Binary Puzzle Candidate Elimination with Complete Line Lists

List completions from the visible clues

We checked BinaryPuzzle.com's worked example of comparing line candidates, then made the exercises below independently. They demonstrate the balance and no-triples checks on one line. Each exercise is a separate partial row, with no crossing columns or completed neighboring rows supplied.

An underscore means a blank. Number columns from left to right; R1C4 means row 1, column 4. A six-cell line needs three of each digit, and an eight-cell line needs four. Copy the clues and any digits already proved into every candidate, keeping their positions fixed.

For a short line, fill its blanks systematically. Three blanks have eight possible fillings. In blank order, write 000, 001, 010, 011, 100, 101, 110 and 111. Putting each filling back into the original row gives a complete list you can check without relying on which patterns happen to catch your eye.

With more blanks, use the quota first. If four blank positions need exactly one zero, place that zero in each position in turn and fill the other three with ones. You have listed every balanced completion without writing twelve unbalanced fillings. Check triples afterward.

• Review the overall Binary Puzzle solving order

Filter all eight fillings of a six-cell row

Start with _ 0 1 _ _ 0. Its two zeros and one one leave one zero and two ones to place in C1, C4 and C5. The table checks all eight fillings, including those with the wrong counts. Read each compact sequence from C1 to C6.

Only 101010 and 101100 survive. Both put 1 at C1. Their values at C4 and C5 differ, so those two cells stay blank. The row becomes 1 0 1 _ _ 0. You have proved one move while retaining both possible endings.

Use the quota before checking an eight-cell line

Now take 0 1 0 _ _ _ 0 _. Three zeros and one one are visible. The four blanks need one zero and three ones. Put the missing zero at C4, C5, C6 or C8 in turn. Those are the only four balanced completions.

The option with zero at C8 makes 111 across C4 to C6. Remove it. Each remaining candidate puts 1 at C8, although the missing zero still has three possible positions. Write 0 1 0 _ _ _ 0 1 and leave the middle segment open.

Place only values shared by all survivors

Compare one position at a time. A blank is forced when every surviving candidate gives it the same value. When candidates disagree, leave it open, even if one digit appears more often. In exercise 2, C4 is 1 in two survivors and 0 in the third. Both values remain possible there.

If you accidentally omit 01001101 from exercise 2, your shortened list appears to prove C4 is 1. The omitted candidate puts 0 there and passes every stated check. Before entering a shared value, check that your list includes every filling allowed by the clues and counts.

The same method handles Takuzu line candidates down a column. Number its positions from top to bottom and compare them in that order. Keep the candidate list beside the grid on paper, so a possible completion remains separate from the values you have entered.

Practise with an unresolved line

Try 0 _ _ 1 _ _ 0 1. Can you fill any blank using only this row? It already has two zeros and two ones, so its four blanks need two of each. List the six ways to choose the two zero positions, then check every three-cell window before comparing survivors.

The six balanced completions are 00011101, 00110101, 00111001, 01010101, 01011001 and 01110001. Delete 00011101 for triples at C1 to C3 and C4 to C6. Delete 00111001 for 111 at C3 to C5. Delete 01110001 for triples at C2 to C4 and C5 to C7.

That leaves 00110101, 01010101 and 01011001. Across those three, each original blank takes both values somewhere. No new cell is forced by these checks. Continue elsewhere on the board and return when a crossing line supplies another confirmed value.

Keep the line proof within its limits

On a full classic board, also reject a candidate that duplicates an already completed row; compare columns with completed columns. A row may match a column. The separate uniqueness guide covers that check. A candidate can also fail a crossing line, so surviving these row checks alone does not establish a complete-board solution.

If your list has no survivors, recheck the copied clues, required counts and earlier entries. If survivors disagree at every blank, keep those cells open and inspect another line.

• Check the Binary Puzzle uniqueness rule

Sources

BinaryPuzzle.com's candidate example supports the comparison method. Its rules specify the line constraints used here. All three exercise rows and their filtering lists were created for this guide and independently enumerated.

• BinaryPuzzle.com's worked line-candidate example
• BinaryPuzzle.com's classic rules