The Latin-Square Three

Onlie went first, because she always found the free ground. She ignored the fences and the sums entirely at first and just hunted for a square hemmed in by its own row, column, and box — the pure no-repeats rule, nothing fancy. "This one," she said, tapping a cell where one through eight were all spoken for in its lines. "Only nine is left. Not a guess — the only place it fits." Lantern down. The single certain nine rippled outward: two neighbouring squares now had one fewer candidate each. "I've opened the door," she said. "Someone read the fences."

Cage stepped in, reading the same grid as a set of penned targets. A three-square fence marked *6× overlapped the region Onlie had just touched. "One, two, three multiply to six," he said, "and Onlie's nine next door forbids a repeat down this column, so the two can't sit here* — it'd clash. Which leaves exactly one way to lay one-two-three in this pen." He set them, humming. "The uniqueness Onlie loves and the target on my plaque agreed. More rules, fewer answers." Two more cells locked, and a cross-sum run now had all but one of its cells filled.

Kaku finished the corner, reading the grid as totals. A run clued *17 across two cells stared at her. "Eight and nine — the only pair that reaches seventeen without repeating," she said, "and Cage's placement already put a nine in the crossing column, so this* cell can't be nine. It's eight. Which makes its partner nine, over there where nine is still allowed." She wrote it in her ledger with the small satisfaction of a balanced book. "The sum named its parts. Onlie's uniqueness and Cage's cage told me the order." The corner, hopeless twenty minutes ago, was solid.