What is the difference between ball sort and water sort?
They are the same puzzle with one difference in how a move works. In ball sort you move one ball at a time. In water sort a pour moves every touching unit of the same colour at once, as far as the other tube has room. Whether a starting position can be sorted is the same in both.
How does a move differ?
| Ball sort | Water sort | |
|---|---|---|
| What moves | The top ball of a tube | All the touching units of one colour at the top of a tube, as many as fit |
| Where it can go | Onto the same colour, or into an empty tube | Onto the same colour, or into an empty tube |
| Goal | Every tube holds one colour | Every tube holds one colour |
| Counting moves | One ball is one move | One pour is one move, however much it carries |
If a board can be solved in one, can it be solved in the other?
Yes. Ito and co-authors proved it in the paper "Sorting Balls and Water: Equivalence and Computational Complexity" (FUN 2022): a position can be sorted as a ball sort puzzle exactly when it can be sorted as a water sort puzzle. The same paper shows that deciding whether a general position can be sorted is NP-complete, which means no fast method is known that works for every board of every size.
Do they take the same number of moves?
No. A single pour can do the work of several single-ball moves, so the move counts differ. A par or a "fewest moves" figure from one game does not carry over to the other, even for the same board.
Which one is Orb Alchemy?
Orb Alchemy is a ball sort puzzle. There is no water in it: you move one orb at a time between glass vials that hold four orbs each. It adds three things of its own. A full vial of one colour is corked for good. Clouded orbs hide their colour until they reach the top of a vial. And a sealed vial stays shut until another vial is complete.
Try a ball sort board
Five colours. Each tap moves a single orb.
Can I use a ball sort solver on a water sort puzzle?
Yes. Because the two puzzles agree on what can be sorted, the Orb Alchemy solver works for both. Enter each unit of water as one orb. The answer comes as single-orb moves, and a water sort player pours the same colours in the same order.
Where next?
- How to play: the rules, with boards to try.
- Unlimited boards: play ball sort online with no ads.
- Today's Daily Brew: three new boards every day.