Balls in Pockets (Balls on Hills)

Question

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.

Puzzle

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Penpa Link

Rules

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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:

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.

Extra

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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!

Answer 1

Solution:

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Logic:

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.

2. 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.

3. 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.

4. 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!

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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! :)