Creek

The numbers are not in the cells — they sit on the corners. Each one counts the shaded cells among the (at most) four squares that touch it, and every unshaded cell must join one connected creek. Click a cell to cycle shaded → creek → empty; right-click erases. Show every corner and the answer has a closed form, so the whole puzzle is in the corners the setter rubbed out.

shaded bank creek (unshaded) 2 two of my four cells are shaded impossible corner, or a stranded creek

Size Rules

A puzzle with a closed form

A corner of the paper touches exactly one cell, so it simply names that cell. An edge corner touches two, an interior corner four. Write C(i,j) for the number on corner (i,j); then C is the mixed second difference of the board, and inclusion–exclusion inverts it:

x[r][c] = Σ_{i≤r, j≤c} (−1)^{(r−i)+(c−j)} · C(i,j)

The answer is the chequerboard-signed running total of the clues. A fully clued Creek board is therefore never ambiguous, never needs the connectivity rule and never needs a search — and count, the weakest rung in the ladder, finishes 100% of them, because repeatedly counting a corner with one unknown cell is that scan. Everything difficult about Creek is in the numbers the setter rubbed out.

Rub out the border and the ambiguity is a rank-one pattern

Erase the whole outer ring of corners and only genuine 2×2 windows remain. Two boards that agree on every 2×2 sum differ by a d whose chequerboard twist has vanishing mixed second difference, so d[i][j] = (−1)^i+j(u_i + v_j): the kernel has dimension w+h−1, and restricted to ±1 it is a catalogue of three shapes — a whole row, a whole column, or a row-and-column cross, flipped in alternation. That is a lot of algebraic freedom and almost no combinatorial freedom: nearly every border-erased board is still unique, and when one is not, the second solution can be read straight off the answer with no search at all.

Five rule sets

count reads one corner: a cardinality constraint on four cells, which is already the strongest thing that scope allows. linear row-reduces the entire clue system once, at build time, and bound-propagates on the reduced rows — this is the triangular scan, rediscovered rather than hard-coded. pair takes two corners one step apart, which share a domino, and enumerates their six cells exactly; this is where subtracting two adjacent numbers lives, and unlike the row reduction it is allowed to know a cell is an integer. connect is the only non-linear rule in the ladder. probe assumes a colour and watches the rest refute it. The middle two rungs are incomparable — each deletes candidates the other cannot — and yet on 3,000 random boards neither ever finished one the other could not.