Is every water sort puzzle solvable? We tested 25,200 boards
The short answer
In our test, yes: every one of 12,600 randomly dealt boards with two empty tubes could be solved, at every size from 3 to 10 colors. Take away one of those empty tubes, though, and the picture flips. With a single spare tube, most boards cannot be solved at all, and at 10 colors fewer than 1 in 100 can.
So the second empty tube is not a small comfort. It is what makes the game fair.
How we tested
We wanted numbers, not opinions, so we let a computer do the work.
- Deal a random board. Each color fills exactly four slots. We shuffled all the colored layers and poured them into the tubes, four per tube, with no checking and no “nice” deals.
- Add the empty tubes: two (the usual setup) or one.
- Search every possible position. A breadth-first search tries every legal pour from every position it reaches. Because it goes level by level, the first solution it finds is also a shortest one, measured in pours. If it runs out of new positions without finding a solution, the board is proven unsolvable.
We dealt 2,000 boards for each size from 3 to 8 colors and 300 boards for 9 and 10 colors, once with two empty tubes and once with one: 25,200 boards in total. Every single search finished, so there are no guesses in the table below.
A pour follows the normal rules: you can pour onto the same color or into an empty tube, and all of the top color that fits moves at once. Ball sort uses the same boards but moves one ball at a time. Researchers have shown that a board can be solved with ball moves exactly when it can be solved with water pours, so these results hold for ball sort too.
The results
| Colors | Solvable, 2 empty tubes | Solvable, 1 empty tube | Shortest solution (average pours) | Shortest solution (worst board) |
|---|---|---|---|---|
| 3 | 100% | 77.0% | 7.9 | 11 |
| 4 | 100% | 50.1% | 11.3 | 15 |
| 5 | 100% | 27.1% | 14.6 | 18 |
| 6 | 100% | 13.5% | 17.9 | 21 |
| 7 | 100% | 5.8% | 21.2 | 25 |
| 8 | 100% | 2.6% | 24.5 | 29 |
| 9 | 100% | 2.0% | 28.0 | 32 |
| 10 | 100% | 0.7% | 31.1 | 35 |
The shortest-solution columns are for boards with two empty tubes. Rows with 9 and 10 colors are based on 300 boards each, so treat their one-tube percentages as rough.
What this means when you play
If you are stuck, the deal is not the problem
With two empty tubes we did not find a single dead board. When a level seems impossible, the position you are in right now may well be dead, but the starting board almost certainly was not. The fix is to undo, not to give up. Our water sort game has unlimited undo for exactly this reason, and a Hint button that finds a working next pour from wherever you are.
An empty tube is your most valuable thing
The one-tube column shows what happens when you fill a spare tube too early: you effectively turn a friendly two-tube game into the much harsher one-tube game. Good players treat the empty tubes as working space, not storage.
- Pour into an empty tube only when it completes a step you can see, such as freeing a color that finishes another tube.
- Try to empty a tube again soon after you use it.
- If you have to choose, fill a tube with a color that already has three layers showing elsewhere. It will not stay there long.
Finish colors early
Every finished tube leaves the game for good, which shrinks the board. A board with 8 colors and two finished tubes plays like a 6-color board with more room.
Do not split a run if you can avoid it
When the top of a tube shows two layers of the same color and the target has room for only one, pouring splits that color across two tubes. Sometimes that is the right move, but often it creates one more thing to tidy up later.
Know your par
Our data gives a simple benchmark: a perfect solution takes about 3 pours per color (2.6 on small boards, 3.1 on big ones). If you solve a 6-color board in 18 to 20 pours, you are playing close to perfectly. The move counter in our game makes this easy to track.
Why the empty tube matters so much
Each pour needs somewhere to go. With no matching color on top of any tube, the only legal moves are into empty tubes. With two spare tubes you can always park two different colors while you dig. With one, you can park only one, and many shuffled boards simply never offer a chain of pours that works.
In general, deciding whether a water sort board can be solved is a hard problem for computers (it is NP-complete, in the language of complexity theory). The boards in phone games are small enough for a computer to check completely, which is how every level in our game is checked before you see it.
Sources and method
- Our study: 25,200 boards dealt at random and solved by complete breadth-first search, four layers per tube, standard pour rules. Data collected on 27 September 2026.
- Ito, T., Kawahara, J., Minato, S., Otachi, Y., Saitoh, T., Suzuki, A., Uehara, R., Uno, T., Yamanaka, K., & Yoshinaka, R. (2022). Sorting balls and water: equivalence and computational complexity. arXiv:2202.09495.
