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Chains, Colouring & Uniqueness

BUG+1

BUG+1 is a uniqueness technique for the end of a puzzle. When every empty cell has exactly two candidates except one cell with three, and each digit appears twice in every row, column and box except one digit that appears three times around that cell, the cell must be that digit. Any other choice would leave a grid that has no solution or several — so it only works on a puzzle with exactly one solution, as every Skudo puzzle has.

What you need to see on the grid

Think of BUG+1 late in a hard puzzle, when singles run out and almost every empty cell is down to two candidates. Four terms carry the method:

BUG
A grid where every empty cell has two candidates and every digit appears exactly twice in each row, column and box. Such a grid never has exactly one solution.
The +1 cell
The single cell with three candidates. It is the only thing standing between the puzzle and a BUG.
The tripled digit
The candidate of that cell that appears three times in its row, its column and its box, where every other digit appears twice.
Uniqueness
The move rests on the puzzle having one solution. Skudo checks every puzzle it publishes for that, so the move is safe here; on a puzzle of unknown origin it is not.

How to apply it, step by step

One example runs through all four steps, near the end of an expert puzzle from the app, at the exact point where its hint needs BUG+1. BUG+1 is about the whole grid, so the first three diagrams draw every cell; all 13 empty cells happen to sit in columns 4 to 6.

  1. Step 1Check that every empty cell has two candidates but one

    Go through every empty cell and count its candidates. Here twelve of the thirteen have exactly two; R6C6 alone has three: 6, 7 and 8.

    Your pencil marks must be complete and up to date: one forgotten candidate, or one missed elimination, and the count means nothing.

    The whole grid. Every empty cell lists two candidates, except R6C6 with three.
    C1C2C3C4C5C6C7C8C9
    R17694153
    R2541639782
    R38231649
    R4635491278
    R51783496
    R6492531
    R73569814
    R8217384965
    R99841327
    • Digit already on the grid
    • The one cell with three candidates

    Every cell is filled in. Nothing is removed yet: this step only reads the grid.

  2. Step 2Find the digit listed three times around that cell

    Count each candidate of the +1 cell in its row, its column and its box. In row 6, 7 appears three times (R6C4, R6C5, R6C6) while 6 and 8 appear twice. Column 6 and box 5 give the same answer: 7 three times, everything else twice.

    Check the rest of the grid too: in every other row, column and box, each digit is a candidate exactly twice. Here it is.

    Row 6, column 6 and box 5 each list 7 three times; every other count in the grid is two.
    C1C2C3C4C5C6C7C8C9
    R17694153
    R2541639782
    R38231649
    R4635491278
    R51783496
    R6492531
    R73569814
    R8217384965
    R99841327
    • Digit already on the grid
    • Row, column and box of R6C6
    • A place for 7
    • Candidate 7, the digit studied

    Every cell is filled in. Nothing is removed yet: this step only reads the grid.

  3. Step 3Place that digit in the cell

    Suppose R6C6 were 6 or 8 instead. Then 7 would vanish from it, and every empty cell would have two candidates with every digit twice in every unit: a BUG. A BUG grid cannot have exactly one solution — its pairs can always be swapped as a whole, or nothing fits at all.

    The puzzle has exactly one solution, so R6C6 is 7. This is the one step in this article that relies on uniqueness rather than on the drawn digits alone.

    Technique: BUG+1

    R6C6 takes the tripled digit: 7.
    C1C2C3C4C5C6C7C8C9
    R17694153
    R2541639782
    R38231649
    R4635491278
    R51783496
    R64927531
    R73569814
    R8217384965
    R99841327
    • Digit already on the grid
    • The one cell with three candidates
    • Candidate 7, the digit studied
    • Digit placed

    Every cell is filled in. Placed: R6C6 = 7.

  4. Step 4Collect what the placement unlocks

    The 7 in R6C6 removes 7 from its row, column and box. Box 2 had two places for 7, R3C4 and R3C6; R3C6 shares column 6 with R6C6, so R3C4 is the only one left: a Hidden Single, and the app's next hint in this puzzle. From here on, singles finish the grid.

    Technique: Hidden Single

    Box 2 after R6C6 = 7. 7 now fits only R3C4.
    C1C2C3C4C5C6C7C8C9
    R14
    R2639
    R371
    R4
    R5
    R6
    R7
    R8
    R9
    • Digit already on the grid
    • Box 2: one place left for 7
    • The cell that must be 7
    • Candidate 7, the digit studied
    • Digit placed
    • Not needed for this step

    Filled in: box 2. The other 72 cells are left blank: they are not needed for this step. Placed: R3C4 = 7.

Common mistakes

Using it with incomplete pencil marks

BUG+1 reads the whole grid. A cell where you never wrote a candidate, or one you forgot to erase, changes the counts and can point at the wrong cell or the wrong digit. Only trust it with complete, current pencil marks.

Picking the wrong candidate of the +1 cell

The answer is the digit that appears three times in the cell's row, column and box — not the cell's first or last candidate. Count all three units: they must agree on the same digit.

Applying it to a puzzle that may have several solutions

The argument needs a single solution. On a hand-made or unverified puzzle it can be wrong; on Skudo's puzzles, all checked for uniqueness, it is safe.

Keep going

  • Unique RectangleThree corners share a pair inside two boxes — the fourth loses it, or the puzzle would have two solutions.
  • Naked PairTwo cells locked to the same two digits clear those digits from every unit they share.

Learn the move, not just the digit

In a real game, when BUG+1 is what opens the next cell, the hint in the app names it and shows the cell it unlocks — the same move as step 4 above.

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