There is a folk answer to how you make Eternity II easier: hand the solver some
correct pieces. The natural next question is how many, and the community's own
hint-geometry discussion already sharpened it to
where. This study asks something more basic that both questions skip past: on our
own generated boards, do the hints help at all?
For a chronological backtracker, the answer is no. Seed one with the five
official clue pieces at their real board positions, then measure it against the very
same board with no hints, and it does worse on every fill order tested, without
exception. On a compact row-major sweep the five correct pieces cost only ten to
twenty matched edges; on the fill order a newcomer would reach for first, "get to
the clues and connect them up", they cost around three hundred and forty-five,
turning one of the best no-hint orders into the worst. The reason is simple once
seen: to a solver that fills cells in a fixed order, a pinned piece is not free
information but a hard constraint it must satisfy the moment it arrives, and
sometimes it cannot.
Each board below fills along a different order, on a loop. Nothing is being solved;
this is only the order the search would visit cells, made visible. The bright cell is
the one just placed, and the trail behind it is the recent wavefront, so you can
see how much boundary each order keeps open as it runs. That open boundary, the
frontier, is what a backtracker pays for: its branching grows with the frontier, and
a small frontier is also what leaves the search room to route around a hostile pin.
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A row-major sweep keeps one thin frontier and rolls it down the board, so when it
meets a pinned piece it can adjust the single row it is building. The hint-seeking
orders do the opposite. Spiralling in or out drags a whole ring as its frontier, and
tracing the hints scatters its wavefront across the board from the very first move,
committing everywhere before it can know whether the commitments are consistent. The
compact sweep survives the five hints; the hint-seeking orders are undone by them.
Two paired comparisons, run on the same generated boards:
- Do the hints help, and which order survives them? Fix the hints (five, in the
shape of Eternity II's own five clues) and vary only the fill order, measured
against the identical board with no hints. This is the study's spine: it shows the
hints never help, and that how much they hurt is set by the fill order and its
open frontier.
- Count. Does adding more hints help? Some, but the naive way of measuring it
is confounded. A block of clustered hints banks a pile of correct edges for free
just by being pinned; that free "floor" flatters clustered layouts on raw score
while saying nothing about whether the board got easier to finish. The
method page defines the
floor and the floor-immune metrics (solved-rate and earned score) that see past it.
Everything here is built and measured from scratch: our own parametric board
generator, our own family of backtrackers, our own canonical scorer, and a beam
solver as a non-backtracker contrast. No community board, puzzle, or engine is used;
only the shape of the five-clue arrangement is borrowed from the list discussion,
as a geometry to test.
Every board is re-scored by one canonical matched-edge scorer that never counts a
border-facing (grey) seam, the same convention as the
benchmark and the
DFS study. The maximum is 480.
Each of the fifteen seeds is a genuinely distinct generated instance, not merely a
different solver seed, so the spread across seeds is real instance-to-instance
variance and is reported as such. No self-reported score is trusted.
The comparisons are worked through on the
findings page; how
the boards are generated, why the colour recipe stays faithful across sizes, and
what every metric means are on the
method page.