This puzzle is a "Balls on Hills" puzzle. It's similar to puzzles like Slitherlink in that your goal is to find a curve given some clues. The intuition behind this puzzle is balls rolling down hills into valleys (pockets).

Unlike Slitherlink, you do not need to find a closed loop - rather, you need to find a curve that partitions the grid into exactly two areas. This can be a loop, but it can also be a curve that starts and ends at the grid boundary.


The Puzzle

Penpa Link


Curve Rules

  1. The solution is a single, non-self-intersecting curve that partitions the grid into two zones (whether by looping or starting and ending at the grid boundary)

  2. The curve travels along grid edges (like Slitherlink) within the grid boundary

  3. There is only one curve that fits the constraints

Balls on Hills 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 curve.

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!).

  3. Balls do not interfere with each other (i.e. they can pass through each other, overlap each other, etc)

A couple visual examples may help: Example of ball mechanics

I've highlighted the deepest path a ball can roll down in green. The example in red is an impossible situation, as the deepest path for the 1-ball has depth 2.


This is the second Balls on Hills puzzle. I was planning on waiting a bit longer to post this, since I posted the first yesterday; however, I realized something a tad frustrating: by the old rules (make a loop instead of "partition into two"), you could never touch the boundary (it would violate uniqueness).

I didn't want this to be the case, because I wanted reasoning near the boundary to be difficult. I've changed the rule to "partition into two", which I think keeps the spirit of the puzzle. By posting this so soon after the first puzzle, I hope it serves to "canonize" this rule change ;) I do have another Balls on Hills that I'm working on, but won't post it for a few days.

There's also a debate on the name of the puzzle, since referencing pockets/valleys would be better suited for the intuition of how to solve these (you should hopefully see what I mean after giving this a go). Feel free to suggest a name. I've kept it as "Balls on Hills" for now because both words end in the same consonant cluster (lls).

In general, since this is a new puzzle type, any comments/suggestions on the format, rules, etc are welcome!

  • $\begingroup$ Ooh, I like this arrows more. Nice improvement! $\endgroup$
    – justhalf
    Commented Jan 3, 2023 at 3:50
  • $\begingroup$ By the new rule, is it no longer possible for the "loop" to touch the boundary at two or more places then? I was still unique in the old rule (due to single loop constraint), but it creates 3+ zones. $\endgroup$
    – justhalf
    Commented Jan 3, 2023 at 4:13
  • $\begingroup$ Thanks! And yeah, by the new rule it has to touch at most twice. Since the clues can't tell the difference between a loop edge and a grid boundary, if the loop touched the boundary in the old rules then it could either travel clockwise or counterclockwise along the boundary and still be valid (which I didn't realize until after I had already designed this one). $\endgroup$
    – BaileyA
    Commented Jan 3, 2023 at 9:50
  • $\begingroup$ I thought about restricting it instead with the condition "smallest loop is the valid one", but this could have weird affects off the boundary too (it interferes with reasoning by uniqueness as it amounts to explicitly endorsing multiple solutions). Another alternative was "loop that shares the least edges with the boundary", but that would encourage some trivial outcroppings (for example, you could fold some of the corners inwards in this puzzle and still have a valid solution - these are the "trivial outcroppings") $\endgroup$
    – BaileyA
    Commented Jan 3, 2023 at 9:53
  • 1
    $\begingroup$ Fair points. Also this means we can use this rules as well for trickier puzzle solving (e.g., if we already know the loop touches the boundary at two points, then there could no more be other places where the loop touches the grid). $\endgroup$
    – justhalf
    Commented Jan 3, 2023 at 10:01

1 Answer 1



enter image description here


  1. Zeroing in

We can start with the simple deductions for the 0s which must have a boundary beneath them (with respect to direction of gravity). 1 > 1 top left must also be separated by a boundary to prevent a 2, and this solves the top left.

enter image description here

  1. Finding loose ends

We now have a situation top left where the curve must continue right to prevent shutting itself off. The 2s in the same direction bottom right must also be separated.

Now consider top left. The 1 down still needs to be caught, and must be caught directly downwards. But to prevent further rolling down the left, it must connect to the side. This gives us out two contact points with the boundary, so everywhere else must connect. This means top right must connect and then go down left for the 1 down.

enter image description here

  1. Join the do... lines?

The top can now be connected as there is only one way to do so with respect to the 1 and 2 ups, and the 5 right. The curve must also then continue down the right to catch the 1 up.

Now consider the bottom 8. It can't go left due to the 5, so must go 8 to the right, meaning it has to reach the top. With this in mind, we can connect the right hand side to form half the curve, so the 8 can reach the top, the 3 and 2 outside can free fall, and the 1 and 3 trapped inside can fall to the appropriate level.

enter image description here

  1. Finishing touches

The 2 on the left has to be caught, and can only be done one way by joining the curve and trapping the 1 up. The 4 up and the 5 up at the bottom must go left, while the 0 must be trapped so we can finish off by staircasing our way to the answer!

enter image description here

Feedback for the puzzle (very enjoyable puzzle type by the way):

  • I really like that the curve can either be a loop, or connect to the sides as it adds an extra aspect to think about.
  • I'm not sold on the term 'curve', as its all straight lines, but can't think of a better term, perhaps 'ring', or keep it as 'loop' clarifying that it can be an open loop as well as closed.
  • 'Valleys' is better than 'pockets' IMO as it makes more sense visualising a boulder rolling down a mountain
  • For the name of the puzzle, I think 'Balls on Hills' could definitely be improved upon. Something simple such as 'Boulders' or 'Valleys', could be better, or even 'Boulders & Valleys'!
  • Alternatively as it's a Slitherlink spin-off the name could be based on that, such as 'Valleylink'. All just suggestions, your creation so completely up to you! :)
  • 1
    $\begingroup$ Awesome job! I liked your term "staircasing". Boulders & Valleys is nice, I think that's my current favorite :) Thanks for the feedback! $\endgroup$
    – BaileyA
    Commented Jan 2, 2023 at 21:29
  • 1
    $\begingroup$ @BaileyA you're very welcome, thanks for the new puzzle type! Always good to see new grid puzzles being created $\endgroup$ Commented Jan 2, 2023 at 21:42
  • 1
    $\begingroup$ Nice job. I solved it first without noticing the rule change about allowing to end the loop on the border. That makes it to have a unique solution indeed. Nice idea. $\endgroup$
    – justhalf
    Commented Jan 3, 2023 at 4:12

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