There are countless color and shape modifications, where the internal mechanism is still a regular cube core, but the look is different. Sometimes it's difficult to decide, whether a twisty toy is just a shape modification, or whether it has a different internal mechanism.
Besides the sometimes strange look and feel, there can be additional challenges: Center piece orientation for odd-layer cubes, placement of inner face pieces for higher order cubes, and parity issues due to undetermined placement or orientation of parts.
Except for the center pieces, all modified cubes can be solved with the same strategies and algorithms as their regular counterparts. Some modifications reduce the complexity, what would make simpler solution strategies possible. But in most cases, the regular solution scheme does a good enough job - especially for funcubing.
For a regular 3×3×3 cube, the only challenge are center piece rotations. There can be pairs of 90° clockwise and 90° counter clockwise rotations, single 180° rotations, and any combination of these.
If the algorithms of your solving scheme do not affect center piece rotation, you can take care of the five centers in the first two layers while solving these. Then you can only end up with a 180° twist in the last layer. Unfortunately, most speedsolving algorithms affect center piece rotation. This can be annoying and confusing, if center piece orientation also affects the shape.
For higher order cubes, the placement of the individual center face pieces can be important. First, you have to take care to put the pieces to the matching places, instead of just any of the four possible places. At the end, swapped pieces can remain, where special center piece parity algorithms are needed.
Parity issues can occur, if the modified cube allows more freedom in placing or orienting the individual pieces. The typical placement parity issues are two swapped edge or corner pieces at the end of the solving attempt. Then you have to look for other edge or corner pieces that can be swapped. On the "Morphix" variants, for example, the typical situation are swapped corner pieces, where you can swap any two of the non-distinct edge triplets. Orientation parity issues can occur, if some pieces have no orientation. On the Fisher Cube, for example, you can end up with a single flipped edge piece, where you can flip one of the non-distinct edge pieces from the middle layer along with the obviously flipped top/bottom edge piece.
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The Octagon Barrel is one of the first modifications on the market. The octagonal faces are considered to be top and bottom faces. It is a cube, where the vertical edge columns are cut at 45° angle to form a single flat face between the vertical center columns. These have four additional colors, the placement of them is not determined by shape or color.
The modification creates four edge pieces without given placement in the vertical middle layer. The orientation does not matter, because they have ony one color and a symmetrical shape. The eight corner pieces get their placement by the two face colors. The triangular side must match the top or bottom color. The rectangular face must match the horizontal layer edge piece color. The orientation of the corner pieces can be derived from shape or color.
I think that this is one of the most useless modifications. It does not introduce any additional challenge, but instead creates parity issues, due to the undetermined placement of the four corner columns.
The Octahedron shape modified 3×3×3 cube was already announced in 1981 by Josef Trajber in one of his books. It's a cube where all eight corners are cut at an angle, down to the middle of the face centers. This makes the actual center pieces look like corners, and the actual corner pieces look like center pieces of a triangular face. The special feature of this puzzle is the fact, that is does not shape shift.
The six center pieces now have an orientation indicated by the four color segments. The twelve edge pieces run "straight" from center to center, but due to the edge line, they still have two colors that indicate the orientation. The eight corner pieces have no orientation, corner orientation is completely out of scope during the solve.
Exactly the same behaviour can be achieved with a sticker modification of a regular 3×3×3 cube: Each of the corner pieces gets one color, each color is extended to half of all adjacent cubie faces, and a quarter of the respective center piece.
Although the octahedron does not shape shift, it's a bit uncomfortable to turn. There is not much grip, and to give an accurate shape, the individual pieces have sharp edges, and will lock frequently. But it's still fun to solve, because of the eight faces, and the same, but somewhat different search for the edge pieces. You still search for edges with two colors, but the colors are twisted 90°.
The 7-color diamond is a shape modification where the symmetry axis runs between two opposite corners. The cut resembles a "rose cut" diamond, giving a hexagonal outline with 13 facets. The vertically adjacent facets share the same color, the flat top (table) has the seventh color (grey).
This shape modification leads to a remarkable layout: The six center pieces remain as they are, they are flat with only one color. One corner piece makes the tip of the diamond with six faces, the opposite corner makes the top with only one color. The other six corner pieces have two faces with the same color, and have to be oriented by shape. The twelve edge pieces are divided in four groups of three identically shaped pieces: The three edge pieces around the tip have three faces of different color. The three edge pieces around the top have four faces of different color. The edge pieces of the remaining two groups have a very similar shape, but are mirrored counterparts. Each piece has four faces with two different colors. The placement is determined by these two colors, so you don't have to worry about the similar, but different shape.
The 7-color Diamond is the ideal shape modification for beginners. You don't have to care about center piece orientation, placement is fully determined by color, only six corner pieces have to be oriented by shape. The shape shifts ony a little, so the cube is easy to hold and to handle.
The Mirror Cube is a variant, where the placement and orientation is fully determined by the thickness of the layers. The core is still a regular cubicle with 19mm edge length, as well as all center piece faces. The center pieces have no orientation by shape, but some have stickers with a brushed look, that make 90° turns obvious.
To form a regular standard sized cube with 57mm edge length, the same amount of thickness has to be added and removed for opposite outer layers. On a "normal" 3×3×3 Mirror Cube, the layer pairs have 17mm and 21mm (±2mm), 13mm and 25mm (±6mm), and 9mm and 29mm (±10mm). That makes at least 4mm difference between all outer layers. The sizes are different enough, so that it's possible to eyeball instead of trying out the placements and orientations.
I like the mirror cube, because of the idea of using layer thickness instead of color. Because of the regular shape shift, it is still good to handle. What I don't like are the stickers with brushed look, which introduce importance on center piece orientation. I'm not sure whether this is part of the design, or a misconception. Because I am a fussy person, I always solve the mirror cubes with respect to the center pieces. I prefer to start with the thickest layer to get less shape shift while solving the cube - especially the least shape shift in the last layer.
The 4-Color Cube is a sticker modification of a regular cube. Each face has two colors at each side of an imaginary diagonal line, splitting each face into two triangles. Triangles that share a full corner piece face have the same colors. That makes twelve half faces of four colors.
Because the two colors run across the center pieces, all center pieces have two colors, and therefore have to be rotated properly. The four corner pieces where the three face parts of the same color meet have no orientation. The other four corner pieces are the ones where the diagonal line runs across each of the three faces, and have three different "edge line" colors indicating the orientation. All edge pieces are located where two same-colored faces meet, and therefore have no orientation. In addition to that, there are four sets of three edges having the same color. This leaves a certain freedom in placing the edge pieces, except that this can lead to a parity error for corner piece placement. This can be resolved by swapping two edges of same color along with a pair of corner pieces. The simple case where two edge pieces seem to be swapped can be resolved by cycling a set of three edge pieces, where two have the same color.
The four color cube has a significant reduction in complexity, where only center piece orientation is a true challenge. However, it is an excellent way to practice for the Mastermorphix, without having to deal with the odd shape. The main difference is that edge piece orientation does not matter, here.
The Apple Cube is a shape modification, where the cube is shaped like an apple. There are two groups of four identical corner pieces, and three groups of four identical edge pieces. Due to the different shape of the upper and lower half, the four center pieces in the middle layer need to be oriented properly.
First I thought that the Apple Cube would be just another useless shape modification, like the Star and Heart shape modifications. But it turned out that it could be used as practice cube for the Morph Egg. There are four centers that need to be oriented, and there are different groups of corner and edge pieces, that have to be identified, placed and rotated by shape. In case you have difficulties solving the Morph Egg, I would highly recommend the Apple Cube. If not, get it because it looks nice.
The Morph Egg is a cube that is shaped like an egg, where the layers run diagonal, i.e. top and bottom of the egg are former corner pieces of the cube.
The Morph Egg has to be solved entirely by shape, where all six center pieces have to be oriented properly as well. Due to the round shape, the Morph Egg does not have much shape shift, but that makes it very difficult finding the matching parts. Basically, it's just about finding a matching piece by trying all orientations of each possibly matching piece. The good part is, that each matched piece reduces the amount of pieces left to try.
To make it simple: There are six center pieces with four edge lines, where there are 12 edge pieces with 24 possible edge lines. Then there are eight corner pieces with 24 edge lines, with 24 possible edge piece edge lines to match. However, the Morph Egg would not be an egg, if there wasn't an "easter egg" feature. If you end up with an "unsolvable" situation, go and find the hidden feature. In solved state, the egg must have smooth transitions between all pieces. I have already seen "solved" Morph Eggs in videos, that did not look solved to me...
During my first solve, I thought that it's just annoying, and that I won't scramble it again. It's hard to explain the complexity, it's definitely a lot more difficult than the Mirror Cube. Compared to the Ghost Cube, trying to match pieces is easier, because the Ghost Cube has to be changed between solved shape and turnable state. However, the parts of the Morph Egg are less distinct. All edge pieces look very similar, and for the corner pieces, smaller ones and bigger ones can be identified. If you have trouble solving the Morph Egg, you can try the Apple Cube. This puzzle also needs a search by shape, but there are only three types of edge pieces, and two types of corner pieces.