Cantle · when nothing is forced
When the board stops moving
You have placed everything that was obvious. Nothing is forced now, and it looks like guesswork. You won’t need to guess: here are three shapes of deadlock on real Hard boards, from the simplest to the most general.
What a deadlock is
Every cell still has a choice
Each empty cell has at least two colors left, and none has exactly one. Two things about this moment are worth knowing.
In 45% of cases the board deadlocks before a single cell has been placed. It is stuck from the start, and you did nothing wrong.
The empty cells always form one connected patch, so there is one place to look.
All three boards below are real Hard boards: eighteen nodes, about forty lines, thirteen cells to fill.
Shape one · cheapest
The triangle
Two joined cells holding the same two colors take that pair away from every cell touching both.
Those two colors will split between them, one each, whatever happens, so a cell touching both can’t use either.
How to find it
- Find two empty cells joined by a line.
- Check whether the same two colors are left in both. To see that, look at their colored neighbors: which two of the four are still free.
- Find a cell that touches both.
Why it holds
The two are joined, so they can’t match, and between them they have exactly two colors. So one takes the first and the other the second. The order is unknown, but both colors get used, and a cell touching both can use neither.
What it gives you
The shared neighbor loses two colors at once. If it had three left, its color is now known.
Cell 0 had three colors: ●, ■, ◆. Two of them are spoken for by a pair that is not even placed yet, and that is enough to leave one.
And then there is nothing left to do. The moment 0 became ◆, the ordinary rule finished the board on its own: all twelve remaining cells followed.
90% of hard boards solved, together with the first rule
Shape two · wider
The witness
The triangle needs the pair to be the same. Often it is nearly the same: two joined cells drawing on the same three colors. That alone settles nothing, but with a witness nearby it is enough.
If a cell touching both has only two colors left, and both come from that same three, then one of the pair is certainly the third, the one the witness doesn’t have.
How to find it
- Find a cell with exactly two colors left. That is the witness. On the board below it is 10, holding ■ and ◆.
- Among its neighbors find two that are joined to each other: here 4 and 14.
- Check that those two draw on only three colors between them. Here each holds ●■◆: three, two of which are the witness’s own.
Why it holds
- Could both of the pair land inside the witness's pair, taking ■ and ◆?
- No. They are joined, so they would take one each, and between them they would strip the witness of both its colors. Nothing would be left for it, and that can’t happen on a solvable board.
- So at least one of the two lies outside the witness's pair.
- Outside ■ and ◆ the only color available to them is ●. So one of them is ●.
What it gives you
You don’t know which of the two, and you don’t need to. It is enough that ● is taken by one of them, so any other cell touching both can no longer be that color. One color struck out.
Why 8 loses it and not the witness 10
This is the easiest place to get lost, because three different roles are in play and all three are just cells on a board.
| Role | Which | What happens to it |
|---|---|---|
| Pair | 4 and 14 | One of them is ●. Which one stays unknown, and that bothers nobody. |
| Witness | 10 | Loses nothing. It only makes the conclusion possible. |
| Target | 8 | Loses ● and is left with one color. |
The whole answer is one line: look at which of them has ● at all.
The witness 10 has ■ and ◆ left. ● is not in it, so there is nothing to take away.
The target 8 has ● and ■ left. ● is in it, so it is the one that loses it. And since it had exactly two colors, one remains: 8 = ■.
And then
Nothing more to do. The moment 8 became ■, the ordinary rule finished the board: all twelve remaining cells followed.
“Can I put ● on 14 instead?”
The most natural question here, and the answer shows why the rule is built the way it is.
The rule says: one of the pair is ●. It deliberately does not say which. So at that moment putting ● on 14 looks just as legal as putting it on 4, and none of these three rules forbids it.
On this board 14 can’t be ●: exhaustive search confirms it. Yet no simple rule can see that. Force 14 to ● and “only one left” hits no contradiction at all. There is one, but it lies too deep to spot.
So the right move is not to decide this at all.
Do the thing the rule gives you for certain: 8 = ■. Right after it, the ordinary rule lays out the whole board in one pass, including 4 = ● and 14 = ◆.
There’s no need to guess which of the pair is sky. Make the one move you’re sure of, and a moment later the board shows it.
95% of hard boards solved, together with the shapes above
The law · both grow from it
Two colors left
A cell with exactly two colors left does not allow any two of its neighbors to take both.
If they did, it would be left with nothing, and that never happens on a solvable board.
This is the statement behind both shapes above. The triangle is the case where two joined cells are themselves two-colored and identical. The witness is the case where the shared neighbor is the two-colored one. Both shapes were this law all along.
One difference makes it feel strange: it strikes out a combination of colors. You learn what two cells cannot be at the same time. At first that seems weak, which is why it goes unnoticed. It is stronger than it looks, because such bans can be combined.
Step 1 · what to find
A cell with exactly two colors left. Call them X and Y. Then any two of its neighbors, A and B. Whether they are joined to each other doesn’t matter. What matters is that both touch this cell.
Step 2 · what to write down
Write down one sentence: “A and B cannot take X and Y between them”.
Two arrangements drop out: A=X together with B=Y, and A=Y together with B=X. Each of those colors is still possible in each cell on its own. Only the pair is impossible. That is what feels odd: you wrote something down, and nothing on the board changed.
Step 3 · turning it into a move
- Take some color of cell A, say C. The question is always the same: can A be C?
- If it can, what would B be then? Run through B's colors. Out go the ones equal to C (if A and B are joined), and the ones banned together with C.
- If nothing is left for B, then A cannot be C. Strike it out.
What it gives you
One color disappears from a cell that neither shortcut could reach. Usually that is enough for the first rule to start working again, and the board fills in from there.
Where neither the triangle nor the witness helps
Roughly one hard board in twenty runs into a place where no shortcut gives a move. Here is such a deadlock, on a real Hard board too. Nowhere on it is there a pair with the same two colors, and no witness. There are nine two-colored cells, but each of them alone says nothing.
Question: can 16 be ●? It had ▲●■ left.
Take its neighbor 7, which has ■ and ◆ left. If 16 really is ●, then 7 is either ■ or ◆. Both options close, but by different cells:
16 = ● and 7 = ■ is banned by cell 5: it holds exactly ● and ■, and both of these cells are its neighbors.
16 = ● and 7 = ◆ is banned by cell 1: it holds exactly ● and ◆, and it touches both as well.
No partner is left, so 16 is not ●. Two options closed by two different two-colored cells, one each. No single shortcut would have done it.
And what did that give? Nothing yet, and that’s fine
The answer to the most important question: cell 16 held ▲●■ and now holds ▲■. Nothing was placed. The board looks the same, and the other twelve cells are as open as they were.
Here is the main difference between the law and the two shortcuts, and the thing I should have said first: the triangle and the witness give you a move, the law only gives you a narrowing. It almost never places a color outright. You make one strike, then another, and somewhere along the way a cell is left with one color. Only then does the first rule switch on.
The result comes all at once, at the end. Until then it feels like standing still, though every step narrows the choice. That is why this rule is hard to love.
How hard this gets
To carry such a board to the end on bans alone, you have to hold more in your head than is comfortable. That is bookkeeping more than insight.
Which is why the triangle and the witness matter so much. They close nineteen boards in twenty, and they’re practical in a real game. The law is worth knowing to understand why those two work, and to know, on the twentieth board, that a way out exists even when you can’t see it.
100% of boards, on all five tiers
The order
What to do, in order
- Place everything forced, while anything moves: three colors among the neighbors means the fourth is the only one left.
- Stuck? Look for a triangle: two joined cells with the same pair. It’s the simplest shape, and on nine boards in ten it’s all you need.
- None? Look for a witness: two joined cells drawing on three colors, and a shared neighbor holding two of that same three.
- Not that either? Then the law. Walk the cells with exactly two colors: no two neighbors of such a cell can take both. Strike out any color left without a compatible partner.
- Go back to step 1. One strike almost always starts a chain that runs to the end.
Where the shapes stop
What this does not promise
It cannot be wrong
This follows from the reasoning itself: a two-colored cell whose two neighbors took both colors is left with none. That never happens on a solvable board, so the ban is always fair.
That they are always enough is measured, with no proof
Two rules were enough for 50,000 game boards, 1000 random planar graphs and 2840 arbitrary ones. Not once did they fail. But that’s a check: to claim “always” you need a mathematical proof.
On an ambiguous board it stops
450 boards were made deliberately ambiguous. The method never struck out a color that appears in some answer, and never named one answer as the right one. It simply stops.
All three boards on this page come from the game's own generator, and every deduction was checked against the puzzle's stored solution. The numbers: 50,000 game boards across five tiers · 1000 random planar graphs up to 50 nodes · 2840 arbitrary graphs · 450 deliberately ambiguous boards · two independent implementations · zero wrong eliminations anywhere.
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