Magnets
The board is already tiled into dominoes. Each domino either becomes a magnet — a + pole on one half, a − pole on the other — or stays neutral. Two cells that share an edge may never hold the same pole, and the margins count the + cells (top and left) and the − cells (bottom and right) of every row and column. Click a half to cycle + here → − here → neutral → blank; right-click cycles backwards.
The clue that can't see half the board
A line carries two numbers — its + count and its − count — and the useful way to read them is as a sum and a difference. The sum counts magnetised cells and cannot tell + from −. The difference can: but a domino lying entirely inside the line cancels out of it — a magnet contributes one + and one −, a neutral contributes nothing, and either way the difference never hears about it. What is left is exactly the cells whose domino crosses the line, one half in, one half out:
plusClue − minusClue = Σ over crossing cells of σ,
σ ∈ {+1, −1, 0}
— a signed flux through the line's boundary. The flux
rule reasons on that identity alone, with interval bounds and one
parity fact: when every crossing cell is known magnetised, the sum's
parity is forced. Here it is, live, for the rows of the board above:
Turn the tiling sideways and the same clues say nothing
That identity has a consequence you can measure. Bias the tiling
generator toward horizontal dominoes (hbias) and rows
gradually lose their crossing cells; at hbias = 1 every
domino lies inside its row, every row's difference clue is
identically 0, and the puzzle dies — at either extreme:
| 8×8, hbias | crossing cells per row | rows with difference 0 | pinned by flux | pinned by probe | exactly one answer |
|---|---|---|---|---|---|
| 0.00 (all vertical) | 8.00 | 28.4% | 1.6% | 3.8% | 0.0% |
| 0.25 | 6.16 | 33.0% | 10.8% | 71.7% | 23.8% |
| 0.50 | 4.51 | 39.8% | 20.4% | 87.3% | 50.0% |
| 0.75 | 2.74 | 46.6% | 15.7% | 80.1% | 35.0% |
| 1.00 (all horizontal) | 0.00 | 100.0% | 2.7% | 7.7% | 0.0% |
Same 32 printed numbers in every row of this table. An anisotropic tiling reads them into one direction only, and uniqueness collapses from 50% to 0% at both ends — parallel dominoes stacked on the same two tracks are exactly the configuration that can trade states silently (see the certificate below).
A generator that cannot dead-end
Answers are drawn with no search and no rejection: visit the dominoes in random order and give each one a state that is legal now. Because a neutral domino conflicts with nothing, the legal set is never empty — neutral is a structural escape hatch, and the no-dead-end property is a theorem about the rules rather than a measured retry rate. The price is a dial: how hard the generator tries to magnetise. Both ends of that dial are easy and the middle is where puzzles live:
| 8×8, magnetise | mean magnetisation | count finishes | probe finishes | exactly one answer |
|---|---|---|---|---|
| 0.0 | 0.0% | 100.0% | 100.0% | 100.0% |
| 0.2 | 20.3% | 40.0% | 53.3% | 53.3% |
| 0.4 | 36.6% | 8.3% | 41.7% | 41.7% |
| 0.6 | 52.6% | 0.0% | 45.0% | 45.0% |
| 0.8 | 65.7% | 0.0% | 56.7% | 56.7% |
| 1.0 | 75.7% | 0.0% | 50.0% | 50.0% |
An all-neutral board is trivially unique (every clue is 0), and a densely packed one is rigid again — ambiguity peaks in the middle, at the density where the adjacency rule has started to bite but not yet to dictate.
Two incomparable rungs under one roof
count (a census per sign, over all cells of the line)
and flux (a signed sum, over the crossing cells only)
are incomparable — neither subsumes the other. But every constraint
of this puzzle lives inside some line: horizontal adjacency
in the row, vertical adjacency in the column, and each domino's
coupling in whichever line the domino lies in. So gac,
which enumerates complete assignments of one line at a time, is a
roof over both rungs — and over pole, the base
bookkeeping, too. The ablation table confirms it with bits, not
vibes:
| board | variant | finished | cells pinned | fixpoints that moved | line assignments built | probes |
|---|---|---|---|---|---|---|
| 10×10 | full | 40.0% | 80.5% | — | 3,424 | 215 |
| 10×10 | −pole | 40.0% | 80.5% | 0 of 20 | 6,071 | 215 |
| 10×10 | −count | 40.0% | 80.5% | 0 of 20 | 3,491 | 215 |
| 10×10 | −flux | 40.0% | 80.5% | 0 of 20 | 3,548 | 215 |
| 10×10 | −gac | 25.0% | 58.6% | 8 of 20 | 0 | 476 |
| 10×10 | −probe | 25.0% | 40.9% | 15 of 20 | 327 | 0 |
Removing pole, count or flux
leaves the fixpoint bit-for-bit identical on every board at every
size — but dropping pole nearly doubles the roof's
enumeration work. The cheap rules are accelerators; only the roof
and the probe change what gets solved.
The √2 census
The independent verification engine walks dominoes one at a time with raw bookkeeping only. On a clue-free 1×2k strip its solution count must satisfy a(k) = 2a(k−1) + a(k−2) — the transfer matrix over {neutral, +−, −+} — which is OEIS A001333, the numerators of the continued-fraction convergents of √2. The engine reproduced all nine terms, horizontally and transposed: 3, 7, 17, 41, 99, 239, 577, 1393, 3363.
Reading a second solution straight off the answer
Under full clues a single domino can never change state silently: flipping a magnet moves the counts of the two lines it crosses, and neutralising it moves every line it touches (the tests prove this exhaustively). The cheapest ambiguity is a pair — two parallel dominoes on the same two tracks trading states so that all four line counts balance. Scanning a finished grid for such a pair explained 37–47% of all non-unique boards at every size, with zero false alarms on unique ones. The probe ladder and true uniqueness agreed on all 680 boards, in both directions.