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Four people stand at positions A, B, C and D and hold two ropes in the initial configuration shown.

enter image description here

These folk may “dance” with these ropes by performing just two moves, multiple times, in any order they like:

  • Rotate as a group 90 degree counterclockwise. (Thus the person in position A moves to position B, the person in position B to position C, and so on.) Call this move “ROTATE”denoted as R.

    Example: Given initial state, on one rotate, below is the state:

    enter image description here

  • Folks in positions D and C(the folks in the northeast and southeast) swap places with D holding the rope up and over C. Call this move a “SWAP.” denoted as T.

    Example: Given initial state, on one swap, below is the state:

    enter image description here

ROTATE - R

SWAP - T

Properties of operations:

  • Operations are not commutative - RRRTT not equals RTRTR

  • Operations are associative - (TR)T equals T(RT)

  • Inverse - R^-1 = R^3 or 3R and T^-1 = RTRTR

    R^-1 (one anti-clockwise rotation) is:

    enter image description here

    3R (three clockwise rotation) is:

    enter image description here

  1. How to represent the state of this tangle with an abstract data type?

  2. How do the operations(ROTATE & SWAP) modify the state of this tangle represented by this abstract data type?

overexchange
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  • Just on terminology, I think the move SWAP might be more clearly defined as 'the folks in the northeast and southeast positions swap places' etc, not least because in the diagram just before that line, it's not D and C in those positions! – AakashM Oct 21 '20 at 09:38
  • Why not just represent this as a string of Rs and Ts? Adding an operation to the string may involve rewriting, e.g. "RRRR" is replaced with an empty string and "TRTRT" is replaced with "RRR". – n. m. could be an AI Oct 21 '20 at 19:26
  • @n.'pronouns'm. I spent time thinking about it, and believe that any string reduces down to a unique minimal one, but I couldn't prove it. Or, put another way, if two strings of R and T do not contain TRTRT or RRRR and they are unequal, there can't be a way to get from the one string to the other through any combination of substitutions back and forth. – btilly Oct 22 '20 at 00:48
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    This is a [word problem for groups](https://en.wikipedia.org/wiki/Word_problem_for_groups) and this one is solvable with [Knuth-Bendix](https://en.wikipedia.org/wiki/Knuth%E2%80%93Bendix_completion_algorithm) — provided that the set of equations is really complete. – n. m. could be an AI Oct 22 '20 at 06:38
  • @n.'pronouns'm. Just wondering, if clockwise rotation is allowed in rational tangle dance – overexchange Oct 22 '20 at 16:56
  • Not sure what exactly you mean. You can express it as a composition of allowed operations, so it must be allowed. – n. m. could be an AI Oct 22 '20 at 17:02
  • @n.'pronouns'm. In rational tangle dance, when they say that one of the allowed operation is `R`, Is it a 90 degree clockwise rotation or 90 degree anti-clockwise rotation? – overexchange Oct 22 '20 at 17:08
  • Either one is expressible via the other so it doesn't really matter. – n. m. could be an AI Oct 22 '20 at 17:13
  • @n.'pronouns'm. My question is, are both anti-clock & clockwise rotations allowed in rational tangle dance? – overexchange Oct 22 '20 at 17:22
  • @overexchange The problem specifies that R is clockwise. But RRR is counterclockwise one turn. So it really doesn't matter. – btilly Oct 22 '20 at 17:23
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    Looks like I got the answer: https://youtu.be/JXGyXtNsu14 – overexchange Oct 24 '20 at 07:46

0 Answers0