Nanro
The grid is cut into regions. Put a number in some cells and leave the rest blank so that every region holds at least one number, a region's number equals how many of its cells are numbered, all the numbers form one connected group, no 2×2 block is entirely numbered, and two equal numbers never touch across a region border. Click a cell to cycle number → blank → open; right-click goes the other way. You never type a digit — the digit is the count, so it writes itself.
The digit is never chosen — it is counted
Rule 2 is the whole personality of this puzzle. A region's number is the size of the set it numbers, so the digits are not free variables at all: an answer is completely described by the set S of numbered cells, and the labels are the function region ↦ |S ∩ region|. That is why this board has no keypad. You mark cells, and every number on screen is a live count of its own region — change one mark and a whole region's digits move together.
The solver keeps the same shape: one two-valued variable per cell, plus one label variable per region that the count pins. Cells are the only free variables, which is what makes the search complete while branching on nothing but "numbered or blank".
A clue is two facts, and one of them is nearly worthless
A printed clue says two separate things: this cell is numbered and this region counts to v. Because the two halves are independent, they can be handed out separately — and then measured separately. Same answers, same clue cells, one half withheld (1,300 boards, 5×5 to 10×10):
| board | full clue | position only | value only |
|---|---|---|---|
| 5×5 | 100.0% | 36.0% | 1.5% |
| 6×6 | 100.0% | 17.5% | 0.5% |
| 8×8 | 100.0% | 8.7% | 0.0% |
| 10×10 | 100.0% | 3.0% | 0.0% |
Share of boards with exactly one answer when every numbered cell of the answer is clued (q = 1) and one half of every clue is withheld.
Printing every region's label and nothing else — the arithmetic with the geometry erased — leaves essentially every board ambiguous: 1.5% unique at 5×5 and 0.0% by 8×8. Printing every position and withholding every value does better but still collapses with size, 36.0% → 3.0%. The reason the position half is not free either is worth saying out loud: "this cell is numbered" never says "and no other cell is". Put the halves back together and the same clue cells pin the answer 100% of the time. The information is not in either half; it is in their conjunction.
The answer decides how many clues the setter may print
A clue can only be printed on a cell that is numbered in the answer — there is nowhere else to put a digit. So the clue budget is not a dial the setter turns; it is a property of the answer. Legal answers number 58–59% of the board (the no-2×2 rule caps that at 75%), and the ladder needs about a third of them (32–34%) of those cells revealed to finish. And the budget has to be spent well: 19 adversarially chosen clues make a 10×10 unique, while 19 random ones did it on 0 of 30 boards.
| board | numbered cells in the answer | count | block | clash | reach | probe |
|---|---|---|---|---|---|---|
| 6×6 | 21 | 17 | 14 | 14 | 9 | 8 |
| 8×8 | 38 | 32 | 25 | 24 | 16 | 13 |
| 10×10 | 59 | 49 | 38 | 37 | 24 | 19 |
Median size of a minimised clue set — reveal until the level finishes, then take back every clue it can do without.
Two readings of the same ladder
Adding rules one at a time, on unfiltered boards with a random fraction q of the answer's numbered cells printed:
| board | q | count | +block | +clash | +reach | +probe | unique |
|---|---|---|---|---|---|---|---|
| 6×6 | 0.8 | 3.6% | 12.0% | 12.8% | 32.4% | 40.8% | 40.8% |
| 6×6 | 0.9 | 24.8% | 38.0% | 41.6% | 58.4% | 65.6% | 65.6% |
| 8×8 | 0.9 | 6.0% | 14.0% | 16.5% | 37.5% | 45.5% | 45.5% |
| 10×10 | 0.9 | 0.7% | 6.0% | 7.3% | 21.3% | 24.7% | 24.7% |
Read that column by column and clash looks like a passenger: it adds 2.5 points at 8×8 and buys back a single clue in the table above. Now take each rule away from the full probe ladder instead (at a slightly denser q per size, so no column is pinned at the ceiling), and the same rule turns out to be carrying nearly a fifth of the board — because the probe does not need a rule to advance, it needs a rule to refute, and clash is one of the cheapest contradictions on the board. Only count reads the clues at all, so removing it takes everything.
| board | full | −count | −block | −clash | −reach |
|---|---|---|---|---|---|
| 6×6 | 64.0% | 0.5% | 49.5% | 52.0% | 44.5% |
| 8×8 | 69.3% | 0.0% | 53.3% | 50.0% | 40.0% |
| 10×10 | 54.0% | 0.0% | 37.0% | 36.0% | 24.0% |