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; piling
them into contiguous rows instead, one measured run needed a hundred (sixty
border plus forty inner) just to bring the search down to tens of billions of
placements. To pin the crossover down ourselves, we ran the same contest on a
board small enough to solve to completion: matching a scattered sixteen-hint
lattice took five contiguous rows, forty hints, about two and a half times
the count, decided by geometry alone.
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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. 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 search came down to a
manageable size: a hundred hints (sixty border plus forty inner rows) still
left a tree of roughly 47 billion placements
(msg 11725).
Neither figure pins the crossover exactly, because a 16×16 board never finishes
in seconds and the raw count is muddied by the free matches a solid block hands
you. So we shrank the puzzle to a size that does finish, an 8×8 built to the
same colour recipe, and measured the real quantity: nodes to a full solve, over
thirty seeded instances per layout. There a scattered lattice of sixteen hints
solves every instance in a few thousand search nodes. Contiguous rows have to
climb to five full rows, forty hints, before they solve every instance in the
same node budget. At the same count, sixteen scattered hints beat sixteen
piled into two rows by more than two orders of magnitude in search nodes, and
the block fails to solve six of the thirty at all. The lattice reaches the
endgame; the block never does until it nearly buries it.
18
scattered hints, solved in minutes
2.5×
more hints, as contiguous rows, to match a scattered lattice (measured to completion)
99%
of search time spent past depth 132
70%
of search time spent past depth 150
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.
Hints only help to the extent they reach that band before the search does.
It also fits why the interior gives 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.
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 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.
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. The five-row crossover
is our own measurement, on our own generated 8×8 boards and backtracker: the
fewest contiguous rows whose solve-rate and median nodes-to-solution both
reach a scattered sixteen-hint lattice, over thirty seeded instances per
layout. Reproduce it with just experiments hint-study-solve.