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Binairo: Try Every Arrangement

The every-arrangement technique says: list every way a line could still be finished, keeping the rules (including not copying a finished line). A square that’s the same colour in all of them is settled. It finds everything one line can tell you, but it’s slow, so it’s for when the quick techniques find nothing.

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This is the last of the Binairo techniques.

The rule

  1. Check that no pair, gap, full count or no-repeats move is left anywhere.
  2. Pick a line with only a few empty squares.
  3. List each way to fill them that keeps the rules: no three alike, half and half, and not a copy of a finished line.
  4. Any square that’s the same in all of them is settled.
Every arrangement: the only way to finish a row A row of six: yellow, empty, empty, teal, empty, yellow. It needs one more yellow and two more teal. Yellow in square 2 would force teal in squares 3 and 5, making three teal in a row, so square 2 is teal; then square 3 must be yellow, or squares 2 to 4 would be three teal, and square 5 is teal. 1 2 3 4 5 6 1
Yellow, two empty squares, teal, an empty square, then yellow. Only one way works: teal, yellow, teal, teal.

In a real puzzle

On the 10×10 every technique page follows, it’s first needed at step 23. Row 2 has four empty squares, columns 1 and 3 to 5, and needs three teal and one yellow for them. Only two ways keep the rules, and both put teal in columns 1 and 3. They differ in columns 4 and 5, so those stay open for now.

Step 23: every arrangement Hard level 2 at step 23, with row 2 highlighted and the squares this step settles marked. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
Step 23: both ways to finish row 2 have teal in columns 1 and 3.

How often you’ll need it

Never on Easy. On 49 of the game’s 50 Medium levels it’s needed, about 5 times per puzzle, and on every Hard level, about 8 times. On the 10×10 here it’s 4 of the 57 moves.

Next steps

Questions

Isn't trying every arrangement just guessing?

No. You don’t fill anything until you’ve tried every way the line could go. A square is only filled if it’s the same in all of them, so it’s as certain as a pair.