Forcing Chains
Short answer: find a cell with three or four shapes left, or a shape with three or four places left in one unit. Exactly one of those possibilities is true, so try each one in turn and follow what it forces. Any mark that every single branch rules out is dead — whichever branch turns out to be the right one, that mark was never going to survive.
What a forcing chain looks like
Every chain technique up to this point argues from a single strand. Simple colouring follows one shape through alternating links; an alternating chain threads two shapes together and reads the two ends against each other. A forcing chain drops the alternation and asks a blunter question instead: here are all the ways this cell can go — do they agree about anything?
Read the three strands one at a time. Put the triangle in the ringed cell and it leaves the cell to its left, which forces the triangle up its column; that cell can then no longer be the plus, which forces the plus along its row to the cell on the right. Put the star in instead and it leaves the cell directly above, which forces the star into the same cell as before — and rules the plus out of it again, landing the plus in exactly the same place. Put the slash in and the cell to its right, which holds only the slash and the diamond, has to be the diamond, which takes the diamond out of that whole column. Three different routes, one shared verdict: that cell on the right is not the diamond.
How to use a forcing chain
- Pick a branching point. Look for a cell with three or four shapes left, or a shape with three or four places left in a row, column or box. Exactly one of those possibilities is true and you do not know which, and that is precisely what makes the argument work. Two possibilities is not enough — see below.
- Follow one possibility at a time. Assume the first one. Every cell it sees loses that shape; wherever that leaves a shape with only one place in a unit, or a cell with only one shape, you have another forced step. Keep going until the branch runs out or reaches something worth noting, and write down what it proved.
- Do the same for the rest. Repeat for every remaining possibility, keeping each branch separate. You are not looking for the branch that is true — you never find that out — you are looking for a claim that turns up in all of them.
- Strike what they agree on. Any mark that every branch rules out can be removed. One of the branches is the truth, and it rules that mark out; so do all the others, so it does not matter which. If the branches disagree about everything, move the branching point and try again.
Keeping several branches apart on paper is the practical difficulty, not the logic. Give each branch its own mark or label next to the pencil marks it touches — a dot, a tick, a letter — and read a mark as dead only when every label reaches it. That labelling is a note-taking aid for working the puzzle out on paper; it is not how form9 is played, and the app has no branch marks of its own.
Forcing chains vs trial and error
They look identical for the first minute and then part company completely. Trial and error assumes one candidate and hopes the grid contradicts itself, which proves that one candidate wrong and tells you nothing if no contradiction turns up. A forcing chain assumes every candidate, expects none of them to break, and reads the answer out of what they have in common. Nothing is ever guessed and nothing is ever undone, which is why this is a deduction and that one is not.
Find Forcing Chains in your own grid
Paste a puzzle and this page will look for Forcing Chains in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
What is the difference between a cell and a unit forcing chain?
Only where the branches come from. A cell forcing chain takes every shape one cell can still hold; a unit forcing chain takes every place one shape can still go in a row, column or box. Both give you a set of possibilities exactly one of which is true, which is the only thing the argument needs, so the rest of the method is word-for-word the same.
Why does it need three branches and not two?
Because two branches is a technique you already know. When a cell has only two shapes left, those two are a strong link, and a chain down one branch, across that link and back up the other is an ordinary alternating chain — the same thing simple colouring and remote pairs are special cases of. Nothing is gained by calling it a forcing chain. From three branches on, no single chain covers the argument.
Do the branches have to be the same length?
No, and they usually are not. In the board above one branch takes three steps and the others five. A branch that reaches the shared conclusion in a single step is still a branch — it just happens that the struck cell was already in view from that possibility.
What if one branch contradicts itself?
Then stop and use that instead: a possibility that breaks the grid is simply wrong, and you can strike it outright. That is trial and error, a stronger result than the one you were looking for, and it is why the detector behind this page refuses to report a forcing chain when any of its branches collapses — the conclusion would be true for the wrong reason.
Is it worth the effort?
Only once everything cheaper is exhausted, which on most grids is never. But when a puzzle is genuinely stuck, a forcing chain will find something no single-strand argument reaches — the board above is deliberately one of those, and none of the easier techniques form9 knows can take the diamond out of that cell. The sudoku solver will do the check for you and name every move it makes.