Jaap's Puzzle Page

Bandaged 2×2×2

Bandaged 2x2x2

The Bandaged 2×2×2 is a 2×2×2 Rubik's Cube where two pairs of corners are fused together to become 2×1×1 blocks. The puzzle therefore has four normal corner cubes and two fused blocks. In the solved position the fused blocks are not parallel.

The number of positions:

There are 7 arrangements that the two fused blocks can have relative to each other. The unbandaged corners can only be permuted in a 4-cycle, so have just 4 permutations. They can only be twisted by the same amount (though in alternating directions), and so have just 3 orientations. Together this gives 7·4·3 = 84 positions that can be reached.

The table below shows how many moves each position of the Bandaged 2×2×2 needs to solve. It shows the puzzle can be solved in no more than 10 moves (5.4167 on average) or 15 quarter turns (7.8095 on average).

Face turn metric
Q
T
M		 012345678910Total
01 1
1 2 2
2 14 5
3 24 6
4 6 6
5 28 10
6 6 6
7 6 6
8 24 6
9 4 4
10 8 8
11 4 4
12 6 6
13 24 6
14 4 4
15 44
Total1361214881288484

Notation:

Hold the puzzle with one fused block along the bottom left edge of the cube. There are only two faces that might be able to be moved - the top face, denoted by U, and the right face, denoted by R. A clockwise quarter turn of a face is denoted by the letter of the face, i.e. U or R. Anti-clockwise turns are denoted by an apostrophe, i.e. U' or R', and half turns by a 2, i.e. U2 or R2.

Solution:

  1. Hold the puzzle with one fused block along the bottom left edge of the cube.
  2. Look at the colour at the front of the bottom-left fused block. If the other fused block contains that colour, then turn the puzzle around so that the bottom-left fused block becomes flipped around to put that colour at the back.
  3. Do whatever turns of the U and R faces you need in order to bring the second fused block to the bottom-right location.
  4. Turn U until the correct piece is at the front-left corner location, though possibly in the wrong orientation.
  5. Depending on whether the front left corner needs to be twisted in order to make its colours match the bottom-left block, do one of the following:
    1. No twist: R'
    2. Clockwise twist: U R2 U2 R U2 R
    3. Anti-clockwise twist: R2 U2 R' U2 R2 U' R'