# The hint study > Give a backtracker five correct pieces for free, in the puzzle's own clue geometry. It turns out not to help, and depending on the fill order it can hurt badly, because a pinned piece is a hard constraint a fixed fill order must satisfy on arrival. A family of fill paths, run on the same hinted boards, single core, measured against no hints at all. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/lab/experiments/raphael-anjou/hint-study/ - Updated: 2026-07-21 - Topics: backtracking, search-space, structure - Reproduce: `just experiments hint-study` - Source: Runnable generator + committed results + scripts (this study's backing directory) — https://github.com/raphael-anjou/eternity2/tree/main/research/experiments/hint-study --- 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](/research/why/hint-geometry) 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. ## See it: five hints, four orders 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. > **[Interactive: HintPathFill]** Rendered on the canonical page (link above); not shown in this markdown export. 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. ## What the study measures 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](/research/lab/experiments/raphael-anjou/hint-study/method) 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. ## How to read the numbers 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](/research/lab/experiments/single-core-benchmark) and the [DFS study](/research/lab/experiments/raphael-anjou/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](/research/lab/experiments/raphael-anjou/hint-study/findings); how the boards are generated, why the colour recipe stays faithful across sizes, and what every metric means are on the [method page](/research/lab/experiments/raphael-anjou/hint-study/method). ## Pages in this section - [How the study is built](https://eternity2.dev/research/lab/experiments/raphael-anjou/hint-study/method) — 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. - [What the study found](https://eternity2.dev/research/lab/experiments/raphael-anjou/hint-study/findings) — The results, worked through: on these boards the five clue-shaped hints never help a backtracker, they range from a mild cost to a catastrophe, and the fill order decides how much damage they do; the scores are bimodal, not a smooth gradient; and the hint-count question is confounded by a free pinned-seam floor. ## Related - [Raphaël Anjou's experiments](https://eternity2.dev/research/lab/experiments/raphael-anjou) — A notebook of Eternity II search experiments, organised into the shared engines they run on, the combination pipelines that chase the score, three studies that take one search paradigm apart a decision at a time, and exact endgame solves. Each has its idea, its best board, and the questions it left open. The best reaches 463 of 480. - [The DFS study](https://eternity2.dev/research/lab/experiments/raphael-anjou/dfs-study) — One question, asked carefully: among depth-first backtrackers for Eternity II, what does each fill order, each heuristic, and the break mechanism actually buy? A family of from-scratch backtrackers, each one change apart, run on the same ten corner-pinned variants, single core, sixty seconds. - [Where you place the hints beats how many](https://eternity2.dev/research/why/hint-geometry) — On a 16×16 puzzle built like Eternity II, eighteen hints scattered across the board solve it in minutes, while the same puzzle needs eighty or more hints piled into contiguous rows to be as easy. Position, not count, is the lever, and it points straight at the endgame.