Chains, Colouring & Uniqueness
Remote Pairs
Remote Pairs use cells that all hold the same two candidates, say 1 and 3, linked into a chain where each cell sees the next. Two cells that see each other cannot both be 1 or both be 3, so the digits alternate along the chain: odd positions hold one digit, even positions the other. A cell that sees an odd and an even position sees a 1 and a 3, and loses both.
What you need to see on the grid
Look for Remote Pairs when the same pair of candidates turns up in many cells across the grid. It is rare — the app needs it only a couple of times in thousands of puzzles — but quick to check. Four terms carry the method:
- Twin pairs
- Cells whose only candidates are the same two digits. Only these cells can be links of the chain.
- The chain
- An order of those cells where each one sees the next — same row, column or box. It needs at least four cells.
- Odd and even
- Number the chain 1, 2, 3 … The digits alternate, so all odd cells hold one digit and all even cells the other — you just do not know which is which.
- The targets
- Cells outside the chain that see an odd cell and an even cell. They see both digits, so neither fits.
How to apply it, step by step
One example runs through all four steps: the pair {1,3}, in a hard puzzle from the app, at the exact point where its hint needs Remote Pairs. Every diagram shows the whole grid. The cells a step needs are filled in; the others are left blank and hatched, so you know you can ignore them. A digit missing from a drawn cell is already placed in its row, column or box, drawn or not.
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Step 1Find the cells that hold the same two candidates
Scan the grid for cells with exactly two candidates and group them by pair. Here nine cells hold exactly 1 and 3: R1C5, R2C4, R2C7, R3C1, R5C6, R5C8, R6C7, R8C5 and R9C9.
A pair that shows up this often is worth a closer look.
The nine cells of the grid whose candidates are exactly 1 and 3. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - A cell holding exactly {1,3}
- Candidates 1 and 3, the digits studied
- Not needed for this step
Filled in: cells R1C5, R2C4, R2C7, R3C1, R5C6, R5C8, R6C7, R8C5 and R9C9. The other 72 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Link them into a chain where each cell sees the next
Start from one of them and hop to another {1,3} cell it sees, then on again. R8C5 sees R1C5 in column 5; R1C5 sees R2C4 in box 2; R2C4 sees R2C7 in row 2; R2C7 sees R6C7 in column 7; R6C7 sees R5C8 in box 6; R5C8 sees R5C6 in row 5. That is a chain of seven. R3C1 and R9C9 see no other {1,3} cell and stay out.
Each hop flips the digit: if R8C5 is 1, R1C5 is 3, R2C4 is 1, and so on. So the odd positions (1, 3, 5, 7) share one digit and the even positions (2, 4, 6) the other.
The chain, numbered from R8C5 (1) to R5C6 (7). Each cell sees the next. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - A {1,3} cell of the chain
- Chain cell, numbered in order
- Candidates 1 and 3, the digits studied
- Not needed for this step
Filled in: cells R1C5, R2C4, R2C7, R5C6, R5C8, R6C7 and R8C5. The other 74 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Remove both digits from cells that see an odd and an even cell
Look for a cell outside the chain that sees one odd cell and one even cell. R8C8 sees R8C5 (position 1) along row 8 and R5C8 (position 6) along column 8. One of those two is 1 and the other is 3, so R8C8 can be neither: it loses both 1 and 3.
Seeing two odd cells, or two even ones, is not enough — they hold the same digit, and the other digit could still go there.
Technique: Remote Pairs
R8C8 sees R8C5 (1) and R5C8 (6), an odd and an even cell. It loses 1 and 3. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - A {1,3} cell of the chain
- Chain cell, numbered in order
- Candidates 1 and 3, the digits studied
- Candidate removed
- Not needed for this step
Filled in: cells R1C5, R2C4, R2C7, R5C6, R5C8, R6C7, R8C5 and R8C8. The other 73 cells are left blank: they are not needed for this step. Removed: 1 from R8C8; 3 from R8C8.
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Step 4Collect what the elimination unlocks
R8C8 had three candidates, 1, 3 and 5. With 1 and 3 gone, only 5 is left: a Naked Single, and the app's next hint in this puzzle.
Technique: Naked Single
Row 8 after Remote Pairs. R8C8 has only 5 left. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 7 8 6 4 5 2 R9 - Digit already on the grid
- The cell that must be 5
- Candidate 5, the digit studied
- Digit placed
- Not needed for this step
Filled in: row 8. The other 72 cells are left blank: they are not needed for this step. Placed: R8C8 = 5.
Common mistakes
Letting a cell with a third candidate into the chain
Every link must hold exactly the pair. A cell with {1,3,5} may be 5, and then the alternation stops there: the cells after it no longer have to flip. R8C8 itself is such a cell — which is why it is a target, not a link.
Clearing a cell that sees two cells of the same parity
Two odd cells hold the same digit. A cell that sees only odd cells is blocked from that digit alone — and even that needs an odd cell to be the digit, which you do not know. Only an odd-even pair proves anything.
Removing only one of the two digits
A target sees a 1 and a 3 at once, so both leave it. Removing one and keeping the other throws away half the move — here it is the second removal that leaves R8C8 with a single candidate.
Keep going
- XY-ChainA run of bivalue cells pins one digit to one end of the chain or the other — every cell that sees both ends loses it.
- Simple ColoringChain one digit's strong links and colour them in turn — one colour is true, so a cell seeing both loses the digit.
Learn the move, not just the digit
In a real game, when Remote Pairs are what opens the next cell, the hint in the app names the technique and shows the cell it unlocks — the same move as step 4 above.