The apparatus behind the hint study: a parametric board generator faithful to Eternity II's colour recipe at every size, the family of fill-path backtrackers, the one canonical scorer, and the piece of arithmetic that keeps the count axis meaningful, the pinned-seam floor.
This page is the apparatus behind the hint study: how the boards are generated, why they stay faithful to Eternity II across sizes, the family of fill paths, and the piece that matters most for reading the results correctly, the arithmetic of the pinned-seam floor, which is what separates a real effect from a measurement artifact on the count axis.
Every instance is generated from scratch, seeded, and deterministic: the same
(size, colours, seed) produces the same board on any machine. A generated board
is a solved board, piece sits at cell at rotation zero, whose piece
IDs are then relabelled by a seeded permutation, so a hint for cell pos pins the
(relabelled) true piece, and a solver cannot simply walk the identity placement.
The colour recipe mirrors the official puzzle's structure exactly. On an board there are
interior seams (the matched edges; ). These split into a frame band, the seams joining two border pieces along the rim, and the deep interior. The five border colours are confined to the frame band and never appear in the interior; the interior colours appear both in the deep interior and on the inward-facing edge of border pieces. This is the defining structural fact of a real Eternity II board, and the generator reproduces it and is tested against it.
It is tempting, and the earlier write-ups did this, to present Eternity II's even colour census as a specially tuned gift: every colour appears an even number of times, so a perfect solution's matches land with zero slack. The even parity is real, but it is not a tuning achievement. It is forced.
A colour painted on interior seams appears on exactly piece-edges, one on each side of every seam. So for every colour ,
Zero-slack parity is therefore automatic for any board built by colouring seams; it says nothing special about Eternity II. The properties that are genuinely load-bearing, and that the generator must get right, are three: border colours confined to the frame band, the per-colour counts kept balanced (so no colour is rare enough to over-constrain), and every piece distinct up to rotation (so a pinned hint names a unique piece). The study's write-up is precise about this where the earlier framing was not.
The generator and the solver are both size-parametric, the board can be or as readily as , which opens a natural follow-up: does the placement effect strengthen as the board grows? Answering that cleanly needs a colour recipe that does not change the puzzle's difficulty as the size changes. Simply keeping the colour counts fixed while growing would make the puzzle structurally easier at large : with more seams and the same palette, each colour repeats more often, so the average cell accepts more neighbours and the constraint loosens. That would confound size with difficulty.
Instead the recipe holds the per-colour multiplicity roughly constant. Writing for the frame-band seam count and for the interior, the number of border and interior colours is chosen as
targeting the multiplicities Eternity II itself uses at (border , interior ). At this returns exactly the official recipe, five border colours and seventeen interior.
Getting this right on small boards took one fix to the generator. Eternity II's palette is interior-dominant, five border colours to seventeen interior, but the default generator caps the border count at five and takes everything else as interior, which on a small board inverts the ratio: at the recipe wants eight colours, and the cap would split them five border to one interior, a near-uniform interior sea that behaves nothing like E2. The generator now accepts an explicit border-colour count, and the recipe holds the interior at roughly three times the border at every size ( two border, six interior; five, seventeen, unchanged). With that, small boards are faithful and fully solvable, which is what the solve-speed comparison on the findings page relies on. The main path and count results on this page are all at ; the board is used only where a full solve is needed.
Each layout is a pure function of the board size, so the same geometry can be drawn, measured, and scaled consistently. The gallery below renders them all from the one shared board primitive.
Ten geometries, one board. Toggle the seams to see the free-score floor: clustered layouts bank a pile of guaranteed-correct edges just by being pinned next to each other, while a spread layout banks none. That floor is why a raw score comparison flatters clustering — and why the study measures reached depth and solved-rate instead.
Here is the subtlety that reshaped the study. Ask "do more hints help?" and the obvious move is to compare final scores at different hint counts. But a hint does two different things at once, and score conflates them:
The second effect is a pure bookkeeping gift. Define the pinned-seam floor of a layout as the number of interior seams with both endpoints pinned:
Because the pins are true-solution pieces, every such seam is guaranteed correct before the solver runs. A solid clustered block contributes of them; five blocks bank correct seams, a quarter of the whole 480, for free. A spread lattice, whose hints never touch, has a floor of zero.
So a raw-score comparison systematically flatters clustered layouts: they start a hundred-plus points ahead on bookkeeping alone, regardless of whether the board became any easier to finish. This is the same family of error as counting rim seams in a partial board, a floor that inflates the number without reflecting progress. The toggle in the gallery above draws these banked seams so the free score is visible.
The study therefore does not rank layouts by raw score on the count axis. It uses two floor-immune metrics:
Both measure whether the search made progress the pins did not hand it. On the path axis, where every compared layout shares the same hints and hence the same floor, raw score is directly comparable and is used.
The question "what are the hints worth?" only has an answer relative to not having
them. So the path axis is run twice on every board: once with the five clue-shape
hints, once with none (baseline_00), and the reported effect is the paired
difference, the hinted score minus the no-hint score on the same generated
board. Pairing per instance removes the board-to-board difficulty variance, which
on these bimodal boards is large enough to swamp the effect if the two conditions
were compared across different seeds. A negative paired difference means the hints
made that fill order worse than it was with a blank interior, which is what the
findings report. All comparisons use the common set of seeds that ran to completion,
so every path and layout is aggregated over the identical instances.
The engine is the study's sibling DFS backtracker,
run strict (no breaks, no propagation) so that the fill order is the only thing
changing. The orders tested are row-major, its bottom-up mirror, spiral-in,
spiral-out, border-first, Verhaard's comb, a clue-rows-first control, and the
study's own hint-seeking order, connect-hints-first.
Why does the order matter so much? A backtracker's cost is governed by the open frontier: the set of already-filled cells still adjacent to an empty one. When the next cell is placed against a frontier of size , the number of partial boards the search may have to consider grows multiplicatively in , branching is exponential in the frontier, not in the board. A single compact sweep keeps to about one row (); an order that opens blobs around scattered hints runs frontiers at once, and
so fragmenting the fill into disconnected regions multiplies, not adds, the cost.
This is exactly why connect-hints-first, the order that seeks the hints, is the
worst performer: reaching the hints early is worth far less than keeping the frontier
small, and connecting scattered anchors does the opposite of keeping it small.
Every board is re-scored by one canonical matched-edge scorer that never counts a
border-facing (grey) seam. The maximum is ( at ). Each of the
fifteen seeds is a distinct generated instance, so across-seed spread is genuine
instance variance. Throughput, where reported, is search-nodes per second and is
never compared across different path orders, since a node under one order is not the
same unit of work as under another. The whole apparatus, generator, layouts,
per-run results, and the grid script, lives in the study's
backing directory,
and just experiments hint-study reruns it.