tiles on which 3 lines wraps over circularly. This puzzle seems much difficult than Tri-Match (flyovers mini), even if without additional conditions in Tri-Match (flyovers mini).
Put 24 tiles shown in the picture above together to form a regular hexagon with matching color of arcs in a circle of radius 1 unit length for each circle. The tiles are not reversible. The patterns are derived from the tiles of coloring edges with 4 colors as cited in Tri-Match (bisectors #1); here, 4 colors of arcs are used instead of coloring edges. This puzzle is comparably easier because of loose constraints of tile placement at boudary edges.
This puzzle can be played as 3 variations of Tri-Match (bisectors) without the condition at vertices by appling the following conditions (2-4). The patterns are reuse of Dodeca Match (I x YOU = ?).
In addition, a condition of uniformizing colors on the boudary edges can be applied for each conditions except 3rd one (though I have not tried yet).
This puzzle is isomorphic to Tri-Match (two tone); the design is converted so that only hexagons appear.
One additional condition can be applied (unsolved): make white (or circle signed) region to be connected. (Note that it is impossible to make both regions to be connected.) Many solutions exist very close to satisfy this, but not found yet. It may be impossible, but why?
Put the above 24 tiles together to form a regular hexagon with matching colors and shapes of discs at all inner edges. The tiles are not reversible. This puzzle is a mix of Tri-Match (two tone) and Tri-Match (triangles), where the patterns are designed so that the matching logic is the same as the former one at vertices and the same as the latter one at edges.
Put 16 tiles shown in the picture above together to form regular triangle of edge length 4 units with matching the pattern. The tiles consists of the tiles of Tri-Match (discs of two tone), but removing 8 tiles on which 3 discs wrap over circularly. Similar to Tri-Match (discs), one additional conditon can be applied (unsolved): Make the stacking order of all discs (including partial discs at the outer edges) uniquely determined from the whole pattern. (i.e. There exists an upstairing path from the bottom most disc to the top most disc which passes through all discs).
Put the above 24 tiles together to form a regular hexagon with matching colors and shape of discs at all inner edges. The tiles are not reversible. This puzzle is the other mix of Tri-Match (two tone) and Tri-Match (triangles). The patterns are designed so that the matching logic is the same as the latter one at vertices and the same as the former one at edges; the inversion of Tri-Match (discs of two tone mini).
(Unsolved) Put the above 24 tiles together to form a regular hexagon with matching the arcs to form circles of radius square root of 3 unit length. The tiles are not reversible. The patterns collects all possible drawings on a regular triangle of 6 arcs (of radius square root of 3, with one endpoint at a vertex and tangent to an edge) of color chosen from one of 2 colors for each (as usual, one color is transparent in the picture). Very difficult. (Can you find a solution?)
(Unsolved) Put the above 24 tiles together to form a regular hexagon with matching colors of discs at every vertices and align every bisectors to be straight. The tiles are not reversible.
More descriptions will come here; under translation.
(Unsolved) Put the above 24 tiles together to form a regular hexagon with matching colors of discs at every vertices and align every bisectors to be straight. The tiles are not reversible.
More descriptions will come here; under translation.
Put the above 24 tiles together to form a regular hexagon with making a triangle at every vertices and align every bisectors to be straight. The tiles are not reversible. This puzzle is very hard.
More descriptions will come here; under translation.
(Unsolved) Put the above 24 tiles together to form a regular hexagon with making a triangle at every vertices and align every bisectors to be straight. The tiles are not reversible.
More descriptions will come here; under translation.
Put the above 24 tiles together to form a regular hexagon with matching the types of lines at edges aligned in straight. (One of the line types is set to be transparent in this design) The tiles are not reversible.
More descriptions will come here; under translation.
Put the above 24 tiles together to form a regular hexagon with matching the colors and making a triangle for every vertices. The tiles are not reversible.
More descriptions will come here; under translation.
Put the above 24 tiles to form a regular hexagon with matching the patterns; a triangle, a circle or nothing for each vertex. The tiles are not reversible.
More descriptions will come here; under translation.
Put the above 24 tiles to form a regular hexagon with matching the colors at each vertex and make circles. The tiles are not reversible.
The tile selection is similar to Tri-Match (two tone), but matching logic differs (must match arcs at 3 edges adjacent to a tile).
Put the above 24 tiles to form a regular hexagon with matching the arcs to make complete (or partial at the edges) translucent discs. The tiles are reversible and the patterns on both faces are mirror plus inversion images each other (i.e. if a disc resides on a side, the other side must be blank, and vice versa).
The selection is similar to Tri-Match (circles and discs), but the center of a tile can differ whether inside or outside of a disc. Thus, the patterns are doubled; 48 patterns.
Put the above 25 tiles to form a regular triangle of size 5 with matching the patterns. The patterns must not expose at outer edges. In the above figure, the inner edges marked with red circles are not matched. The patterns are collections of the patterns satisfying the following conditions: (1) Each edeg can have one of 4 kinds of patterns or no pattern, (2) The pattern on the edges must be different each other among the same tile. This puzzle seems very hard.
The following figure demonstrates a variation of the above puzzle to form a regular hexagon of size 2, where the tile with no pattern on all 3 edges removed. (Of course, the patterns must not expose at the outer edges.) This variation seems fairly easier than the above one.
Put the above 25 tiles to form a regular triangle of size 5 with matching the patterns (the patterns must not expose at the outer edges). The tiles are reversible and the patterns on the 2 faces are mirror image each other. There can be 45 patterns made when coloring edges of tiles with 5 colors with allowing 2 or more edges of the same tile can have the same color. If patterns are used instead of colors, namely pattern A, B, C, D, X, where A/B and C/D are mirror image each other, and X is symmetric (in the above figure, X corresponds to tile's edges having no pattern), 5 tile patterns: X/X/X, X/A/B, X/B/A, X/C/D, X/D/C become symmetric and the rest 40 tile patterns become 20 pairs of mirror image. Thus, if tiles are reversible, 25 kinds of tiles are made. This puzzle can be made to a mechanical puzzle by shaping edges instead of patterning.
The following figure also demonstrates a variation of the above puzzle to form a regular hexagon of size 2, where the tile with no pattern on all 3 edges removed like as Tri-Match (pyramid #1a). (Of course, the patterns must not expose at the outer edges.)
Arrange the above 24 tiles to form a regular hexagon of size 2 with matching colors an join lines smoothly (including any two tiles at opposite position for every vertices).
This puzzle is similar to Tri-Match (bisectors) that tile selection are the same and the design is almost the same but lines are wavy, which introduces another matching logic. Like "bisectors" series, it is also very hard but (a solution exists). In the above figure, a black wavy line is not somoothly connected at the vertex marked with blue circle.
This design is similar to Tri-Match (circles and discs) but changed patterns on the vertices to triangles.