Eternity II is not a puzzle that happens to be hard. It is the output of a
recipe: a short list of design rules that, followed together, produce the
hardest edge-matching puzzle a given number of pieces can make, while still
guaranteeing that a solution exists. The remarkable part is that the recipe
was not leaked or published by the designers. The community reconstructed it,
ingredient by ingredient, within weeks of launch, mostly in one August 2007
post by Brendan Owen titled, fittingly, "Design the hardest puzzle"
(msg 1947).
This page walks through that recipe: what each rule is, what it costs anyone
trying to solve the puzzle, and where each claim comes from. The measurement
of the sharpest single ingredient (the colour count sitting exactly on the
hardness peak) has its own page; here it
takes its place among the others.
The design problem was stated on the mailing list six years before anyone
had to solve it. In February 2001, with Eternity I barely cold and Eternity II
still a rumour, guenter stertenbrink asked the group how one would design a puzzle
carrying a £5M prize so that it had only about a 1% probability of being
solved within ten years (msg 15).
Replies discussed scaling Eternity-I-style difficulty and even hiding
cryptographic problems in the edges.
That is exactly the tightrope a prize puzzle must walk. Make it too easy and
the prize is lost; that is what happened to Eternity I, which fell in 2000
because it had vastly more solutions than its designer believed. Make it
literally impossible and the contest is fraud. The target is a puzzle that
provably has a solution, positioned so that no realistic amount of computing
finds it within the contest window. Eternity II's designers had watched
Eternity I die, and the recipe below reads like a point-by-point response.
The first rule in Owen's derivation: use a compact board, the 16×16 square,
rather than anything elongated or irregular
(msg 1947). A compact shape
maximises the share of interior joins, where uncertainty is highest, and
leaves no thin arms or corridors that a solver could exhaust cheaply and use
as an anchor. Owen later verified the consequence experimentally: on
comparable 16×16 designs with unbalanced palettes there is always some region
that is cheaper to tile first (for a 2/19 split, starting in the middle is
over a hundred times cheaper than a row scan), but on E2's actual
parameters no such region exists. The design has, in his words, "no weak
areas to start tiling from"
(msg 5263, design comparison
msg 5243).
Every one of the 256 pieces is unique, and none is symmetric under rotation
(msg 1947). In 2010 the
community counted the design space to see how deliberate this is: with 5
frame and 17 interior patterns there are roughly 21,000 possible piece
designs, including forms like aaaa, abab and aabb that repeat under
rotation. The real set pointedly avoids all of them
(msgs 8014–8034).
The consequence is the absence of freebies. A duplicate pair would let any
solution be rewritten by swapping the two pieces, doubling the solution
count for free; a rotationally symmetric piece would collapse orientations
and shrink the decision space. Denying both keeps the expected solution
count pinned where the designers wanted it and hands the solver exactly
zero symmetry to exploit: every placement is a full, independent decision
among 4 orientations of distinct pieces.
There is a second, quieter reason to bar symmetric pieces, and Owen measured
it. A symmetric piece is not just structurally redundant, it is easier to
place, because it fits more contexts. He built a set of 289 pieces (all 17
of the 90-degree-symmetric forms, all 136 of the 180-degree-symmetric ones,
and 136 random asymmetric pieces), tiled a small rectangle every possible way,
and counted how often each piece appeared across all 759 million solutions.
The asymmetric pieces showed up 2.08 times as often as the
180-degree-symmetric ones and 4.14 times as often as the
90-degree-symmetric ones
(msg 2076). So symmetric pieces
are the hard-to-tile ones, and a puzzle that included them would hand a
solver exactly the uneven-tileability handhold that
flat colour frequencies are meant to remove.
Banning them keeps every piece about equally hard to place, with no easy ones
to save for last.
The 22 colours split into 17 interior colours and 5 that appear only on the
joins between border pieces, never inside
(msg 1947). This turns the
frame into its own sub-puzzle whose difficulty can be tuned independently of
the interior, so that neither part is a soft entry point: the same
balancing act as the compact shape, applied to the palette. The five
frame-only colours are also the rare ones, quarantined at the rim where
their scarcity cannot create over-constrained interior cells that
propagation would feast on. That visible signature has
its own page.
When Owen digitised his set on launch day he found the edge-colour
distribution "as flat as could be": 24 edges of each of the 5 border-join
colours, 48–50 of each of the 17 interior colours
(msg 1054).
This single ingredient is the one that killed the Eternity I playbook.
Eternity I was cracked largely through piece-difficulty ordering: its piece
tileabilities varied enormously, so solvers could save the easiest pieces
for last and let the statistics carry them home. Two weeks after launch Owen
showed E2's flat distribution makes that approach worthless: when every
colour is equally common, every piece is about equally tileable, and no
ordering heuristic gets traction
(msg 1667). As doc_s_smith,
one of the people who actually solved Eternity I, put it in reply,
most-constrained-position selection became "our only other hope"
(msg 1722).
The last ingredient sets the colour counts themselves. Owen worked backwards
from the requirement "about one expected solution": setting the expected
number of interior tilings to 1 and solving for the interior colour count
gives
I=(196!⋅4196)1/392≈17.14
Round to 17, add the separately-tuned 5 border colours, and you have
Eternity II's exact palette
(msg 1947). Owen backed the
derivation with simulations, defended 17+5 against the neighbouring 16+8
design from the same one-expected-solution family, since it better balances
the tileability of border and interior pieces and leaves no cheap way in
through the frame (msg 2426),
and followed up with a table of hardest parameters for every board size, a
general recipe of which E2 is the 16×16 row
(msg 2164).
One expected solution is not an arbitrary aesthetic. It is the setting where
solutions are as scarce as they can be while still existing: the peak of
the phase transition, the point where search is provably at its worst. That
measurement, and the published analyses that later confirmed Owen's number,
live on the hardness-peak page. The flip
side proves the knob is real: a deliberately loosened 16×16 design
discussed on the list has around 10^42 expected solutions, though Owen
warned that even that one is no pushover
(msg 4968).
Christopher Monckton invented the Eternity franchise and put up the prize,
but his original idea for the sequel was a 1001-piece three-dimensional
puzzle. Owen relayed this while noting that "the actual design is from Alex
and Oliver" (msg 2697). Alex
Selby and Oliver Riordan are the two mathematicians who won Eternity I by
discovering it had far more solutions than intended; Monckton hired the
people who beat him. The group suspected this weeks before launch
(msg 716), saw it confirmed in
an official Tomy leaflet (the inventor met the E1 winners on breakfast
television and asked them to work on E2's development;
msg 901), and finally sourced
it to a Times article: Selby and Riordan designed the puzzle-generating
program (msg 3373).
That provenance explains the recipe's precision. The one team on earth with
first-hand experience of how a prize puzzle fails statistically was paid to
make sure the failure mode was gone. Every ingredient above (flatness, no
duplicates, balanced palettes, one expected solution) closes a door that
Selby and Riordan themselves had walked through in 2000.
The only detailed account of the generation process in the archive is
second-hand and should be read as such. Dave Clark, founder of the
eternity2.net distributed project, recounted a long phone conversation he
had with Monckton on 26 July 2007: the puzzle was generated from entropy
typed at a keyboard by the contest judges (some 200 inputs), regenerated
until the judges were satisfied, then printed exactly once and vaulted.
Monckton described the random-number generator to him as using "gaussian
residues of powers of appropriately chosen primes", which Clark took to be
Selby and Riordan's own implementation
(msg 4177). The thread briefly
entertained the idea of attacking a cryptographically weak generator; a
reply pointed out that if the construction was Blum–Blum–Shub-like it would
be provably hard (msg 4180).
Nothing came of the angle, but the account stands as the archive's best
primary-adjacent source on where the 256 pieces actually came from.
A puzzle designed never to be solved still has to prove it can be. The
Tomy launch material stated that nobody, not the inventor and not the
designers, knows the solution: the generator printed it "between pages of
random text whilst all parties were out of the room", and the output was
sealed in front of witnesses
(msg 901). After the contest
ended unclaimed, Owen gave the statement still quoted today: "I am sure Alex
and Oliver created a solution when Chris paid them to generate a practically
impossible puzzle", and that it lies hidden amongst reams of printed text
locked in a safe, as insurance against any legal challenge to the contest's
good faith (msg 8823).
That safe is the recipe's final ingredient. The designed solution is what
lets the puzzle sit at one-expected-solution instead of zero: existence
guaranteed by construction, discovery priced beyond any contestant's reach.
Every generic shortcut you might reach for was anticipated and
priced in nearly two decades ago. Piece-frequency ordering died with the
flat distribution. Symmetry and duplicate tricks have nothing to grab.
There is no soft region to tile first, no palette imbalance to lever, and
the colour count sits at the exact setting where search is worst. The
community's twenty years of records (467 in 2008, 470 in 2021, none since)
are the empirical readout of a design that worked precisely as intended.
That does not make the puzzle impossible: one solution certifiably exists,
printed and locked away. It means the remaining gap is not a tuning problem.
Anything that closes it will have to be an idea the designers could not
anticipate. That is, in the end, what this wiki is for.