Kojun

The clue you draw before you write a digit

Every other clue this series has measured is a printed number. Kojun has those too — the givens — but its first clue is drawn, not written: the partition itself. A region's outline fixes its digit range (the size), its internal slope (every in-region vertical edge points downhill), and sometimes its whole content: a region shaped like a column of height k reads k, k−1, …, 1 from the top before a single given is printed. Here is what the shapes of the board above say on their own:

Regions that cannot coexist

The digit classes of a finished board — all cells holding v — are global independent sets: equal digits never share an edge, whatever region they belong to. Every region contains a 1, and every region of size ≥ v contains a v, so small regions pack the board with low digits until their classes no longer fit. The all-singletons partition is the theorem-grade wall (every cell must hold 1, and two of them always touch: no board exists at any size ≥ 2). But the collapse starts long before that:

One partition, one fill attempt, nothing redrawn. The tension runs against the setter: the small regions that say the most (56% of cells pinned by shape alone at sizes 2–5) are exactly the ones that stop existing.

The verticality dial — and a second wall

The vertical rule reads only in-region vertical edges, so the same 4–7-cell region budget carries more information laid tall than laid flat. But the far end betrays you: two column-shaped regions side by side are both forced to k…1, and their equal digits touch — self-solving shapes cannot sit next to each other. The dial raises what the shape solves and simultaneously kills the board's existence:

The last column is measured on the 6×6 partitions that still admit a board — at vbias 1 only 7 of 90 tries did. The shapes that pay for themselves are the shapes that barely exist.

One roof over the ladder

once (a census of homes per digit) and order (a geometry of edges) are incomparable rungs. But every deletion the lower rules can justify lives inside a single region or crosses one boundary edge to a pinned cell — exactly what region, full support enumeration of one region at a time, already checks. So the roof subsumes all three rungs, as a theorem; the ablation table confirms it in bits, not vibes:

The cheap rules are accelerators — dropping neigh raises the roof's enumeration bill by a third. On this naked stream even the roof goes quiet (the probe re-derives its deletions), and only givens wake the middle of the ladder: the shipped bank still fills all five grades at every size.

The multinomial census

The independent verification engine deals whole regions one at a time — its search space is literally Π kᵢ! — with raw bookkeeping only. On a single isolated p×q rectangular region it has nothing to obey but the columns, each strictly decreasing downward, so its count must land on the multinomial (pq)! / (p!)^q — choose each column's digits, and their order inside is forced. It reproduced every term we asked for, including 3×3 = 1,680 and 2×5 = 113,400.

Reading a second solution straight off the answer

A single cell can never change value silently — its region would gain a duplicate and lose a digit (the tests prove it exhaustively over every cell and every alternative). The cheapest ambiguity is a swap: two ungiven cells of one region trading digits with every law intact. Scanning a finished grid for such a pair needs no search at all, and it explains a measured share of all non-unique boards — with zero false alarms on unique ones.

Measured: 98.3% of non-unique 4×4 boards and 100.0% at 6×6 and 8×8 are explained by a single swap, with 0 certificates on unique boards. The probe ladder and true uniqueness agreed on all 680 stream boards, in both directions, and the two counting engines agreed on 482/482 solution counts.