Definition:

The distance of X, denoted Dist(X), is the number of pieces between X and X-1 going counter-clockwise around the loop. For example, Dist(2) is the number of pieces between 2 and 1.

Example:

Given the following configurations:

4, 5, 10, 2, 19, 20, 23, 15, 18, 1, ….

Where reading left to right is considered counter-clockwise around the loop.

Then Dist(2) = 5;

Note: When the game is solved Dist(Y) = 0 for all pieces Y.

Definition:

Let R denoted the permutation that moves each piece counter-clockwise one piece. S will denote a spin move. In cycle notation:

R = ( 1, 2, 3, 4, 5, …, 19, 20)

S = ( 1, 4 )( 2, 3)

Reachable States

Let X = [SRSR^(-1)SRSR^(3)]

Y = X^5

Z = YR^(-1)

In cycle notation Z = (1, 2), that is it interchanges 1 and 2 while fixing every other piece.

By placing any piece in the left most positions of the spinner and applying Z we interchange that piece with the piece directly to its right. Thus every configuration is reachable from the initial configuration by applying this method.

Algorithm

Placing pieces 1-16:

Let X = 2, i.e. begin by placing piece 2;

• While X is not piece 16 do:
• While Dist(X) is greater than 3
• Apply R until an application of S permutes X.

NOTE: This will place X in the right-most position of the spinner

• Apply S
• Apply R^(-3)
• Determine Dist(X)
• Now Dist(X) is less than or equal to 3
• Place X in the left most position of the spinner
• If Dist(X) = 3
• Apply R^(-3)S
• Else, if Dist(X) = 2
• Apply R^(-1)SR^(-1)S
• Else, if Dist(X) = 1
• Apply R^(-1)SRSR^(-1)S
• X = X + 1 (we now look at the next piece and repeat these steps)

Placing pieces 17-20:

Recall the procedures X = [SRSR^(-1)SRSR^(3)]

Y = X^5

Let X = 17, that is X is piece 17

• While the puzzle is not solved
• While Dist(X) > 0
• Place the piece directly to the left of X in the left most position of the spinner.
• Apply Y
• X = X + 1 (we now consider the next piece)