Pick one digit, find every house where it has exactly two places, and join each such pair of cells. Color the joined cells in two alternating colors. One color holds the digit in every one of its cells and the other holds it in none. A color that meets itself in a house is the false one, and a cell that sees both colors cannot hold the digit.
How to spot it
Simple coloring follows one digit across the grid. Pick a digit and look for houses where it has exactly two places. Two places like that are a strong link, because one of them holds the digit and the other does not. The X-wing is built from two of them, and the W-wing from one.
Links that share a cell join into a chain. Give the first cell one color and the cell at the other end of each link the other color, and keep going until no link is left. You get two groups of cells, and the digit sits in all of one group and in none of the other.
You do not yet know which group is which. Two checks work anyway. If two cells of the same color share a house, that color is false. And a cell outside the chain that sees both colors cannot hold the digit.
Find the links on one digit
Take the 2. It has exactly two places in rows one and three, in columns one, three and seven, and in boxes one, three and four. Those eight links share cells, so they join into one chain of seven cells. Row eight and box nine have two places too, but both join the same two cells in row eight, and nothing else touches them.
Color the chain in two
Start anywhere. Row one, column one gets blue. Its links lead to row one, column eight, row four, column one and row three, column three, and those three get purple. Their other ends get blue again, and so on. Every link ends up with one blue end and one purple end. Three cells are blue and four are purple.
Look for a color that meets itself
Row four, column one and row four, column seven are both purple, and they share row four. If purple were the true color, row four would hold two 2s. So purple is false. No two blue cells share a house.
Strike every purple 2
Purple is false in every cell at once, so the 2 comes out of all four purple cells. Those are row one, column eight, row three, column three, and both purple cells in row four. Nothing else changes yet.
Read what the strike left behind
Row one, column eight had 2 and 4. Now it is a 4, a naked single. Each blue cell is now the only place for a 2 in one of its houses, so all three are hidden singles. So is row four, column nine, the last 2 left in row four. Before the strike this grid had no single of either kind. Nothing earlier on the path removed a mark either, from pointing pairs through to the three wings.
Why one color holds every 2
A house holds each digit exactly once. So when the 2 has only two places in a house, one of them is the 2 and the other is not.
Now follow the chain. Suppose row one, column one is the 2. Then the other end of each of its links is not, which makes the next ends the 2 again, and so on around the chain. Suppose instead that row one, column one is not the 2. Then everything flips. Either way, all the blue cells agree with each other and all the purple cells agree with each other. One color is the 2 everywhere and the other nowhere.
Two purple cells in row four cannot both be 2, so purple is not the true color, and blue is. In the solution all three blue cells hold the 2.
A column with three places
Column eight looks like it could extend the chain. It has a 2 in row one, which is purple, and a 2 in row five. Join them and row five, column eight turns blue.
It is not a link, because column eight has a third place for the 2, in row eight. Treat it as one anyway and blue now meets itself, since row five, column eight and row five, column three share row five. You would decide that blue is false and strike the 2 from the three blue cells. Those are exactly the three cells where the 2 belongs.
Before you add a link to a chain, count the places for the digit in that house. It needs exactly two.
The 2 has two places in column three. One of them holds it, so the two cells take opposite colors.
The 2 has three places in column eight. The one in row eight breaks the link, and in the solution it is the 2.
A cell that sees both colors
The second rule does not need a clash. It works on any chain, even one where no color meets itself, which is the more common case.
Look at row five, column nine. It holds a 2, it is not part of the chain, and it sees one cell of each color. Row five, column three is blue and shares its row. Row four, column seven is purple and shares its box. One of the two colors is true, so one of those two cells holds the 2, and either way that 2 is in a house with row five, column nine. It loses its 2 without your knowing which color is true.
On this grid the two rules agree. The clash says blue is true, which puts the 2 in row five, column three, and that alone would take the 2 out of the rest of row five.
Pick a digit and join every pair of places where it has exactly two in a house. Color the chain in two, alternating along each link. If one color meets itself in a house, strike that color everywhere. Otherwise strike the digit from any cell that sees both colors. Three places in a house is not a link.
Where this sits on the path
Simple coloring places nothing. It removes candidates and leaves the placing to the singles. On the grid above it made five singles at once.
It opens the chapter on chains and coloring. The X-wing used two links and the W-wing one. Simple coloring uses as many as one digit has, joined into a chain.
The X-chain comes next. It follows one digit as well, and it can also step between two cells that see each other, since those two cannot both hold the digit. Later in the chapter, 3D Medusa carries the same two colors across both digits of a cell with only two candidates, so one cluster can pass from digit to digit.
Row five, column nine loses its 2 because it sees both colors. Which other 2 on this board can you strike for the same reason?
For each candidate cell, list the blue cells it shares a house with, then the purple ones. It needs at least one of each.
الخلاصة
- ابحث عن
- One digit with exactly two places in several houses, where the links share cells and join into a chain.
- النطاق
- One digit. The cells of the chain, and any other cell holding that digit that sees cells of the chain.
- النتيجة
- Either every cell of one color loses the digit, or the digit comes out of cells that see both colors.
- العلامة
- A digit that keeps turning up in pairs, two places in a row here, two in a box there.
- إذا لم تنجح
- If no color meets itself and no cell sees both colors, try another digit. A house with three places for the digit is not a link.