This puzzle is a loop deduction puzzle, like Slitherlink and Masyu. It is based on the idea of balls rolling down hills - the clues here give the distance that a ball can roll down.
Standard Loop Deduction Rules
- There is a single, closed, non-intersecting loop.
- There is a unique solution.
- There is a ball at every number. The value of the number indicates the deepest depth the ball could roll down to. It must roll down to this depth along at least one path, but it does not have to roll down to this depth along every path.
- Arrows indicate direction of gravity for the ball.
- Grid boundary stops the ball as if it were part of the loop, regardless of whether there is a loop segment at the boundary.
Mechanics of Rolling Balls
- A ball can roll down any corner.
- A ball cannot roll "up" (against gravity) a wall or along two "horizontal" (perpendicular to gravity) segments in a row (the hill would be too shallow!).
Here are a few examples of the ball rolling mechanic:
The example in red is an impossible situation - the number indicates the deepest a ball could roll, but that ball could roll a depth of two if it went right. The other examples have the path of the deepest roll traced out in green.
Note that balls can overlap.
It should be solvable using logical deductions rather than guessing - you may need to prove by contradiction, but you shouldn't need to think far ahead to do so. You shouldn't have to use the uniqueness condition in your deduction.
I found this fairly difficult to solve, even though I had the advantage of knowing what the solution would be. A decent part of the difficulty, in my opinion, comes from the fact that it's hard to reason about the boundary of the grid, since clues don't directly inform you about whether there is an edge there. Often in loop deduction puzzles, it's easiest to start from the edge and work your way inwards.
Hope you enjoy it! Please let me know if anything is unclear.