# Balls on Hills - Loop Deduction Puzzle

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.

## The Puzzle

Standard Loop Deduction Rules

1. There is a single, closed, non-intersecting loop.
2. There is a unique solution.

Clue Rules

1. 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.
2. Arrows indicate direction of gravity for the ball.
3. 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

1. A ball can roll down any corner.
2. 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.

## Extra

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.

• Nice puzzle! And welcome to Puzzling! I wonder if someone can make a Penpa link for this Jan 1 at 12:29
• Thanks! I'm not sure how to make the arrows nice in Penpa, so it's quite ugly, but here you go :) (Edit: There was a mistake in first link, forgot an arrow) Jan 1 at 12:33
• There can be an edge that is on the grid boundary, right? Like the left edge in your top left example? Jan 1 at 12:44
• Is it allowed for balls to overlap? In other words, can a ball be in the same final resting position as another ball, or do the balls stack on each other? Jan 1 at 12:51
• Now that I think more about it, I think the phrase "balls on hills" doesn't really quite match the mechanism itself. It gives the imagery that each cell has a height, and that the balls need to find a path rolling down (where down is the direction that goes into the screen from your eyes). Probably this mechanism is better visualized as pachinko machine or pinball, where down is where the gravity arrows point to, and the loops form like pockets/barriers for the balls to fall into. This is how I visualized it in my head while solving it. Jan 2 at 7:10

Solution:

Explanation:

First, we know that in the direction of 0 is going there should be an edge. Also a number can't go directly to another number that goes in the same direction that is not exactly one less. So 1->1-> needs to be separated by edge, as well as two occurrences of 4->2->

Next,

We focus on lower right 3 going left. The two next possible edges should not be edges (I put X in the spots). If any of those were an edge, the 3 will join the 4 at the bottom of it.

(If there were an edge at R5 C3-4 we will get this to prevent 3 to join the 4, and the 1 Down at R6C3 and the 4 Left at R6C5 clash. Either of them will be blocked, since we will need to continue the loop there.)

Next,

The 1 at R5C2 must go to the right to reach depth 1, since it can't go to the left since it will join the 2. So R5C4-R6C4 must be blank too. And to continue the loop, it must go around the 0 and to the right.

Next,

Column 2, R5-R6 if it's an edge, then the edge will continue to the left and down to allow for the 2 Down to reach depth of 2. But this edge will necessarily block the 4 Left at R6C5. So the edge to the left of R5C3 must be present, to prevent 1 to go to the left and joining the 2.
Similarly, bottom of R3C4 must be an edge since there is no way to reach depth 4 if going down there.

Next,

Then continue the loop on the right. This is the only way to not block the 3 Down and 1 Right, while avoiding creating a loop around the 2 in the center.

Next,

Now R4 C1-2 must be edge, to confirm the depth 2 and 3 going left.
Also the edge below 5 Down should be there, since otherwise it will be stuck inside at depth 4.

Finally,

To close the loop there is really only one way. The other way will block either the 5 Down or 4 Right.

• Awesome job! Your second step was nice; I didn't realize that so my solution was more drawn out. Congrats :) Jan 1 at 13:45
• @BaileyA I missed one case in that part, but I just fixed it, the core concept is the same. Thanks for creating this cool puzzle! You made this mechanics yourself? Jan 1 at 13:59
• Yes - I love puzzles like Slitherlink/Masyu/Corral so I wanted to try my hand at making up some loop deduction mechanics myself. I didn't check if anyone has made a similar puzzle type before, though. I'm glad you liked it :) Jan 1 at 14:02
• " a number can't go directly to another number that goes in the same direction that is not exactly one less." I don't follow. How do you know that? Jan 7 at 21:31
• "So 1->1-> needs to be separated by edge, as well as two occurrences of 4->2->" I don't follow. Why can't they both just move 1 space? Jan 7 at 21:32