Multi-House Rule of 45
Two rows total 90, three total 135 — bands of houses have whole cages to subtract where a single house has only fragments.
How it works
- 1The rule of 45 scales: two rows hold 1-9 twice over, so they total 90; three total 135.
- 2That reaches deductions one house cannot. A cage straddling two rows is wholly inside the two-row band even though it is inside neither row on its own.
- 3Take rows 4 and 5, where every cage either straddles the two rows or crosses out of the band, and not one sits inside a single row. The cell you are chasing is in that band, and its column and its box have no whole cage inside them either.
- 4That kills every single-house form at once. With nothing whole to subtract, each of those houses has all nine cells left over — not the one the innie rule places, nor the two to four the multi-cell form works on — and the outie form spills far more than a single cell.
- 5The cages sitting entirely within the BAND are another matter: they come to 83 across seventeen cells. The eighteenth belongs to a 3-cell cage totalling 19 whose other two cells drop into row 6 — loose enough to allow it anything from 2 to 9, so its own arithmetic leaves it eight marks, and each of those eight still has other homes in its row, its column and its box. No single of either kind reaches this cell.
- 6The band holds 90, so that cell is 90 − 83 = 7, solved outright. This is the technique that opens boards where single-house innies have dried up.