Logic Masters Deutschland e.V.

Once upon a time, digits lived peacefully in the Kingdom of Hyperbola. One day, Hex (he is a digit 6) found his neighbor Nova (she is a digit 9) holding a party. He wanted to join, but was told the party is open for square numbers exclusively! Desiring so badly, Hex decided that he could pretend to be a digit 9 like Nova! But how?

You see, in the 2D Kingdom of Hyperbola, digits can't simply rotate themselves like us in the 3D world. Instead, they can only move parallelly along their neighborhood. Given this, can you help Hex find a path to both Nova's house and her appearance?

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Some math background (not necessary for solving the puzzle):

This puzzle is inspired by the famous Gauss-Bonnet theorem from differential geometry. Shortly, it tells that when someone travels in a non-flat surface, their direction will change not only when they turn, but also when they go around curvatures or "holes". This puzzle's grid has a negative curvature concentrated at the center point, zero curvature elsewhere, and no holes anywhere. So when Hex goes around the center point, his direction will change in an opposite way he goes around. (If he goes clockwise, the direction changes anti-clockwise.)

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Rules:

1. (Hyperbolic Sudoku) Fill the grid with digits 1-9 without repeats in any row or box. A "box" is a 3x3 square and a "row" is 6-cells-long and slightly folded at box edges.

2. Draw a path from Hex (the given 6) to Nova (the given 9) that moves orthogonally throught centers of cells. The path can't enter any cell more than once, except for the cell marked by an X ("crossing"), where it can enter more than once.

3. The path is a kropki line. Adjacent digits on the path must be either consectutive or in 1:2 ratio. For the purpose of this rule ONLY, the killer cage is considered as one single cell, and its value is the sum of digits inside.

4. The greater-than signs are valid.

5. After parallel translation along the path, Hex (the given 6) must perfectly overlap with Nova (the given 9). The following image illustrates how parallel translation works:

Click to Play on Penpa with answer check. Note: If you play on penpa, please use "Number" mode and turn on "Draw on Edges", make sure to insert digits at the middle of a cell! If you want to draw lines, please switch to "Edge" and "Free".