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## Options

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## Help

You are looking at Conway's *rational tangles*: the tangles you could get
from two pieces of string by repeatedly picking two neighboring
ends to twist over each other.

But in this game, you have only two kinds of moves:

- ⤮ / X — Twist 'em up!
- ⟲ / Z — Turn 'em 'round!

Get a friend to scramble it for you—or, if your friends are social-distancing, check the ⚙ menu for an auto-scramble button. Then try untangling it. Good luck!

In memory of John Horton Conway (1937–2020), dwelling in particular on the Canada/USA Mathcamp performance that this game is an adaptation of—during which he wrapped the tangle in paper that he later tore open with his teeth.

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## Spoilers

### ordered by increasing spoilage

- One of Conway's theorems was that two rational tangles are
*ambient isotopic*(can be turned into each other without moving or passing the ends) if and only if they represent the same fraction via some algorithm he devised. - I'm drawing these tangles constrained to a dumpling surface, which dodges a lot of magic. In particular, it makes the string's average slope a fraction.
- By "slope", I really mean the number of times a string flips over the top or bottom, divided by the number of times it flips around the left and right sides.
- What does each move do to the slope?
- To see this in action, you can make it display fractions.
- Mathematicians should think about (a) what has the torus as a
double cover branched at the half-or-whole-integer points, and
(b) what SL(2,
**Z**) does to it.