Shakashaka

Click a white square to drop a triangle in; click again to spin it round the four corners and then back to empty. A number on a black square counts the triangles in the four squares touching it edge-on. When you are done, every connected piece of leftover white has to be a rectangle — square-on or tilted 45°. Check marks the exact corners where that has gone wrong. Everything runs in your browser; nothing is uploaded.

A rule about whole regions that you can check at one point

"Every connected piece of white is a rectangle" sounds like something you can only judge once you can see a whole piece. You can't see one from a single square, so it looks global. It isn't. It is exactly the same as a test you can run at one corner of the grid.

Four squares meet at a corner. Each contributes a 90° wedge, and each wedge is cut in two by the square's own diagonal — eight 45° sectors in a ring. Walk the ring and measure each run of white. A run of two is a right-angled corner; a run of four is a straight edge; all eight means you are deep inside white. Any other run is fatal: an odd one is a 45° or 135° spike, and a run of six is a 270° inside corner, which no rectangle has.

Checked in both directions across every board up to 7 × 7 — 21,508 of them — and on scattered black-cell layouts besides. Not one disagreement either way. There is a reason rather than a coincidence: outlawing the 270° run outlaws inside corners, a closed boundary with no inside corners has to turn exactly four times, and four right-angled turns is a rectangle. Two rectangles touching at a single point stay two separate regions, because regions here join edge to edge, so they never pinch into one bad shape.

Verified exhaustively to 7 × 7, which is where enumeration stops being cheap — not proved for every size.

"Four corners" is not the test

The obvious way to check a region is a rectangle is to trace its outline and count the turns. Four turns, four corners, rectangle. That is wrong, and it is worth showing how wrong, because it was a real bug in this page's own checker — staircases of half-squares were sailing through as rectangles. What caught it was the corner rule above disagreeing.

A right trapezoid has four corners too, and a staircase of half-squares traces one exactly. On a 3 × 3 the shortcut waves through 3,013 boards when 3 are legal — 99.9% of its yeses are wrong, and it gets worse as the grid grows, not better. The fix is to insist the corners be right angles, which is a dot product of zero, and which also stops a square-on edge from meeting a 45° one.

Which is what makes the puzzle solvable at all

Being local is not a curiosity; it is the reason the solver on this page returns instantly. Each free square has five states, so a search that checks nothing until the end walks 5 + 5² + … + 5ᵏ nodes. Checking corners as it goes cuts that by a factor of at least 23,251:

The right-hand column is measured. The left-hand one is the closed form for the untouched tree, checked against the real unpruned search from zero free squares up to six — past that, running it honestly would mean a billion nodes, which is the point.

The numbers are a garnish

Every puzzle here is built to have exactly one solution, and then every clue it turns out not to need is taken back off. What is left is striking: most black squares end up carrying no number at all. The arrangement of the holes is the puzzle; the digits only tidy up.