Four dancers stand at the corners of a square. Each one holds one end of one of two long ropes and never lets go. At the start, the ropes look like this:
A D
\_________/ rope 1 joins A and D (top)
_________
/ \ rope 2 joins B and C (bottom)
B COnly two moves are allowed:
- R (Rotate): the whole group turns 90 degrees counterclockwise. The dancer at A moves to B, the one at B moves to C, the one at C moves to D, and the one at D moves to A.
- S (Swap): the dancers at C and D swap places. D holds the rope up and C passes under it.
A fact you may use without proof (John Conway's theory of rational tangles). Every tangle these dancers can make has a label x, which is a rational number or the symbol ∞. Two tangles are the same, up to sliding the rope around with the ends held fixed, exactly when they have the same label. The labels behave as follows:
- The starting position (two horizontal ropes, as drawn) has label 0. The ropes count as untangled exactly when the label is 0 again.
- S changes the label from x to x + 1. It leaves ∞ as ∞.
- R changes the label from x to −1/x. It sends 0 to ∞ and ∞ to 0.
Moves are carried out in the order they are written, from left to right.
Starting from the untangled position, a dance master calls out this 18-move dance:
SSSRSSRSRSSSSSSRSSQuestion. What is the shortest sequence of R/S moves that the dancers can carry out next to untangle the ropes, meaning to bring the label back to exactly 0?
Answer format: there is exactly one shortest untangling sequence. Give two integers separated by a comma: its length L, and the sum of the positions (numbered 1 to L from the left) at which the letter R occurs in it. For example, the sequence RSSRS would be reported as 5, 5, since its R's are at positions 1 and 4.