Put the solvers aside and measure the object itself. The official set is 256
square pieces, four coloured edges each. Had those pieces been drawn casually,
some structural slack would survive somewhere: a repeated piece, a symmetric
piece, a colour in oversupply, an orientation bias that costs nothing, a
clique of pieces that prefer each other. Slack of that kind is what solvers
eat. This page checks each of those doors directly against the piece set
alone, and every one of them is shut, exactly, not approximately.
Five of the six measurements below are committed as one reproducible topic.
Each recomputes from the published instance in under a second, and the binary
asserts its own expected values, so reproducing the numbers is the same act as
checking them.
| Axis | Measurement | The extreme it sits at |
|---|
| Rotation symmetry | 256 of 256 pieces have all four rotations distinct | 1,024 piece-rotations, nothing to quotient |
| Twins | 251 distinct edge-colour multisets; 5 twin pairs and 114 near-twin pairs in 32,640 pairings | no duplicates, coincidence at the 0.4% level |
| Fixed-orientation cap | at most 307 of 480 joins can match if no piece rotates | rotation is structurally necessary |
| Colour budget | all 22 colours have even side counts; pairing capacity sums to 480 | exactly sufficient, zero slack |
| Palette split | 17 interior colours, 5 border-join colours | 17 is the one-expected-solution setting |
A piece that looked the same after a quarter or half turn would collapse
orientations and shrink the decision space, and the standard reduction,
searching one canonical form per orbit, would collect the discount. Measured:
every one of the 256 pieces has a full rotation orbit of size 4, so the puzzle
genuinely has 1,024 distinct piece-rotations and orbit-canonical search gains
nothing at all. The community's census of the design space shows this is a
choice, not an accident: with this palette the space of possible piece designs
includes rotation-repeating forms, and the real set avoids every one of them
(msg 8025). The
design recipe carries the intent side of that
story, including Brendan Owen's measurement that symmetric pieces tile
unevenly enough to hand back exactly the difficulty-ordering signal the
designers were removing.
Ignore edge order and ask which pieces carry the same four-colour budget. The
256 pieces produce 251 distinct multisets: exactly 5 pairs coincide, pieces
(2,3), (5,14), (7,51), (109,110) and (171,181) in the research numbering, and
in each pair the shared colours sit in a different cyclic order, so no piece
repeats, even up to rotation. Loosening the question to near-twins, pairs
sharing 3 of their 4 same-position edges in the stored orientation, adds 114
pairs in 79 groups. Against the 32,640 unordered pairs the set offers, even
that loose form of coincidence occurs at a fraction of a percent. The door
this shuts is the free doubling: a genuine duplicate pair would let every
solution be rewritten by swapping the two pieces, and there is none.
Freeze every piece in its published orientation and ask how many of the 480
joins could possibly match. Colour by colour, horizontal joins can use at most
the smaller of the east-facing and west-facing supplies, and vertical joins
the smaller of the north-facing and south-facing supplies. The sum caps at
307 of 480. That is a counting bound, not a search result: no arrangement
of unrotated pieces, anywhere on the board, can beat it. A pipeline that fixes
orientations early therefore concedes at least 173 joins before search
begins, and 307 sits far below every high board on the
records page. Both numbers here count matched edges
between adjacent pieces with the outer rim excluded, the convention used
across this site.
Count piece sides per colour. Every one of the 22 non-grey colours has an even
total, and the pairing capacities, half the side count per colour, sum to
exactly 480, the geometric number of joins on the board. The supply is exactly
sufficient for a perfect board: no colour runs short, which would certify the
designed solution impossible, and no colour runs over, which would leave slack
for partial boards to spend. Zero slack also means zero leverage. No counting
argument on colour supply alone can prune anything, so the hardness lives
entirely in which pieces carry which colours, not in how much of each colour
exists. The sharpest thing supply reasoning does yield is the
border balance invariant, and that condition
is necessary, never sufficient.
The 22 colours divide into 17 interior colours and 5 that appear only on the
joins between border pieces. The separation itself is automatic: with a solid
grey rim, a border piece's coloured edges only ever meet other border edges or
the interior, so the two pools never mix, as the
rare-colour geography page explains.
What was chosen is the counts. Owen's launch-year derivation recovers 17.14
interior colours from the requirement of about one expected solution
(msg 1947), the scarcest a
solution can be while still existing, and the
hardness peak page measures that this is the
worst possible place for search. The full reconstructed recipe, ingredient by
ingredient, is on the design recipe page.
One more measurement belongs to the thesis but is not yet part of the
committed topic. Build the graph whose nodes are the 256 pieces, weighted by
how many rotation-adjacencies each pair supports, and read its spectrum. The
graph splits cleanly into exactly two blocks, the 60 frame pieces and the 196
interior pieces, and beyond that split it shows no community structure at any
scale: no cliques of mutually compatible pieces, no cheap sub-puzzle to carve
out and solve first. The clustering is single-scale, and the one visible
cluster boundary is the frame-versus-interior line every solver already knows
about. The spectral numbers behind this description are deferred until their
computation is committed alongside the other five, so read this axis as a
description for now, and the five above as exact.
Each axis closes a standard door.
- Symmetry reduction. Nothing to quotient: all 1,024 piece-rotations are
distinct.
- Duplicate tricks. No free doubling of solutions: 5 near-misses in
32,640 pairings, none of them a true duplicate.
- Orientation shortcuts. Fixing rotations early concedes 173 of 480 joins
by a counting argument, before any search runs.
- Supply arguments. Budgets are even and exactly sufficient, so counting
colours prunes nothing.
- Cheap sub-communities. Nothing softer than the frame-versus-interior
split exists to start from.
This page is the measurement companion to two neighbours. The
design recipe reconstructs, from the
launch-year record, why the set was built this way; the
theorem sweep collects the laws proved about
the same object. And the extremes here are the ground floor of the section's
walls: no forced moves is the same flatness
felt cell by cell during a build, and
sigma-cycles is what the absence of small
exchanges looks like between finished high boards. For which method dies on
which wall, the walls-and-methods map is
the index.
The topic directory holds a self-contained Rust binary that loads the bundled
official instance (256 pieces, no hints) and recomputes every number above:
the rotation-orbit census, the twin and near-twin counts, the
fixed-orientation matching cap, and the per-colour parity and pairing
capacity. The run is deterministic, completes in well under a second, prints
one JSON document, and exits non-zero if any expected value fails; the
committed results file is byte-stable across runs. Piece ids in the twin list
follow the research numbering of the set. The two score-like numbers on this
page, 307 and 480, count matched edges between adjacent pieces with the outer
rim excluded, the same convention as the records page;
nothing here scores a candidate board, and nothing on this page is a record
claim.