**About the Game**

The *i*QUBE can be described as the fusion between two
of the most popular puzzles: the 130 years old *Sliding
Tile Puzzle* and the 30 years old
*Rubik's cube*. It was created by a Swiss
mathematician, Dr Mayoraz, to be as elegant and conceptually
simple as its illustrious ancesters, while remaining a very
challenging puzzle. It consists of rolling dice that need to
be placed in a specific position and orientation.

**Can one reach any possible configuration of the
cubes on the board?**

In the *Sliding Tile Puzzle*, only half of all
possible configurations can be reached. For example, in an
given configuration, one cannot permute two adjacent tiles
leaving all other tiles unchanged. As for the
*i*QUBE, if the orientation of each cube is ignored
(imagine all faces painted in the same color), the i-cube is
identical to the Sliding Tile game. Hence, the limitation
of the Sliding Tile same holds for the *i*QUBE.

Now let us bring in the additional complexity of the
orientation of each cube. Imagine a single cube. It can be
oriented in 24 ways: 6 ways to pick the top color, and 4
ways to pick the front color for each top color. Playing
with a dice on a table, you will find out that only 12 of
these 24 orientations can be reached by pivoting the dice on
the table and bringing it back to the starting location.
For each orientation, the one that would be obtained by
turning the dice by 90 degree leaving two opposite faces
unchanged, cannot be reached by rolling moves. The beauty
of the *i*QUBE is that one can reach any configuration
where the location of the cubes corresponds to a possible
configuration of the tile game, and where all cubes are in
one of the 12 possible orientations. In particular, one can
place all the cubes in the same possible orientation and in
their original location.

**How many possible configurations does
the iQUBE have?**

Following the above discussion, the number of possible
configurations of a 3-by-3 *i*QUBE is:

9! * 12^{8} / 2 = 78,015,878,922,240 = 7.8 *
10^{13}

which reads 78 billions or 78 trillions, depending where you are from.

This number of configurations grows quite rapidly with the
size of the game. A 4-by-4 *i*QUBE takes

16! * 12^{15} / 2 =
161,178,937,602,476,749,086,523,392,000 = 1.6 *
10^{29}

However, it is worth noticing that the number of configurations is not the main criterion to determine the difficulty of a puzzle. A puzzle can have a very simple algorithm to find at each step a move that brings it closer to the solution, in which case the total number of configurations is not a barrier for an easy resolution of the puzzle.

The beauty of the *i*QUBE is that even a 2-by-3, with
5 cubes (i.e. 5 moving pieces only), has "only" 6! *
12^{5} / 2 = 89,579,520 configurations, but is
already very challenging to solve.