Cage Splitting

How it works

1. 1Work out a region's leftover cells the usual way — subtract the cages sitting wholly inside it from 45, 90 or 135.
2. 2Now check whether those leftover cells all belong to ONE cage. If they do, you have just split that cage: its portion inside the region has a known total.
3. 3That matters because cells of a single cage hold distinct digits by the cage rule — even when they span two houses — so the combination reasoning applies to the split part.
4. 4Example: columns 4 and 5, which hold 90 between them. Every cage inside the pair straddles both columns, so neither column on its own has anything whole to subtract — each sees all nine of its cells left over, and says nothing at all.
5. 5The band's whole cages come to 75 across sixteen cells. The two cells left over are the band half of a 4-cell cage totalling 25, the other half sitting in column 6.
6. 6Two leftovers is one too many for the band's own innie rule, which places a lone cell and nothing else, and its outie form wants a single cell lying outside, where this cage leaves two.
7. 7That rule then stops, and for a reason: cells left over from a BAND sit in different houses, so in general nothing forces their digits to differ — and differing digits are exactly what the combination table assumes. These two are one cage's cells, and a cage never repeats a digit, so the table applies to them after all.
8. 8They add to 90 − 75 = 15, so they are 6+9 or 7+8 and nothing else — 1 through 5 go from both. Reading the cage on its own could never have said that: 25 across four cells admits every digit from 1 to 9.