Simple Coloring in Sudoku, Step by Step

The idea behind coloring

Coloring works on one digit at a time, and that narrow focus is exactly what makes it manageable. In many units that digit has only two possible cells left, and those two form a conjugate pair, meaning exactly one of them is the digit and the other is not. If you color one cell of such a pair, its partner is forced to take the opposite color. Chaining these pairs across the board gradually splits a whole network of cells into two alternating groups. Within a single chain the two colors are perfect opposites: one color is entirely true and the other entirely false, and the only thing you do not yet know is which is which. That is the quiet power of the method. You are not tracking nine digits at once or juggling complicated candidate lists; you are following a single digit through the grid and letting a simple rule, opposite colors on conjugate pairs, do the bookkeeping for you. When the chain finally stops growing, the two colored groups hold a strong logical relationship that you can cash in for eliminations.

Building the chain

1. Pick a digit and find every unit where it has exactly two candidate cells.
2. Start at one such cell, give it color A, and give its conjugate partner color B.
3. Follow the links outward; each time a colored cell shares a conjugate pair with an uncolored one, the partner takes the opposite color.
4. Keep going until the chain stops growing, then read off the two colored groups.

A worked example

Take the digit five. Suppose box one has five only in r1c1 and r3c3, so color r1c1 as A and its partner r3c3 as B. Now column three has five only in r3c3 and r7c3, so r7c3 must be the opposite of B, which is A. Row seven, in turn, has five only in r7c3 and r7c8, so r7c8 becomes B. Your chain now reads color A at r1c1 and r7c3, and color B at r3c3 and r7c8, four cells alternating cleanly across two boxes. Exactly one of those colors marks the true fives; either the A cells are all fives and the B cells are not, or the reverse. You have not yet solved anything, but you have organised four scattered candidates into a tidy two-sided structure, and that structure is about to hand you an elimination that ordinary scanning would never find.

The two ways it pays off

There are two possible conclusions from a finished chain. The first is rare but decisive: if two cells of the same color ever land in the same unit, that color is impossible, because a unit cannot hold the digit twice, so every cell of the other color immediately becomes the digit. The second is far more common and worth mastering. Any uncolored cell that can see both a color A cell and a color B cell cannot hold the digit at all, since one of those two colors is the true one and already claims the value within that cell line of sight. Return to the example and look at r7c1. It sees r1c1, an A cell, straight down column one, and it sees r7c8, a B cell, across row seven. If A is the true color then r1c1 is five and column one is taken, so r7c1 is not five; if B is the true color then r7c8 is five and row seven is taken, so again r7c1 is not five. Either way the answer is identical, and you erase five from r7c1. The two colored cells doing the work do not even share a unit; the chain reached right across the grid to trap the victim.

Keeping the colors straight

Two highlighters, or two note colors in an app, are all the equipment you need. The single discipline to keep is that colors alternate strictly along conjugate pairs and nowhere else. Never color a cell just because it feels related; only a genuine strong link, a unit with exactly two candidate cells for the digit, is allowed to pass the color along. If a chain ever seems to contradict itself, retrace it and check that every step really was a conjugate pair, because one loose link will poison the whole conclusion. It is worth the care, because coloring reaches eliminations that naked pairs and simple scanning cannot, and it needs no exotic pattern recognition, only patience and two colors. Start with digits that already appear as tidy conjugate pairs in several units, since those chains grow the fastest and reveal the most. In Shapedoku each value is a distinct glowing shape, so following a single shape conjugate pairs across the grid is easy on the eye, and turning on notes lets you tag the two groups without losing your place. Once coloring clicks, a whole tier of harder boards suddenly opens up.