# Linking the clues early does not help, it hurts

> Joining clue pairs with early piece corridors feels like adding constraints. Counting says otherwise: a 1-wide corridor between even the two closest clues admits about two quadrillion distinct fillings, so it excludes almost nothing while spending pieces the endgame will need. In a controlled A/B, every corridor arm lost to its own control at every board size.

- Canonical page (with interactive figures/demos): https://eternity2.dev/research/why/clue-corridors/
- Updated: 2026-07-22
- Topics: structure, search-space
- Reproduce: `cd research/topics/clue-corridors/compute && cargo run --release --bin corridors > ../results/corridor_counts.json`
- Source: Official clue placements (cells 34, 45, 135, 210, 221; instance data at e2.bucas.name) — https://e2.bucas.name/

---
The five clues are the only cells of the official puzzle whose contents you
know for certain. Stare at the board for a minute and an idea offers itself:
connect them. Lay a short chain of pieces from clue to clue, let each chain
become a fixed spine, and hope the spines constrain everything that comes
after. I liked the idea enough to measure it, and the measurement is a clean
negative twice over. An exact count shows that a clue-to-clue corridor admits
about two quadrillion distinct fillings, so committing to one constrains
almost nothing; and in a controlled A/B, every arm that laid corridors early
finished worse than the same solver without them, at every board size tested.

## Five islands, and the bridges between them

The clues sit at cells 34, 45, 135, 210 and 221, that is (x, y) = (2, 2),
(13, 2), (7, 8), (2, 13) and (13, 13): four inset two cells from the corners,
one just left of centre. Their pairwise Manhattan distances run from 10 to
22. No two clues are close to adjacent, so the shortest possible bridge
between any pair is a chain of ten placements.

What a chain placement is worth depends on how many sides it has to match.
With $n = 196$ interior pieces and $C = 22$ declared interior colours, a cell
matching $k$ already-placed neighbours has an expected
$b_k = 4n / C^k$ legal candidates: $b_1 \approx 35.6$, $b_2 \approx 1.62$,
$b_3 \approx 0.074$. A placement only prunes, in expectation, once $b_k$
drops below 1, which requires $k \ge 3$ matched sides. A 1-wide corridor is
built entirely from $k = 1$ placements: each new piece touches only its
predecessor, and inherits about 36 legal candidates at every step.

## What one corridor is worth: two quadrillion fillings

The branching table is a uniform model, so the checker also counts exactly on
the real piece set. If anything the real set is looser: the number of
(piece, rotation) pairs presenting a given interior colour on a given side
averages 46.1 (range 43 to 49), against the model's 35.6. Build the transfer
matrix $T[a][b]$ counting inner pieces that show colour $a$ on one side and
colour $b$ on the opposite side, raise it to the corridor length, and the
matrix powers give exact walk counts. For the length-10 corridor between the
closest clue pair, with both endpoint colours fixed by the clues, that count
is $2.59\times10^{15}$ walks (the mean over colour pairs). Walks may reuse a
piece; applying the mean-field distinctness correction
$\prod_{j=0}^{9}(1 - j/196) = 0.792$ leaves $2.05\times10^{15}$ fillings
using distinct pieces. My first pass at this count, with a slightly tighter
piece model, gave $1.9\times10^{15}$; the committed checker settles on
$2.05\times10^{15}$, and the conclusion does not move between the two.

A commitment that can be satisfied two quadrillion ways is not a constraint
in any useful sense. Laying it excludes a vanishing fraction of the search
space, while it withdraws ten pieces from the pool at $k = 1$, exactly where
the arithmetic says the search gets nothing back. Route two or three clue
pairs and the bill is 10 to 20 pieces spent before a single genuinely
constrained cell has been filled.

## The A/B: every corridor arm lost

Counting says the corridor buys nothing; it takes an experiment to show that
it costs. The study's original run compared three arms that differ in nothing
but the corridor phase: a control (clues pinned, then a max-contact greedy
fill with restarts), a 1-wide corridor arm, and a 2-wide ribbon arm, each
laying its clue-to-clue routes before the identical fill. It ran on scaled
instances with faithful clue geometry, at board sizes N = 8 to 16, with 12
paired seeds per cell and 20 seconds per single-core run.

| Board | Control mean | 1-wide corridor | 2-wide ribbon |
|---:|---:|---:|---:|
| 8×8 | 92.50 | -4.67 | -3.00 |
| 10×10 | 144.17 | -9.58 | -5.50 |
| 12×12 | 209.83 | -13.58 | -6.75 |
| 14×14 | 285.67 | -12.00 | -7.83 |
| 16×16 | 377.17 | -17.67 | -8.08 |

Scores are matched edges, and each delta is an arm against its own control on
paired seeds. Every corridor arm lost to its control at every size, with
paired Wilcoxon $|z| \ge 2.80$. The harm grows as the boards, and with them
the corridors, get longer: a mean loss of 4.7 edges at N = 8 becomes 17.7 at
N = 16, where the two score distributions separate outright (the control's
worst seed scored 373; the corridor arm's best managed 365). And widening
the corridor to a 2-cell ribbon, which lets its second rank land at 2
contacts instead of 1, roughly halves the damage at every size. That is
exactly the width dependence the counting predicts, which is what ties the
mechanism to the measurement.

## What this rules out, and what it leaves open

The negative is precise: laying 1-wide (or 2-wide) clue-to-clue corridors as
an early commitment hurts, and more corridor hurts more. The same arithmetic
that kills them also points at where constraint is real: cells placed with 2
or more contacts, since only $k \ge 3$ prunes outright and $k = 2$ comes
close. Growing compact clue-anchored regions, where most cells arrive with
several contacts, is a different move and is untouched by this experiment.

The result also fits what the hint studies keep finding. On a related
16×16 puzzle, [hint position beats hint count](/research/why/hint-geometry)
because a hint's value lies in reaching the part of the board the search
struggles with; and in [the lab hint study](/research/lab/experiments/raphael-anjou/hint-study),
the five official clues alone never helped a chronological backtracker on the
boards tested. A corridor is the extreme way to misspend that gift: it cashes
the five fixed cells immediately, in the cheapest region of the search, and
pays for it with the piece pool. Like the negatives in
[the theorem sweep](/research/why/theorem-sweep), this one comes priced: for
clue linking to pay, its cells would have to arrive with three contacts, and
a 1-wide path never does.

> **Note**
>
> The counting side reproduces exactly: the checker in the topic directory recomputes the clue geometry, the branching table, the colour supply and the transfer-matrix corridor counts from the official piece set in about a second, and its output is committed as results/corridor_counts.json. The A/B table is the study's original measurement: the packaged rerun of the three arms is specified in the topic's reproduction plan and its tables are not yet committed, so read those deltas as the record of one run, with the fresh-run confirmation still pending.

## Related

- [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 piling hints into contiguous rows needs a hundred just to bring the search down to tens of billions of placements. Position, not count, is the lever, and it points straight at the endgame.
- [The hint study](https://eternity2.dev/research/lab/experiments/raphael-anjou/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.
- [The theorem sweep: thirteen structural laws](https://eternity2.dev/research/why/theorem-sweep) — One research arc, thirteen families of structural theorems: ring purity, the 479 parity floor, the 470 wall as a phase boundary, flux invariants, the entropy area law, and the impossibility results that price every standard shortcut. This page is the map.