Norinori

Click a cell to cycle blank → cross → shaded. Crosses are your own notes. Shade exactly two cells in every outlined region, and make every shaded cell touch exactly one other shaded cell side by side — so the shading forms dominoes. Dominoes may cross region borders. Everything runs in your browser; nothing is uploaded.

A region does not own its domino

The tempting way to read Norinori is region by region: this region is this shape, so its two shaded cells must go here or here. It does not work, because a region's two shaded cells are often paired with cells belonging to somebody else. Across 400 layouts per configuration, this is how often a region's pair sits entirely inside it:

The share that cross runs from 16.9% to 44.2% — it nearly triples across these twelve configurations, so quoting a single figure would be choosing one. Two things hold everywhere: crossing rises as regions get smaller, since a small region has less room to hold a domino inside itself, and regions of equal size cross more than uneven ones at every matched configuration.

Either way, a region's options are never a function of its shape on its own. They depend on what is next door, which is exactly why the puzzle cannot be taken apart into independent pieces and worked one region at a time.

Shade half the board and there are no puzzles left

Two shaded cells per region means you do not get to choose how much of the board is shaded — it is fixed at twice the number of regions. More regions, more shading. And dominoes must be pairwise apart, so each one sterilises the cells around it, and eventually there is nowhere left to put them.

At 16 regions of 4 cells — half the board shaded — none of the 56 layouts built had a solution. Not rare: zero. The solvable share falls at every step down the table, so this is a ceiling rather than noise. Fewer layouts get built at the dense end because a layout containing a one-cell region is rejected before it is ever tested: such a region cannot hold two shaded cells, so it is not a Norinori at all.

Why the boards above stop at 7 × 7

A puzzle needs exactly one solution, and layouts with exactly one are scarce. Testing 3,000 random layouts per configuration:

Solvable layouts are everywhere; uniquely solvable ones are not, and the gap widens fast. At 6 × 6 roughly one layout in 300 qualifies, at 7 × 7 about one in 700, and at 8 × 8 this search found none at all.

No uniquely-solvable 8x8 layout appeared in 3,000 random tries at three densities. That is a statement about this generator, not about the puzzle: hand-made 8x8 and larger Norinori certainly exist. Three other approaches did worse and are worth recording. Choosing the shading first and drawing regions around it left 300 or more solutions every time. Repairing a layout by moving cells across borders to kill each alternative in turn never converged. And evening out the region sizes — which sounds tidier — produced no uniquely solvable layout at all, because a small region pins its dominoes and a uniform one does not.