A digit forcing chain follows one candidate both ways. Assume it is true and fill in the singles it forces; then assume it is false and do the same. Whatever both branches agree on is true.
How to spot it
Look for a candidate that forces a digit at once whether it is true or false. A candidate in a cell with only two candidates is a good start: if it is false, the other one is true, so the false branch also gets going straight away. Good starts sit next to tight houses, where a digit has only two places in a row, column or box.
Row six, column one has a 2 and a 7. Test its 2. If the 2 is true, column nine has one place left for its 2. If the 2 is false, the cell is the 7, and row one has one place left for its 7. Both of those places are in column nine, so it is worth checking what the two branches agree on there.
Pick one candidate
Row six, column one holds a 2 and a 7. Test the 2. Follow it twice from this same grid, once as true and once as false, and fill in only the naked and hidden singles each branch forces.
Suppose the 2 is true
Then the 2 leaves column one, row six and box four. Row six, column nine loses its 2, and so does row five, column one.
Column nine's 2 goes to row five
Column nine had two places for a 2, rows five and six. Row six, column nine has lost it, so row five, column nine is the 2. Its 1 and its 7 go.
Now suppose the 2 is false
Start again from the full grid. Without its 2, row six, column one has only the 7 left, so it is the 7. That 7 leaves column one, row six and box four. Row one, column one loses its 7, and so do row five, column one, row six, column two and row six, column nine.
Row one's 7 goes to column nine
Row one had two places for a 7, columns one and nine. Row one, column one has lost it, so row one, column nine is the 7. Column nine now holds its 7, so row five, column nine loses its 7.
Both branches agree
If the 2 is true, row five, column nine is the 2. If it is false, row one, column nine is the 7, in the same column. Either way row five, column nine is not the 7.
Strike the 7
The 2 is either true or false, so the 7 in row five, column nine goes for real. The branches agree on one more candidate. If the 2 is true, row five gets a 2. If it is false, row six has only one place left for its 2, column nine, inside box six. So row five, column seven loses its 2. Erase both branches. Only the two strikes stay.
Why it works
The 2 in row six, column one is either true or false, and there is no third choice. Each branch shows what follows from one of them. When both branches remove the same candidate, it goes whichever way the 2 turns out.
Neither branch has to break. In the solution the 2 is false and the cell is the 7. The true branch is the wrong one, yet it stops after two digits with no single left and nothing broken. So Nishio can't strike either candidate of this cell: the 2's trial never breaks, and the 7 is true. The forcing chain still gets two strikes out of this one 2.
Each branch is a short chain
Written out, the two branches read like this. If the 2 is true, row six, column nine is not the 2, so row five, column nine is the 2, so it is not the 7. If the 2 is false, row six, column one is the 7, so row one, column one is not the 7, so row one, column nine is the 7, so row five, column nine is not the 7. Each step is forced by the one before, and both chains end on the same fact.
What a branch may use
Each branch uses only naked and hidden singles, the same two moves Nishio uses. The in-game hint builds its branches that way too. Keep to singles and every step of a branch is easy to check.
How it differs from Nishio
Nishio tests one candidate to a contradiction. If the trial breaks, the candidate goes. If it doesn't, Nishio learns nothing.
A digit forcing chain follows one candidate both ways, to a common result. It never needs a branch to break. If the true branch does break, the candidate is false, and that is a Nishio result. If the false branch breaks, the candidate is true. The hint tries Nishio first.
The two methods can reach the same candidate. Nishio also strikes the 7 in row five, column nine. Assume it is the 7, and row one's 7 has one place left, column one. Column one then takes the 7 from row six, column one, which leaves it the 2. Row six then takes the 2 from row six, column nine, and column nine has no place left for a 2. That trial needs two forced digits, and you have to pick the right candidate to test.
Start from one candidate, true and false. Both branches remove the same 7.
Start from the candidate. Two forced digits, then column nine has no place for a 2.
What the in-game hint shows
The in-game hint may show a different or simpler step first. On this grid it opens with a claiming group on the 2s. Column nine has its 2s only in box six, so row five, column seven loses its 2, the second candidate this forcing chain strikes.
Next come simple coloring on the 7s, a pointing pair and a Y-wing. They take the 7 from row five, column one, row six, column nine and row one, column one. Singles follow, and the third of them puts the 7 in row one, column nine, so row five, column nine loses its 7.
The hint looks for this result by testing both candidates of a two-candidate cell. On the starting grid, no step the hint checks before Nishio strikes that 7 on its own. Nishio does, as shown above. The game checks Nishio before forcing chains, so the hint seldom shows one. A different route can still be right.
After the strikes
Row five, column nine keeps its 1 and 2, and row five, column seven keeps its 1 and 7. No single follows from the two strikes alone. In the solution row five, column nine is the 1.
Testing every candidate of one cell is a cell forcing chain, later in this chapter.
Keep the branches apart
On paper the two branches must not mix. Follow the true branch to its end, note what it removed, and erase it. Then follow the false branch from the same starting grid, and compare the two notes.
Pick one candidate, ideally in a cell with two. Follow it as true and as false, with singles only, both times from the same grid. Any candidate that both branches remove is gone. Neither branch has to break.
Where this sits on the path
Digit forcing chains come right after Nishio in the chapter on chains and coloring. Nishio follows one candidate as true and needs the grid to break. A digit forcing chain follows one candidate both ways and needs only agreement.
Cell forcing chains come next, and unit forcing chains close the chapter.
On a different grid, row two, column one holds a 3 and an 8. Follow its 3 as true and as false, with singles only. Which candidate do both branches remove?
Start with the 3 true. Row two, column four has only a 3 and a 4.
الخلاصة
- ابحث عن
- A candidate that forces a single at once whether it is true or false, often in a cell with two candidates.
- النطاق
- One candidate, followed as true and as false. Both branches start from the same grid and use only naked and hidden singles.
- النتيجة
- Any candidate that both branches remove is gone. The tested candidate stays.
- العلامة
- Two short branches that land in the same house.
- إذا لم تنجح
- If the branches agree on nothing, erase them and test another candidate. If the true branch breaks, the candidate is false, and that is Nishio.