A unit forcing chain takes one digit in one house and follows every place it can go there, one branch per place, with singles only. The digit must sit in one of those places, so whatever all the branches agree on is proven.
How to spot it
Start with a digit that has two places left in one row, column or box. Then check what each place forces at once. A good place sits in a cell with two candidates, or leaves another house one spot for some digit, so its branch gets going straight away.
Row three has its 9 in only two cells, column three and column nine. A 9 in column three leaves row one, column three with one candidate. A 9 in column nine leaves column nine one place for its 5. Both of those land in row one, so the two branches may well meet there.
Find a digit with two places
Row three still needs a 9, and only two of its cells can take one, column three and column nine. Try each place in turn, each time from this grid, and add only the naked and hidden singles that follow.
Put the 9 in column three
Then row three, column three is the 9, and its 2 goes. That 9 rules out the other 9s in row three, in column three and in box one. Row one, column three loses its 9, and so does row three, column nine.
Row one, column three is the 2
Row one, column three had only a 2 and a 9. With the 9 gone, it is the 2. That 2 rules out the other 2s in row one. Row one, column five and row one, column nine both lose their 2.
Now put the 9 in column nine
Go back to the starting grid. This time row three, column nine is the 9, and its 2, 5 and 8 go. That 9 rules out the other 9s in row three, in column nine and in box three. Row three, column three loses its 9, and so do row one, column eight, row one, column nine and row nine, column nine.
Column nine's 5 goes to row one
Column nine had two places for a 5, rows one and three. Row three, column nine is now the 9, so row one, column nine is the 5. Its 2 and its 8 go.
Both branches agree
If the 9 is in column three, row one, column three is the 2, in the same row as row one, column nine. If the 9 is in column nine, row one, column nine is the 5. Either way, the 2 in row one, column nine cannot stand.
Strike the 2
Row three's 9 is in one of those two places, so the 2 in row one, column nine is gone for good. Wipe out both branches and keep just the strike.
Why it works
Row three needs a 9, and it has only two cells that can take one. So one of the two branches is what really happens on this grid. A candidate that both branches remove is false wherever the 9 turns out to be.
No branch has to fail for this to work. The 9 really sits in column three, so the column-nine branch is the false one. Even so it fills seven cells, runs out of singles and leaves the grid intact. So Nishio can't strike either 9 here. The 9 in column three is true, and the trial with the 9 in column nine never breaks. The forcing chain still gets its strike.
The two branches as chains
In words: if the 9 is in column three, row one, column three is not the 9, so it is the 2, so row one, column nine is not the 2. If the 9 is in column nine, row one, column nine is the only place left for column nine's 5, so it is not the 2. Every link follows from the one before it, and the two chains finish in the same cell.
Three or four places
A house can give a digit three or four places. Then there is one branch for each place, and a candidate goes only if every branch removes it. Branches stick to naked and hidden singles, as Nishio's trial does. With more branches there is more to keep apart, so a digit with two places is the easiest start.
How it differs from digit forcing
A digit forcing chain follows one candidate as true and as false, while a unit forcing chain follows every place one digit has in one house.
When the digit has only two places, the two methods meet. Test the 9 in row three, column three as true and as false. Its false branch puts row three's 9 in column nine by a hidden single, and from there it runs like the second branch above. So digit forcing reaches the same strike, one single later.
With three or four places, a false branch still leaves the digit two or more places in that house, so no hidden single for it follows there. Unit forcing still has one branch for each place.
Unit forcing and Nishio
Nishio assumes one candidate and looks for a broken grid. A unit forcing chain never needs a branch to break. If one does, that place can't hold the digit, and its candidate goes. That is a Nishio result.
Nishio can reach the same 2 here. Assume row one, column nine is the 2. Row one, column three loses its 2 and is the 9. Column nine's 5 now has one place left, row three. Then row three has no place for its 9. Row three, column three shares a column with that 9 above it, and row three, column nine is the 5. The trial is two forced digits long, but it only works if you guess which candidate to assume.
Start from one digit in one house. Both branches remove the same 2.
Assume the 2 itself. After two forced digits, row three has nowhere for its 9.
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 pointing pair on the 2s. Box eight has its 2s only in column six, so row three, column six loses its 2.
Next come another pointing pair on the 9s, a naked pair in row three and simple coloring on the 4s. Then come three singles, the last of which puts the 2 in row one, column three. That takes the 2 from row one, column nine.
The hint's own unit forcing search tries every place a digit has in a house where it has two to four. On the starting grid, no step the hint checks before Nishio strikes that 2 on its own. Nishio does, as the comparison shows, and so does the hint's digit forcing search. Four moves from the next chapter strike it too. The game checks unit forcing last of all, so the hint seldom shows one. A different route can still be right.
After the strike
Row one, column nine keeps its 5, 8 and 9. No single follows from the strike alone. The solution puts a 5 there.
Other houses on this grid lead to the same strike. Row three's 5 also has two places, columns five and nine, and both of its branches take the same 2.
Working on paper
Run one place at a time. Take it as far as the singles go, write down its removals, and rub it out before you start the next place from the original grid. Then compare the lists.
Pick a digit with two places in one house. Place it in each spot in turn, add singles only, and start both times from the same grid. A candidate removed in every branch is gone, even if no branch breaks.
Where this sits on the path
Unit forcing chains close the chapter on chains and coloring. They come after digit forcing chains and cell forcing chains. All three run singles-only branches from the same grid and keep what the branches agree on. They differ in where the branches start.
The last chapter opens with BUG+1, which counts candidates instead of testing them.
On a different grid, row nine has only two places for its 6, columns seven and eight. Try the 6 in each place, adding singles only. Which candidate is gone in both branches?
Start with the 6 in column seven, then look up that column.
Итог
- Ищите
- A digit with only two places in a row, column or box, where each place forces a single at once.
- Область
- One digit in one house. One branch for each place, each from the same grid, with only naked and hidden singles.
- Результат
- Any candidate that every branch removes is gone. The digit's places in the house stay.
- Признак
- Two short branches that end on the same cell.
- Если не сработало
- No common removal? Clear the branches and pick another digit or house. If one branch breaks, that place can't hold the digit, which is a Nishio result.