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Puzzle Quarry

Twin Elimination: Putting Rule Three to Work

The third rule sits idle for most of a puzzle and then decides it. Here is the moment it becomes the only thing that will move the grid.

One more filter on the same list

Mechanically this is line analysis with an extra step. List the legal endings for a line as before — quota respected, no run of three — and then strike out any that exactly reproduce a row already completed elsewhere in the grid, or a column, if it is a column you are working on. What is left is a shorter list, and a shorter list agrees about more.

Often it leaves exactly one ending, and the line is simply finished. That is the version most players meet first and it feels less like a technique than like a coincidence. It is not: it is the third rule doing precisely the job it was written for.

The one thing to be careful about is what counts as a twin. Only a line that is completely full can rule anything out. Two half-written rows that agree so far are not twins and may never become them, and treating them as if they were is the one way this technique can produce a wrong answer.

Before — row 4 has two legal endings
123456
1100110
21
31
40110
510
60
After — one of them is already taken
123456
1100110
21
31
4010110
510
60
Row 4 reads _ _ 0 1 1 0, and both 0 1 0 1 1 0 and 1 0 0 1 1 0 obey the quota and make no run of three. But row 1 is already 1 0 0 1 1 0, and no two rows may be identical — so the second ending is out and the first is forced. Line analysis alone would have found nothing here.

cells the reasoning read the cell it decides what it wrote

Why it comes last

Because it needs raw material that does not exist early. At the start of a puzzle no line is complete, so there is nothing to compare against and the rule is inert. It only starts paying out once several lines have been finished by other means — which is to say, once the cheaper techniques have already done their work.

That is also why an expert grid feels different rather than merely harder. It moves along under pairs and counting like anything else, then reaches a point where the board is two-thirds full and nothing at all will place, and the only way forward is to notice that a row you finished twenty moves ago has quietly forbidden something.

And it is why expert is the top of the published ladder here. Above it there is only assumption — following a symbol until it breaks — which will crack any grid whatsoever and therefore says nothing about how hard a grid is. Puzzles needing it are discarded during generation rather than sold as a harder level.

Columns are where it hides

Rows get compared and columns get forgotten, because rows are what the eye reads. In practice the technique fires about as often vertically as horizontally, and a stuck expert grid where every row has been checked is very frequently one column comparison away from moving.

A useful habit once a grid is half full: whenever you complete a line, take two seconds to read it as a pattern and register it — not to memorise it, just to have looked. When some other line later offers two candidate endings, one of them will occasionally ring a bell, and that bell is this technique.

Where this sits on the ladder

Common questions

What is row elimination in a binary puzzle?

Ruling out a candidate completion for a line because it would exactly duplicate a line of the same kind that is already finished. It applies to columns just as much as to rows.

Can I compare a half-finished row with another half-finished row?

No. Only a completely full line forbids anything. Two partial rows that agree so far are not duplicates yet and may diverge in the cells still empty.

Why does the third rule seem useless for most of a puzzle?

Because it needs a completed line to compare against, and early on there are none. It is inert until other techniques have finished several lines, then becomes the only thing that will move an expert grid.

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