Rule of Parity
A region's total fixes how many odd digits it holds, mod 2 — which can force a single undecided cell either odd or even.
How it works
- 1Only odd digits change a sum's parity, so the number of odd digits in any set of cells always matches the parity of their total.
- 2Take a two-row band: 90, less 63 for the cages sitting wholly inside it, leaves six cells that must total 27. Neither row on its own has a whole cage inside it, so every single-house form of the rule of 45 has nothing to subtract.
- 3Every cage rung is stuck on those six as well. The band's own innie rule places a lone leftover cell and here there are six; the multi-cell innie rule takes four at most and the split-cage form five, so six is past both; they come from several different cages, so there is no one cage to split; and spanning two rows they are not even obliged to hold distinct digits, which is what the combination table needs.
- 4Count the odds anyway. Two of the six carry only odd marks, three carry only even ones, and one is undecided at {2,7} — so the six hold either two odd digits or three.
- 527 is odd, so the number of odd digits must be odd: three, not two. That decides the swing cell — strike its 2 and it is 7. Parity works on regions far too large to enumerate combinations for, which is exactly where the other cage techniques give up.