Dogic

The 12-color Dogic

The Dogic is an icosahedron-shaped puzzle like the Rubik's cube. The 5 triangles meeting at its tips may be rotated, or 5 entire faces (including the triangles) around the tip may be rotated. It has a total of 80 movable pieces to rearrange, compared to the 20 pieces in the Rubik's cube.

History

The 10-color Dogic

The Dogic was patented by Zoltan and Robert Vecsei in Hungary on 20 October 1993. The patent was granted 28 July 1998 (HU214709). It was originally sold by VECSO in two variants under the names "Dogic" and "Dogic 2", but was only produced in quantities far short of the demand; for a long time, the only feasible way to acquire it was via second-hand channels such as eBay.

In 2004, Uwe Mèffert acquired the plastic molds from its original manufacturer at the request of puzzle fans and collectors world-wide, and made another production run of the Dogics. These Dogics were first shipped in January 2005, and are now being sold by Meffert in his puzzle shop, Meffert's.

Description

The basic design of the Dogic is an icosahedron cut into 60 triangular pieces around its 12 tips and 20 face centers. All 80 pieces can move relative to each other. There are also a good number of internal moving pieces inside the puzzle, which are necessary to keep it in one piece as its surface pieces are rearranged.

The 10-color Dogic, with the two possible layers of rotation slightly twisted.

There are two types of twists that it can undergo: a shallow twist which rotates the 5 triangles around a single tip, and a deep twist which rotates 5 entire faces (including the triangles around the tip) around the tip. The shallow twist moves the triangles between faces but keeps them around the same tip; the deeper twist moves the triangles between the 5 tips lying at the base of the rotated faces but keeps them on the same faces. Each triangle has a single color, while the face centers may have up to 3 colors, depending on the particular coloring scheme employed.

The current Dogics sold by Meffert come in four varieties: the 12-color version originally sold under the name "Dogic", the 20-color version originally sold under the name "Dogic 2", a new 5-color version, and a new, easy-to-solve black-and-white version. These varieties differ only in the coloring scheme; the underlying mechanism is the same. The 12-color version has 3 colors per face center, and in the solved state has 5 triangles of the same color surrounding each tip. The other versions have the same color for all 3 color indicators on the face centers, and in the solved state the triangles match in color with the faces rather than the tips.

Solutions

The solutions for the different versions of the Dogic differ.

The 12-color Dogic is the most challenging version, where the face centers must be rearranged to match the colors of the face centers in adjacent faces. The triangles must then match the corresponding colors in the face centers. The face centers are mathematically equivalent to the corner pieces of the Megaminx, and so the same algorithms may be used for solving either. The triangles are relatively easy to solve once the face centers are in place, because the 5 triangles per tip are identical in color and may be freely interchanged.

The 10-color Dogic is slightly less challenging, since there is no unique solved state: the face centers may be randomly placed relative to each other, and the result would still look 'solved'. However, it may still be desirable to put them in aesthetically-pleasing arrangements, such as pairing up faces of the same color, as depicted in the second photograph. The triangles are slightly more tricky to solve than in the 12-color Dogic, because adjacent triangles in the solved state are not the same color and so cannot be freely interchanged.

The 5-color and 2-color Dogics are even less of a challenge, since there is a large number of identical pieces. These simpler versions cater to those puzzle fans who aren't yet at the skill level to manage the full complexity of the 12-color Dogic.

Number of combinations

Due to different numbers of visually-identical pieces in the two versions of the puzzle, they each have a different number of possible combinations.

12-color Dogic

There are 60 tip pieces and 20 centres with 3 orientations, giving a maximum of 60!·20!·3²⁰ positions. This limit is not reached because:

1. Only even permutations of centres are possible (2)
2. The total twist of the centres is fixed (3)
3. Some tip pieces are indistinguishable (5!¹²)
4. The orientation of the puzzle does not matter (60): all 60 possible positions and orientations of the first corner are equivalent because of the lack of face centers.

This leaves ${\frac {59!\cdot 20!\cdot 3^{19}}{2\cdot 5!^{12}}}\approx 2.2\times 10^{82}$ positions for the 12-color Dogic.

10-color Dogic

There are 60 tip pieces and 20 centres (whose orientations don't matter), giving a maximum of 60!·20! positions. This limit is not reached because:

1. Only even permutations of the centres are possible (2)
2. Ten of the centres are visually identical to the other ten (2¹⁰)
3. Some tip pieces are indistinguishable (6!¹⁰)
4. The orientation of the puzzle does not matter (60)

This leaves ${\frac {59!\cdot 20!}{2\cdot 2^{10}\cdot 6!^{10}}}\approx 4.4\times 10^{66}$ positions for the 10-color Dogic.

See also

• Rubik's Cube
• Pyraminx
• Skewb Diamond
• Impossiball
• Megaminx
• Combination puzzles
• Mechanical puzzles

References

• Jaap's Dogic page, which includes solutions and some brief historical data.
• The Magic Polyhedra Patent Page