Klein's Foggy Glasses
(Published on 3. October 2024, 21:43 by DubiousMobius)
Who hasn't wanted to Yin Yang on a Klein bottle at some point? This was largely inspired by Sfushidahardy's incredible Flat Cosmos. Big thanks to ThePedallingPianist and Phil the Hat for help with formatting, as well as everyone who helped test this.
The rules are long, but a lot of that comes from clarifying the strangeness Klein topology and wraparound fog, so don't let your first glance throw you off.
Rules:
Place the digits 1-9 in each row, column, and box in the 9x9 sudoku grid.
Twisted Space: The grid behaves like a Klein bottle. The top and bottom row wrap around, and are considered adjacent. The first and last columns also wrap, but also undergo a twist, so that r1c1 is adjacent to r9c9, and r2c1 is adjacent to r8c9, etc. The border coloring reflects this.
Yin and Yang: Shade some cells in the grid so that the shaded and unshaded cells each form an orthogonally connected set. No 2x2 region of the grid may be entirely shaded or unshaded. This includes 2x2 regions that wrap around.
Sightline arrows: The digit in a cell with an arrow counts the total number of cells belonging to its yin or yang region in the direction the arrow points before reaching a cell of the opposite shading. The arrow cells are counted in this total. If a cell has more than one arrow, its digit will be the sum of all cells counted this way. The arrow cell is only counted once in this case.
A 2-cell arrow is a 2 digit number with the leftmost digit acting as the tens digit and the rightmost digit acting as the unit digit. Both arrow cells are counted in its total.
A sightline arrow that crosses the twisted edge of the grid diagonally will have its vertical component flipped as it twists. For example an arrow in r2c1 pointing southwest would first look at r7c9, and then might continue northwest to r6c8, etc.
Maze-Runner Circles: Maze-runner circles prevent the digit that appears inside them from appearing nearby within the same Yin Yang region. If the digit N appears in a cell with a circle, the shortest path of orthogonally connected cells through its Yin Yang region to any other instance of N must be at least N cells, including both end-points.
Fog: The grid has been partially covered in fog. Placing correct digits will remove the fog from the surrounding cells.
A moat has been provided for taking notes. Placing correct digits in the moat will also clear the surrounding fog, and thus may be required in the solve. The digits in the moat must be filled to trigger answer check.
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If you enjoy yourself and would like more Yin Yangs in strange places you can try a Mobius Strip or Torus as well. Have fun, and thanks for playing!
Click the puzzle to play.
Solution code: The shortest maze-runner path from r4c7 of the 9x9 to another instance of that digit, including both endpoints.
Comments
on 20. October 2024, 20:41 by DubiousMobius
Aesthetic update, and rule clarification
on 17. October 2024, 02:50 by aqjhs
the maze-runner circle rule only applies between a circled digit and another instance of it in the same region, *not* between two uncircled instances of a circled digit in the same region.
