Nishio tests one candidate. Assume it is true, fill in only the naked and hidden singles that follow, and look for a contradiction: a cell with no candidate left, a house with no place for a digit, or a house that needs the same digit twice. If the trial breaks, the candidate is false.
When to try it
Nishio has no pattern to spot. It is what you try when the patterns run out. On this grid, every technique the in-game hint tries before Nishio finds nothing, from the singles through coloring and chains.
Pick a candidate whose trial will be short. Good starts sit where the grid is tight: a row, column or box with only two places for a digit, or a cell with only two candidates. Then the assumption forces a digit at once, and that digit forces the next.
Assume the 5
Row five, column two holds a 2, a 5 and an 8. Assume it is the 5. Its 2 and 8 go, and so does every other 5 in row five, column two and box four.
Row five needs a 2
Row five had two places for a 2: this cell and row five, column eight. With a 5 here, the 2 must go in column eight, so that cell's 1 goes.
Row five needs a 1
Row five also had two places for a 1, columns one and eight. Column eight now holds the 2, so the 1 goes in column one. That cell loses its 5, 7 and 8.
Follow both columns down
Row eight, column one has a 1 and an 8. Column one now holds the 1, so it is an 8. Row nine, column two has a 5 and an 8. Column two holds the assumed 5, so it is an 8 too.
Box seven needs two 8s
Both cells sit in box seven, and a box holds the 8 only once. The trial has broken: a house needs the same digit twice.
Strike the 5
The 5 led to a contradiction, so row five, column two is not a 5. Strike it and erase the rest of the trial. Only that one result stays.
Why this is not guessing
A guess writes a digit in and keeps solving, hoping the grid works out. Nishio is a test with a fixed rule. You assume one candidate, apply only naked and hidden singles, and stop. If the singles break the grid, the assumption was false and the candidate goes. That is a proof, the same kind 3D Medusa makes when a color puts one digit twice in a house.
If the trial runs out of singles without a contradiction, it proves nothing. The candidate may still be false, but this test can't show it. Erase the trial and test another candidate.
The in-game hint uses Nishio. It assumes each candidate in turn, fills in singles only, and strikes the candidate when the grid breaks.
Three ways a trial breaks
- A cell with no candidate. Every digit in some cell has been removed.
- A house with no place for a digit. A row, column or box still needs a digit, and none of its cells can take it.
- A house that needs a digit twice. Two cells of one house are left with only the same digit. Write either one in and the other cell is empty, so this is the first kind, one step early. It is what happened here.
Which candidate to test
On this grid 44 candidates fail their trial, but the trials differ a lot in length.
The in-game hint checks cells from the top left and reports the first candidate that fails. Here that is the 5 in row four, column one. Its shortest trial needs eight forced digits before row seven needs two 9s. The 5 in row five, column two fails after two.
Short trials start where the grid is tight. Row five had only two places for a 2 and two for a 1, and both ran through a cell with only two candidates.
Two forced digits, then two cells of box seven both need an 8.
Eight forced digits, then two cells of row seven both need a 9.
After the strike
Row five, column two keeps a 2 and an 8, and the 2 is its digit in the solution. No single follows from this one strike. Each trial removes one candidate, and the next trial starts from the thinner grid.
The two 8s in box seven were never the problem. In the solution, row nine, column two is an 8 and row eight, column one is a 1. The 5 forced that 1 away.
Nishio and forcing chains
A forcing chain also starts from an assumption and follows what it forces. Nishio is the simplest kind: one start, singles only, and it needs a contradiction to prove anything.
The other forcing chains in the in-game hint run the same singles-only trial from every candidate of a cell, or from every place of a digit in a house. Whatever all the branches agree on is true, so they can prove things without a contradiction. The hint tries Nishio before any of them.
Assume a candidate in a tight spot and fill in only the singles it forces. Stop at the first cell with no candidate, house with no place for a digit, or digit needed twice in a house, and strike the candidate. A trial that ends quietly proves nothing.
Where this sits on the path
Nishio comes near the end of the chapter on chains and coloring. 3D Medusa tests a whole color at once. Nishio tests one candidate, and checks it with nothing but the naked single and the hidden single from chapter two.
Digit forcing chains come next. After unit forcing chains close this chapter, the last chapter opens with BUG+1, which counts candidates instead of testing them.
Row nine, column nine holds a 1, a 5 and an 8. Test each one with singles only. Which can you strike?
Start with the 5. Row nine, column two has only a 5 and an 8.
Recap
- Look for
- A grid where the patterns have run out, and a tight spot: a digit with two places left in a house, or a cell with two candidates.
- Scope
- One candidate at a time. The trial uses only naked and hidden singles.
- Result
- If the trial breaks, that one candidate goes. If it doesn't, nothing changes.
- Tell-tale
- An assumption that forces a digit at once, and another digit from that one.
- If it fails
- Erase the trial and test another candidate. A quiet trial proves nothing either way.