The ORBIX puzzle is a spherical puzzle with 12 button/lights which may be on or off in a given state. (The background gif of this web page depicts an ORBIX-like puzzle with six button/lights total, where each button has four neighbors. Three of the buttons on the front are on, two are off.) Although, ORBIX has four puzzles in all, we shall focus on the mathematics of the first puzzle, Puzzle 1.
Pressing a button affects the button's five neighbors. The pressed button remains unchanged. This may be formulated graph-theoretically as follows. Consider the graph G of the icosahedron (the vertices are the vertices of the icosahedron and the edges are the edges of the icosahedron) and label the vertices 1, 2, ..., 12. Define a "state" to be a 12-tuple (x1, ..., x12) of 0's ("off") and 1's ("on"), one for each vertex. We call xi the state of vertex i and say the i-th vertex is on (resp., off) if xi=1 (resp., xi=0). A "move" is to select a vertex and then add 1 modulo 2 to the state of all its five neighbors. The game is to start with a given state and, after a succession of moves, to achieve the "all on" state (1,1,...,1). Such a move will permute the set S of all possible states and may be regarded as a permutation puzzle .
There is an analog of this game where G is replaced by any connected graph or connected digraph. For example, if G is the square graph
*-------*-------*
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*-------*-------*
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*-------*-------*
then a similar game has been marketed under the name "Merlin's magic square" (see
[P]). Related topics are Gray codes (see [G]) and cellular automata (see [S1]
or [S2]).
As you can see, ORBIX is a puzzle whose solution, like the Rubik's clock, depends on binary arithmetic. Let F2 = {0,1} denote the field with two elements, which we identify with the integers modulo 2. We can label the lights of the ORBIX as follows: 1 is the battery light, 2 is the light between the battery and the speaker. The remaining lights are labeled clockwise downward as for the 12 faces of the megaminx. We denote an on light by 1 and an off light by 0. Pressing a light swaps the 5 neighboring lights on/off. Thus pressing a button may be represented by a 12-vector of 0's and 1's with exactly five 1's, i.e., a "binary code word of weight 5 in F212".
The vector associated to the 1st light will be denoted e1, to the 2nd light will be e2, and so on.
The numbering of the lights on the orbix:
1 = northpole 12 = southpole
view from above: 7
11 8
2
6 3
1
5 4
10 9
Light number neighbors antipodes
1 2,3,4,5,6 1,12
2 1,3,6,7,11 2,9
3 1,2,4,7,8 3,10
4 1,3,5,8,9 4,11
5 1,4,6,9,10 5,7
6 1,2,5,10,11 6,8
7 2,3,8,11,12
8 3,4,7,9,12
9 4,5,8,10,12
10 5,6,9,11,12
11 2,6,7,10,12
12 7,8,9,10,11
The adjacency matrix of this graph is the 12x12 matrix
A=(aij),
where aij=1 if i and j are neighbors (connected by a single
edge) and aij=0 otherwards. In this case,
/ 0 1 1 1 1 1 0 0 0 0 0 0 \
| 1 0 1 0 0 1 1 0 0 0 1 0 |
| 1 1 0 1 0 0 1 1 0 0 0 0 |
| 1 0 1 0 1 0 0 1 1 0 0 0 |
| 1 0 0 1 0 1 0 0 1 1 0 0 |
A = | 1 1 0 0 1 0 0 0 0 1 1 0 |
| 0 1 1 0 0 0 0 1 0 0 1 1 |
| 0 0 1 1 0 0 1 0 1 0 0 1 |
| 0 0 0 1 1 0 0 1 0 1 0 1 |
| 0 0 0 0 1 1 0 0 1 0 1 1 |
| 0 1 0 0 0 1 1 0 0 1 0 1 |
\ 0 0 0 0 0 0 1 1 1 1 1 0 /
(A note for the curious: this matrix has determinant
625=54 and
eigenvalues 5, 51/2, -51/2,
51/2, -51/2, 51/2,
-51/2, -1, -1, -1, -1, -1.)
A move consists mathematically of choosing a vertex i then
adding (modulo 2) row i of A to the state vector.
[G] M. Gardner, "Binary gray codes", in Knotted donuts and other mathematical entertainments, W. H. Freeman and Co., 1986 (originally published as a Scientific American article)
[P] D. Pelletier, "Merlin's magic square", American Math. Monthly, 94(1987)143-150
[S1] K. Sutner, "Linear cellular automata and the Garden of Eden", Math. Intelligencer 11(1980)49-53
[S2] K. Sutner, "The sigma-game and cellular automata", Amer. Math. Monthly 97(1990)24-34
[V] S. Vajda, Mathematical games and how you play them, Ellis Horwood, New York, 1992
Created 3-3-97. Last modified 4-29-98.