# Where you place the hints beats how many > 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. - Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/hint-geometry/ - Updated: 2026-07-10 - Topics: structure, search-space, backtracking - Source: Joe's hint-density study and Peter McGavin's 18-hint solve (groups.io thread 'A method to prune E2 search space by 17-30%+', msg 11725) — https://groups.io/g/eternity2/message/11725 - Source: Peter McGavin: 18 scattered hints solve the 16×16 E2-like puzzle in under 15 minutes (groups.io msg 11746) — https://groups.io/g/eternity2/message/11746 - Source: Peter McGavin: the optimised backtracker traces to Mike's 2007 post (groups.io msg 3098) — https://groups.io/g/eternity2/message/3098 --- Give a solver some correct pieces for free and the puzzle gets easier. The obvious question is how many you need. The better question, it turns out, is *where* they go. On a 16×16 puzzle built with Eternity II's exact colour recipe, eighteen hints placed in the right spots solve it in minutes; the same puzzle wants eighty or more hints, if you pile them into contiguous rows, to be that easy. A fourfold difference in count, decided entirely by geometry. ## Two ways to spend the same eighteen hints > **[Interactive: HintGeometryDiagram]** Rendered on the canonical page (link above); not shown in this markdown export. The left board is the real layout Peter McGavin used: eighteen hints on a regular lattice, every third column on a few odd rows. His plain scan-row backtracker solved Joe's 16×16 E2-like puzzle (five border colours, seventeen interior colours, the same distribution as the official puzzle) in under fifteen minutes on a single core, walking a search tree of 41,160,067,167 placements. Dropping to fifteen hints still worked; the search just grew to several hours. Pile eighteen hints into the first rows instead, the way a top-down scan naturally accumulates them, and they buy almost nothing: the hard part of the board is still completely open. Joe had come at it from the other side, seeding whole contiguous rows from a known solution, and needed far more before the puzzle fell:

scattered hints, solved in minutes

88+

contiguous-row hints for comparable ease

99%

of search time spent below depth 132

70%

of search time spent below depth 150

## Why position wins: the hints have to reach the endgame The two right-hand numbers explain the left-hand ones. Joe instrumented his backtracker over a billion iterations and found the work is not spread across the board at all: 99% of it happens after depth 132 of 256, and 70% after depth 150. Nearly all the pain is in the back half of the fill, and most of it past the three-fifths mark. A block of contiguous hints at the top is spent exactly where the search was never going to struggle. It shortens an easy beginning and leaves the expensive tail untouched. Scattered hints do the opposite: dotted through the board, including down into the region the search reaches last, they pre-empt the choices that would otherwise blow up deep in the tree. This is the same fact the record boards wear on their surface. A near-perfect board packs all its damage into the band of rows the search finished on, because [whichever rows you fill last are where the puzzle makes you pay](/research/why/mismatch-geometry). Hints only help to the extent they reach that band before the search does. It also fits [why the interior gives no forced moves](/research/why/no-forced-moves): with every interior cell still accepting scores of neighbours, a hint's value is not local propagation but global constraint, cutting off whole subtrees the search would otherwise have to walk. A hint far from the hard region cuts off subtrees that were cheap anyway. ## What it does and doesn't say This is a result about a 16×16 puzzle built to Eternity II's colour recipe, not about the official puzzle, whose five fixed clues are a different, much smaller gift in different places. What transfers is the shape of the lesson, and it is the same one the [prune-versus-speed](/research/why/prune-vs-speed) argument makes from the other direction: what matters is changing where the search spends its effort, and the effort lives in the endgame. A handful of hints aimed at that endgame is worth a great many aimed anywhere else. > **Note** > > The counts, the 41-billion-node tree, and the depth statistics are Joe's and Peter McGavin's measurements, reported on the eternity2 groups.io list in January 2026; the scattered layout shown is decoded from Peter's posted board (msg 11746). The optimised backtracker Peter used traces back to Mike's 2007 post (msg 3098). These are community results on a specific E2-like puzzle, recorded here with attribution rather than re-derived. ## Related - [Where the mismatches live](https://eternity2.dev/research/why/mismatch-geometry) — A near-perfect board doesn't scatter its few errors evenly. It packs them into one band of five rows and leaves all the rest flawless. Which band is decided by the direction the search filled the board, and you can see the mirror on the real record boards. - [No forced moves](https://eternity2.dev/research/why/no-forced-moves) — The usual way to crack a logic puzzle is to find a spot where only one piece fits, place it, and repeat. That lever doesn't exist here: every interior piece has between 73 and 137 possible neighbours, and not one is ever pinned to a single option. - [Why a faster computer doesn't help](https://eternity2.dev/research/why/prune-vs-speed) — The single most important idea in hard combinatorial search: shrinking the space you search beats searching it faster, by an exponential margin. Eternity II is engineered so you can barely shrink it at all.