3D Medusa Sudoku Technique: Colouring Across Digits

3D Medusa is a colouring technique that works on several digits at once. You give candidates one of two colours, following strong links on a digit and the two digits inside every two-candidate cell, so that exactly one colour ends up true. To spot an elimination, look for a candidate that sees both colours, or a cell that holds both colours: everything else in that cell can go.

If you know simple colouring, you already know half of it. Simple colouring stays on one digit. Medusa lets the colours jump between digits through bivalue cells, which is where the "3D" comes from.

The two kinds of link

Medusa colours along strong links only. There are two kinds:

• Same digit, one house: a row, column or box where the digit has exactly two candidates. One is true, so they get opposite colours.
• Same cell, two digits: a cell with exactly two candidates. One is the answer, so its two digits get opposite colours.

Start from any candidate, colour it orange, and keep flipping colours as you cross each link. When you run out of links, you have a cluster where either every orange candidate is true or every blue one is.

The rules for eliminating

1. Two colours in one cell. If a cell holds an orange candidate and a blue candidate, one of them is the answer. Every other candidate in that cell goes.
2. Colour trap. An uncoloured candidate that sees an orange copy of its digit and a blue copy of its digit is false.
3. Mixed trap. An uncoloured candidate that sees a coloured copy of its digit, in a cell that holds the opposite colour on another digit, is false. Whichever colour wins, it is either seen or its own cell is taken.
4. Colour wrap. If one colour appears twice in the same cell, or twice for the same digit in one house, that colour is false. Remove every candidate of that colour and keep every candidate of the other.

A worked example

This position is from a real puzzle in our library. The cluster is small: six coloured candidates on the digits 7 and 9, in four cells at the corners of rows 1 and 3, columns 3 and 9.

Colour it step by step, starting with the 7 in R1C3 as orange.

1. 7 in R1C3 to 7 in R3C3 (column 3). Column 3 has a 7 only in R1C3 and R3C3. The 7 in R3C3 is blue.
2. 7 to 9 inside R3C3. R3C3 has only 7 and 9. The 9 in R3C3 is orange.
3. 9 in R3C3 to 9 in R3C9 (row 3). Row 3 has a 9 only in R3C3 and R3C9. The 9 in R3C9 is blue.
4. 9 in R3C9 to 9 in R1C9 (column 9). Column 9 has a 9 only in R1C9 and R3C9. The 9 in R1C9 is orange.
5. 7 in R1C3 to 7 in R1C9 (row 1). Row 1 has a 7 only in R1C3 and R1C9. The 7 in R1C9 is blue.

The colours agree all the way round, so there is no wrap. Now look at R1C9. Its candidates are 4, 5, 7 and 9. The 9 is orange and the 7 is blue. Whichever colour is true, R1C9 ends up as 7 or 9, so 4 and 5 can be removed from it.

Simple colouring cannot find this. The 7s alone and the 9s alone each form a short chain, but neither says anything about R1C9's other candidates. It takes the bivalue cell R3C3 to tie the two digits into one cluster.

Puzzle 201190 from the Sudoku Royale library. Givens (row by row, dots are blanks): .....8.2..1..9.6..6.....3....5.197.8.4.8.3.56.6...21...735.4...12................. The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

How to colour a grid by hand

1. Start where links are dense. A bivalue cell whose digits both have strong links in its houses tends to grow a large cluster.
2. Use two pencil colours or two marks. On paper, circle one colour and underline the other. In an app with candidate colours, use two highlight colours.
3. Colour the whole cluster before looking for eliminations. A half-coloured cluster misses wraps.
4. Check the four rules in order. Cells with both colours and colour wraps are the easiest to see. Traps take a scan of every uncoloured candidate near the cluster.
5. If you find a wrap, it solves a lot. One whole colour becomes true, which usually places several digits at once.

How Medusa relates to other techniques

Many Medusa eliminations can also be written as chains. The example above is the same logic as a short chain from R1C9 around the loop, and an XY-Chain or X-Chain is a single path through a Medusa cluster. Colouring is often faster by hand because one cluster gives you every such path at once.

See our list of sudoku techniques in learning order and the overview of advanced sudoku strategies for where Medusa sits. Almost locked sets are the next family after chains and colouring.