Solutions and Group Theoretic Elements of the Masterball

The Masterball has been called by some, the Rubik's cube of the '90's. This innovative spherical puzzle is divided into eight columns and four sections. The Masterball also comes in a wide array of patterns. The simplest variation, called the Duomaster, consists of alternating black and white columns. The focus of this paper will be the Geomaster, consisting of eight columns of different colors. Several other pictorial variations of the ball are available to include USA Master, Tennismaster, Soccermaster, Circusmaster, Dragonmaster, and Catmaster. Before beginning a discussion of solution strategy or any further group theoretical results, some cursory definitions and notations are necessary:

• longitudinal: a line passing through the north and south poles of the Masterball, dividing the ball into easterly and westerly columns; 8 longitudinales divide the ball into 8 columns
• latitudinal: a line running east/west on the Masterball dividing the ball into polar and equatorial sectors; 3 latitudinals divide the ball into 2 polar regions (a north and a south) and two equatorial regions
• facets: in total the longitudinals and latitudinals divide the Masterball into 32 polar and equatorial pieces which can be referred to as facets; the ball is made up of 16 polar facets and 16 equatorial facets. A facet is denoted by a two digit number composed of its row number first, followed by its column number, i.e., 13 would be the facet in the first row, third column, etc.
• permutation groups: permutation groups on the Masterball puzzle are composed of swaps of two or more of the ball's facets or columns; it should be fairly obvious that polar facets can only be swapped with other polar facets, and equatorial facets can only be swapped with other equatorial facets; We assign G to be the group which concerns itself with swaps of the 32 facets disregarding orientation, G^o to be the group of swaps of the elements where a change of orientation is considered a swap, F to be the group which concerns itself with swaps of the 8 columns disregarding orientation, and F^o to be the group of swaps of columns where change of orientation is considered a swap.
• orientation: orientation can be defined by imagining a north or south-seeking arrow on each facet of the Masterball. a permutation which swaps a northerly to a southerly orientation, and vice versa
• column moves (f₁,f₂,f₃,f₄,f₅,f₆,f₇,f₈): the column moves are denoted f, for flip, as they are characterized by a flip of one hemisphere of the ball about the ball's equator By selecting a column, denoted as f₈ as a reference, and continuing to number around the ball in a clockwise fashion, f₁, f₂,f₃...,f₈, all 8 possible column moves can be defined. (note that f₁=f₁^-1)
• rotations (r₁,r₂,r₃,r₄): a rotation can be defined as the horizontal shift of a polar or equatorial region from left to right across the front of the ball, where the subscripts 1,2,3,4, correspond to the specific region. 1 denotes the north polar region, 2 denotes the north-most equatorial region, 3 denotes the southerly equatorial region, and 4 denotes the south polar region. It is important to note that all rotations are orientation-preserving moves.
• F^o (The column orientation-preserving group): As alluded to in the definition of permutation group, the orientation group is the permutation group that permutes columns while maintaining their orientation.

We can now discuss the solution of the Geomaster puzzle.

The Masterball can be solved using a combination of a method known as fishing and the use of two additional moves. The first four columns of the ball can be solved by fishing, a move defined as r_m*f_n*r_m^-1*f_n... . Once these first four columns are in place, the clever execution of set-up moves allows for the ball to be solved using only a 2-7 polar swap and a 2-7 equator swap, as defined by David Joyner as 3-6 swaps, as below:

Note: We refer to Joyner's 3-6 swap as the 2-7 swap based on differing initial orientations of the ball.

The Masterball group, G=< f₁,f₂,f₃,f₄,f₅, f₆,f₇, f₈,r₁,r₂,r₃,r₄> is a subset of S32. We can provide a simple proof to show that G is a subset of S₃₂, by showing that G= S₁₆ x S₁₆.

So, result: G=S₁₆ x S₁₆ (Joyner)

Proof

Since the Masterball is made up of 32 facets, the permutation group G, on these facets, must be a subset of S₃₂. If we agree that we can take the ball apart and put the pieces back in any position, we have generated S₃₂, saying that any swap is legal. However, as stated before equator facets can only be swapped with equator facets and polar facets can only be swapped with polar facets. Since there are sixteen polar facets, the group of polar swaps, G_p is a subset of S₁₆. Similarly, since there are 16 equatorial facets, the group of equatorial swaps, G_e is a subset of S₁₆. To prove that G_p and G_e do each generate S₁₆, we need only to recall a theorem from group theory. We recall:

Using this result, we can provide a similar proof concerning the column group, F, disregarding orientation.

Result: F=S₈.(Joyner)

Proof

Since the Masterball is composed of eight columns, the column group must be a subset of S₈. Analogous to the proof of G=S₁₆ x S₁₆, we can prove F=S₈ if and only if we can show a two cycle on the columns. We define a move which swaps columns 7 and 8, producing the two cycle (f₇ f₈), as:

When we consider the column orientation group, that is the permutation group on the columns where reverse of orientation is considered a swap, we arrive at a more complicated result.

Result: |F^o|=576.

Sketch of Proof

We will not provide a complete proof of this result, rather a sketch of the central concept behind the proof. We begin by imagining the Masterball as a two-tone disk, like a hockey puck with the top painted red and the bottom painted blue. We now assign numbers 1,1',2,2',3,3',4,4' counterclockwise around the disk at the ends of the lines created by the longitudinales. Considering only flips, it should be intuitively obvious that primes cannot be swapped with non-primes and vice-versa. Since we can swap only primes with primes and non-primes with non-primes, it would appear that the group can be written as the product of two subgroups of S₄. But we know |S₄|=24. So |F^o|=|S₄| x |S₄|= 24 x 24 = 576. Therefore, |F^o|=576.

• D. Joyner, A Puzzle Problem, preprint, 1997.
• D. Joyner and A. Southern, The Masterball Puzzle, 1996.
• S. Robinson, The Mathematics of Bell Ringing, Capstone paper, Spring 1997.