# Torsion Twirl (MIT Mystery Hunt 2020)

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Torsion Twirl
MIT Mystery Hunt 2020
Yesterdayland
Author(s)Shelly Manber
Statistics
No. solves22
No. total guesses53

Torsion Twirl is a puzzle from the Yesterdayland round of the 2020 MIT Mystery Hunt.

## Puzzle Elements

Flavortext - I really enjoyed riding the torsion twirl, but I got so dizzy… These dancers curve effortlessly! They told me the trick is they always get back exactly where they started in finitely many steps, they just have to make sure they know how many. “It’s all about the order,” they said.

Video - The puzzle is presented as eight short videos of two dancers turning. One dancer is marked with an X and the other with a Y.

Hint in Title - The "torsion" in Torsion Twirl refers to torsion points on elliptic curves.

Hint in Flavortext - I really enjoyed riding the torsion twirl, but I got so dizzy… These dancers curve effortlessly! They told me the trick is they always get back exactly where they started in finitely many steps, they just have to make sure they know how many. “It’s all about the order,” they said.

Identification (Performing Arts - Dance) - The dancers in each video perform a different kind of turn some number of times; the turn can be identified, using the enumeration below each video as a guideline.

Knowledge Required (Mathematics - Abstract algebra) - At the bottom are a number of equations for elliptic curves. Each video's dancers turns some finite integer number of times; they are labeled with X and Y, thus uniquely identifying a point on the plane. Each point lies on exactly one of the elliptic curves in question, which allows a bijection between the two. The solver then needs to find the order of this point according to the elliptic curve group law. This can be determined manually by drawing tangent lines to the curve, but the puzzle recommends using PARI/GP to help with the calculations.

TO DO Example

Indexing - The order of the point on that curve can then be indexed into the turn used in its video. Ordering the resulting letters using the video numbers reveals the answer.