Simple Colouring — one digit, two shades

The short version

Take one digit. Ignore the other eight.

Find a unit — a row, column or box — where that digit has only two possible cells left. Those two cells are tied together: one of them takes the digit, the other cannot.

Now follow the tie. The cell at the other end of that pair sits in some unit where the digit again has only two spots, so it ties to a third cell, and so on. Every time you step across a tie, the truth flips.

That gives you two shades. All of shade A is true, or none of shade A is. Shade B is the exact opposite.

Now look for a cell that sees a member of both shades. Whichever way the chain falls, one shade is true and fills its cells with the digit — so that cell cannot take it.

Nothing was assumed. Every step says: this unit has room for the digit in exactly two cells, so if one is out, the other is in.

Why the shades alternate

Call the tie a strong link: in this unit the digit has exactly two homes, so exactly one of them is the digit.

unit:  [ A ] ... [ B ]
    A true  → B false
    A false → B true

Follow two links in a row and you are back to true. That is why colouring works as a single picture: the chain does not tell you which way it goes, but it does tell you that the cells of one shade always agree with each other.

This is the whole difference between colouring and guessing. A guess has one branch. Colouring carries both branches at once, in two colours, and only uses the places where they agree.

A real board

The digit is 8, and three units tie it into a chain of four cells:

• Column 8 has room for 8 in exactly two cells: R2C8 and R8C8.
• Box 9 (bottom right) has room for 8 in exactly two cells: R7C7 and R8C8.
• Row 7 has room for 8 in exactly two cells: R7C4 and R7C7.

That gives one chain: R2C8 — R8C8 — R7C7 — R7C4. Paint it from the top:

• Shade A: R2C8, R7C7
• Shade B: R8C8, R7C4

Now test both cases, and watch them collapse into the same conclusion.

• If shade A is true: R2C8 = 8. R2C4 sits in row 2 with it, so R2C4 cannot be 8.
• If shade A is false: then shade B is true, so R7C4 = 8. R2C4 sits in column 4 with it, so again R2C4 cannot be 8.

Either way, R2C4 loses the 8, and its candidates drop from {2, 4, 8} to {2, 4}.

That is the entire technique: no cell in the chain is filled in, and the answer still comes out.

How to find one

You do not scan for colours. You pick a digit that looks thin and follow it.

1. Count the spots. Sweep the grid for a digit that has exactly two possible cells in several units. Sixes and sevens in a crowded board are good candidates; a digit with three or four spots everywhere is not.
2. Join the pairs. Draw a line between the two cells of every such unit. You are not looking for anything yet — just building the chain.
3. Check for open ends. A chain that stops without closing is what you want. A four-cell loop is an X-Wing; skip it.
4. Paint. Alternate the shades along the chain from one end.
5. Look for a cell that sees both shades. It must see at least one cell of shade A and at least one of shade B. That cell loses the digit.

The last step is where people give up too early. The target is usually not in the chain, so you have to check every empty cell that holds the digit — not just the ones you painted.

The second rule

There is a rarer conclusion worth checking, because when it fires it moves a whole shade at once.

If two cells of the same shade can see each other, that shade cannot be true as a whole — it would put the digit in both cells, and two cells that see each other cannot share a digit. So the shade is false, and the digit comes out of every cell in it.

same shade, two cells in one unit
    → that shade is false
    → the digit leaves every cell in it

This one is rare because chains usually do not fold back on themselves. Check it anyway: it costs one look, and the payoff is several eliminations instead of one.

Colouring, X-Wing, fish

These three all look at the places where a single digit can go, so they get mixed up constantly.

An X-Wing is a colouring chain of four cells that closes into a loop. A Swordfish is not a colouring chain at all — the units hold three spots each, so there is nothing to alternate.

That is the trade: colouring accepts almost any shape, and pays for it by proving less.

The traps

Painting cells that are not in the chain. A cell holding the digit is not automatically part of the structure. Only the cells joined by two-spot units get a shade. Mixing in a cell that merely has the candidate turns the whole picture into nonsense.

Using a unit with three spots. The tie has to be exact. If the digit can go in three cells of that row, nothing follows from one of them being false.

Checking only the painted cells. The victim is almost always outside the chain. The chain exists to prove something about cells you have not painted.

Forgetting that the shades are all-or-nothing. A shade is not "one of these is the digit". It is "all of these are, or none of these are". Cells inside one shade never contradict each other unless you find a pair that can see each other — and that is the second rule, not a broken chain.

Colouring two digits at once. Two colours, two digits, four shades: it is now impossible to read. Finish one digit before starting the next.

Where to use it

Colouring sits between the fish and the chains. Try it when a digit has several two-spot units on the board and the fish techniques have nothing to bite on, because the chain is allowed to bend in ways a rectangle is not.

It is also the cheapest way into chain thinking. XY-Chain walks a line of two-candidate cells; colouring walks a line of two-spot units. Same idea, different object, and colouring is easier to see because you only track one digit at a time.

If the chain gives you nothing and the board is still stuck, the next stop is forcing — but that is a different kind of work, and most Expert boards fall long before it.

How often does this technique actually show up? See its measured numbers →