Country Road
The grid is cut into countries. Draw one closed loop through cell centres that visits every country exactly once. A number says how many of that country's cells the loop uses. And two cells that face each other across a border may not both be empty. Drag from cell to cell to lay road; click a cell to mark it empty.
Contract the countries and the puzzle is a Hamiltonian cycle
Shrink each country to a single node and join two nodes when their
countries share a border. "Visits every country exactly once" then
says the loop's country order is a cyclic sequence hitting every node
once — a Hamiltonian cycle on the region adjacency
graph. Every Hamiltonian graph is 2-connected, so a partition
whose region graph has a cut vertex has no solution at
all, for any numbers whatsoever. Slice the board into
horizontal strips and the region graph is a path: there is no puzzle
there, and you can prove it in O(V+E) before printing
anything. The solver ships that test, and the brute-force counter
agrees — 0 solutions, every size.
The same picture names the rungs. Each country is crossed by exactly
two used border edges, because the loop enters once and leaves once —
which as a relaxation is a 2-factor on the region
graph. The gap between a 2-factor and a Hamiltonian cycle is the
subtour that TSP practitioners spend their lives eliminating, and
macro is that elimination: every Hamiltonian cycle of the
region graph is enumerated once, then filtered by the borders decided
so far. Drop it and the joint fixpoint loses 607 decided bits over 40
boards while the search pays 39% more assumptions.
And then the Hamiltonian cycle turns out to be the easy half
That is the part worth measuring rather than assuming. Take the ambiguous boards — the ones with a second solution — and ask what the second solution changes. At 6×6, 92.2% of them walk the very same country order; at 8×8, 85.3%. The macro problem is almost never where the ambiguity lives. It lives inside the countries, in which cells of a country the road actually uses, which is exactly what the numbers are supposed to pin down and mostly do not: with every country numbered, only 18.0% of 6×6 boards are unique.
The correlation runs backwards, too. Boards whose region graph has a unique Hamiltonian cycle — no macro freedom at all — are unique 7.1% of the time; boards with 5 to 16 cycles are unique 30.8% of the time. The confound is country size: small countries make the region graph dense, so lots of macro cycles, and simultaneously leave almost no room to route inside a country. Hold size fixed and the effect is entirely size's: average country 2.5 cells gives 40.0% unique at 6×6, 6.0 cells gives 3.3%.
The generator never guesses at a partition
Partition-first is a trap you can measure: at 6×6 with four countries, 13.3% of random partitions have a cut vertex and only 83.3% admit any loop. So the loop is grown first — by inflation from a 2×2 square, which cannot fail — and the partition is read off it: cut the loop into consecutive arcs, and the arc order is a Hamiltonian cycle on the region graph. Every blob of empty cells goes to one country whole, which is precisely the border rule. Legality is structural; only uniqueness has to be bought.