A cell forcing chain follows every candidate of one cell. Each candidate is tried as the cell's value, with singles only, and whatever all the branches agree on is proven.
Start with two candidates
A cell with two candidates, say 2 and 7, holds one or the other. Following the 2 as its value is the same as following the 7 as false. So on a two-candidate cell, a cell forcing chain is a digit forcing chain, and the in-game hint counts it under that name. Cell forcing earns its own name at three candidates.
On this grid, row two, column two holds only 2 and 7. If it is the 7, the 7 in row one, column one goes, as both sit in box one. If it is the 2, column two's only 7 is in row one, and that 7 sees row one, column one along the row. Both outcomes remove the same 7. A claiming group on the 7s removes it here too, so this example is only about the shape.
Row two, column two is the 2 or the 7. Both outcomes take the 7 from row one, column one.
Row eight, column seven holds 2, 5 and 7, so it needs three branches. All three take the 2 from row seven, column six.
How to spot it
Look for a cell with three candidates whose digits have few homes close by, so each branch forces a single quickly. Then find a cell that all the branches touch.
Above row eight, column seven, row seven's 5 fits in only two cells, and one of them is row seven, column six. That cell also holds a 2, a 7 and a 1, so each branch may fill it.
A cell with three candidates
Row eight, column seven holds 2, 5 and 7. Give it each in turn, always from this grid, and fill in only the naked and hidden singles that one forces. Watch row seven, column six.
Try the 5
Put the 5 in row eight, column seven. It removes the 5 from row seven, column nine, which shares box nine with it. Row seven's 5 is left with a single home, column six. So row seven, column six is the 5, and its 2 goes.
Try the 2
Return to the original grid and make row eight, column seven the 2. Row eight, column one had only 2 and 7, so it is the 7. Row nine, column nine had the same pair and shares box nine with the 2, so it is a 7 as well.
Row seven's last 7
Those two 7s see every 7 left in row seven except the one in column six. Row eight, column one sees row seven, column one in box seven. Row nine, column nine sees columns eight and nine of row seven in box nine. Only column six remains, so row seven, column six is the 7, and its 2 goes.
Try the 7
Last, start again from the original grid with the 7 in row eight, column seven. Row nine, column nine shares box nine with it, loses its 7 and is the 2. Row four's only 4 is in column three, a single in every branch, so put it in too.
Column three's 2, then row four's
Column three now has one place for a 2. Row four, column three is the 4, and row nine, column three sees the 2 in row nine, column nine. So row five, column three is the 2. In row four, the cell in column two shares box four with that 2, which leaves column six for row four's 2. Row four, column six is the 2, and it sees row seven, column six down the column. That 2 goes as well.
Strike the 2
Whichever candidate row eight, column seven holds, the 2 in row seven, column six goes. Clear the branches and keep only the strike.
Why it works
Row eight, column seven is a 2, a 5 or a 7, so one of the three branches describes the real grid. A candidate removed in all three is false whichever digit the cell holds.
No branch has to break. The solution puts a 7 in row eight, column seven, so the 2 and 5 branches are both wrong, yet each one runs out of singles with the grid intact. That is why Nishio can't strike any of the three. The forcing chain only needs the branches to agree.
Four candidates
A cell with four candidates works the same way with four branches. Each extra branch is one more chance for agreement to fail, so start with three.
Next to digit and unit forcing
A digit forcing chain tests one candidate twice, once as true and once as false, while a cell forcing chain gives one cell each of its candidates in turn. With two candidates the two methods coincide, as shown above. With three, a false 2 still leaves the cell a 5 or a 7, so a false branch forces less.
A unit forcing chain tries every spot one digit has in a single row, column or box, while a cell forcing chain keeps to one cell and changes the digit. On this grid they meet. Column seven's 2 has three places, rows one, two and eight, and a unit forcing chain on that 2 strikes the same 2 in row seven, column six.
Cell forcing and Nishio
Nishio tests one candidate and waits for the grid to break. Assume row seven, column six is the 2. Row seven, column five then has only its 1. That 1 shares box eight with row nine, column six, so row nine's only 1 is in column three, and row six, column three keeps just its 7. Now row five, columns three and five can each hold only a 2, and one row can't take two of them.
The trial is three forced digits long, but you must pick the right candidate to test. A cell forcing chain picks a cell instead and never assumes which digit it holds.
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 hidden single. Row four's only 4 goes in column three.
Next come another hidden single, two pointing pairs on the 7s, simple coloring on the 1s, a pointing pair on the 1s and a W-wing. Singles follow, and the seventh of them puts the 2 in row four, column six. That takes the 2 from row seven, column six.
The hint's own cell forcing search tries every candidate of a cell that holds three or four. On the starting grid, no step the hint checks before Nishio removes that 2 on its own. Nishio does, as the trial below shows. The hint's digit forcing search and every move of the next chapter leave it alone. The game checks cell forcing next to last, so the hint seldom shows one. A different route can still be right.
After the strike
Row seven, column six keeps its 1, 5 and 7. No new single comes from the strike, and the solution puts a 1 there.
The same three branches also strike the 1 and 2 in row four, column three and the 1 in row six, column five. Plain hidden singles on the starting grid remove those too. Row two, column nine holds 2, 7 and 9, and its own cell forcing chain takes this 2 as well, along with fourteen other candidates.
Working on paper
Write each branch as a short list of the singles it forces, starting every list from the original grid. A candidate crossed off in every list is the one to strike.
Pick a cell with three candidates where each digit forces a single quickly. Run one branch per candidate from the same grid, singles only. Strike what every branch removes, even when no branch breaks.
Where this sits on the path
Cell forcing chains are the second forcing chain in the chapter on chains and coloring. They follow digit forcing chains, which cover the two-candidate case, and come before unit forcing chains, which close the chapter. All three keep only what their branches agree on. What changes is the starting point: one candidate, one cell, or one digit within a house.
On a different grid, row five, column two holds 1, 6 and 7. Run one branch for each, with singles only. Which candidate does every branch remove?
Every branch puts a 7 somewhere in row five or in box four.
Recap
- Look for
- A cell with three candidates where each one forces a single or two straight away.
- Scope
- One cell. One branch per candidate, all started from one grid and limited to naked and hidden singles.
- Result
- A candidate struck in every branch is false. The cell itself keeps all its candidates.
- Tell-tale
- Three short branches that each leave the same cell without the same digit.
- If it fails
- Nothing shared? Erase the branches and move to another cell. A branch that breaks shows its own candidate is false, which is Nishio's job.