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Technique 1 of 8

The Overlap Method (Simple Boxes)

The most productive opening move on any grid. When a run is long relative to its line, its leftmost and rightmost placements overlap, and every cell in the overlap is certainly filled.

The overlap method — often called simple boxes — is the first thing to try on every fresh line, and on large grids it does more work than any other single technique. The idea: a run has to sit somewhere in its line. Slide it as far left as it can go, then as far right, and compare. Any cell that is covered in both extremes is covered in every placement in between, so it must be filled in the solution.

For a single clue there is an exact formula. A run of length k in a line of length n leaves n − k cells of slack, and the overlap contains exactly 2k − n cells, centered in the line. Whenever that number is positive you can fill immediately — no other information needed. A clue of 8 in a line of 10 hands you six cells before you have looked at a single crossing line:

8leftmost placement
8rightmost placement
8overlap: fill these
Clue 8 in a line of 10. The run covers cells 3-8 no matter where it sits, so 2x8 - 10 = 6 cells are certain.

The method extends to lines with several clues. Pack all the runs to the left (one empty cell between neighbours), pack them all to the right, and overlap each run against its own two copies — first run against first run, second against second. With 4 4 in a width of 10 the packing leaves one cell of slack, so three cells of each run are forced:

4 4packed left
4 4packed right
4 4overlap per run
Two runs of 4 in a width of 10. Each run overlaps itself by 3 cells. Note the middle cell stays unknown — it belongs to a different run in each extreme, so it is never guaranteed.

One caution, visible in the example above: overlap is computed per run. A cell covered by the first run in one extreme and the second run in the other proves nothing. When you use the technique as a sweep — every row, then every column — you will usually seed the grid with enough certain cells for glue and edge logic to take over.

Try it on a real board.