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Eternity puzzle

The Eternity puzzle is a complex tiling puzzle invented by British author and puzzle designer Christopher Monckton, consisting of 209 unique irregular pieces, each formed by combining twelve 30-60-90 degree right triangles into a single polyform known as a 12-polydrafter, designed to interlock edge-to-edge to form a large dodecagon without gaps or overlaps.^[1]^[2] Launched in June 1999 by the Ertl Company, the puzzle eschewed traditional imagery in favor of uniformly green pieces on both sides, with no tabs or blanks, relying instead on precise geometric matching of edges that could be rotated in 12 possible orientations per piece, resulting in an estimated 10^500 possible arrangements, most of which are invalid.^[3]^[4] Marketed as one of the world's most challenging jigsaws, it quickly captured 20% of the British board game market upon release and included a £1 million prize for the first verified solution submitted within four years, along with smaller cash incentives for partial solves using provided starter pieces and grids.^[3]^[5] The puzzle was solved in May 2000—far sooner than anticipated—by Cambridge mathematicians Alex Selby and Oliver Riordan using computational methods that exploited parity constraints in the pieces' edge orientations, earning them the full prize, which was paid out in October 2000; an independent manual solution was also achieved shortly afterward by German puzzler Guenter Stertenbrink, though it did not qualify for the award.^[2]^[5] The rapid cracking of Eternity not only led to financial repercussions for Monckton, including the loss of his family home to cover the prize payout, but also inspired a sequel, Eternity II, with an even larger $2 million prize that remains unsolved to date.^[3]

History

Invention and Development

The Eternity puzzle was conceived by Christopher Monckton, a British viscount, journalist, politician, and puzzle enthusiast, who spent approximately 14 years developing the concept into a fully realized product.^[3] The idea originated in the mid-1980s during Monckton's research on hydro-dynamics at the British Library, where he became fascinated by the potential for creating a highly challenging edge-matching tiling puzzle.^[3] Drawing inspiration from traditional edge-matching puzzles and mathematical tiling problems, Monckton aimed to design a highly challenging brainteaser.^[1] Development began in earnest in the 1990s, with Monckton focusing on irregular polyomino-like pieces composed of 30-60-90 degree triangles, known as polydrafters, to ensure both aesthetic appeal and computational complexity.^[6] The process involved iterative prototyping to craft 209 unique pieces that could tile a dodecagonal frame without relying on colors or images for guidance—all pieces feature solid colors with edge patterns for matching.^[3] By early 1999, the design was finalized, emphasizing a balance between human solvability in theory and extreme difficulty in practice, with an estimated 10^500 possible arrangements.^[6] The puzzle was launched in June 1999 by the Ertl Company, marking the culmination of Monckton's vision for an intellectually enduring challenge.^[7]

Release and Initial Reception

The Eternity puzzle was released in June 1999 by the Ertl Company at a price of £29.99, achieving initial sales of over 500,000 units worldwide.^[8]^[9] It was marketed aggressively as the "world's hardest puzzle," with a £1 million prize incentive announced at launch to stimulate sales and capture public imagination, drawing comparisons to high-stakes games like the National Lottery.^[10]^[11] The launch garnered positive media coverage in UK outlets including The Guardian and BBC News, which highlighted the puzzle's innovative edge-matching mechanics involving 209 unique pieces and its potential as an unsolvable challenge for enthusiasts.^[10]^[12] Early solver attempts by puzzle fans generated buzz but produced no quick successes, heightening anticipation and contributing to strong initial demand.^[13] Primarily launched in the UK, the puzzle rapidly expanded to international markets, bolstered by its proprietary design featuring uniquely shaped pieces that prevented straightforward replication.^[8]^[14]

Design and Components

Piece Characteristics

The Eternity puzzle consists of 209 unique pieces, comprising 12 border pieces with straight outer edges that form the perimeter of the dodecagonal target shape and 197 internal pieces featuring irregular shapes for interlocking via edge matching. The pieces are numbered from 1 to 209.^[1]^[4] Each piece is a unique irregular polygon with up to 12 sides, constructed as a 12-polydrafter—a compound of twelve 30-60-90 right triangles—ensuring they interlock without overlaps or gaps when correctly assembled.^[1]^[2] The pieces are made of thick plastic, double-sided with identical plain green shading and no pictorial image to guide placement.^[15]^[4] The edge-matching system relies on 19 distinct colors applied as segments along the edges of each piece; adjacent pieces must align these colored segments exactly to fit. Each piece can be rotated in 12 possible orientations corresponding to 30-degree increments, though border pieces have orientation limits due to their straight outer edges.^[4]^[16] No two pieces are identical in shape or edge configuration, with internal pieces based on a triangular composition to promote precise tiling.^[1]^[2]

Overall Structure and Objective

The Eternity puzzle challenges solvers to assemble 209 unique polygonal pieces into a large slightly irregular dodecagon, an equiangular 12-sided polygon serving as the target shape. Each piece is composed of 12 unit 30-60-90 triangles, making the total area of the assembled puzzle equivalent to 2508 such unit triangles. The completed dodecagon measures approximately 18 inches (46 cm) in diameter, providing a compact yet intricate tiling layout.^[1]^[4] The primary objective is to arrange all 209 pieces to cover the dodecagon precisely, without any gaps or overlaps, while ensuring that the colors or patterns on every adjacent internal edge match perfectly. This edge-matching requirement uses 19 distinct colors applied to the edges of the pieces. Border pieces, numbering 12 in total, must be positioned along the outer perimeter to define the dodecagon's boundary, with their external edges left unmatched to form the smooth polygonal outline.^[1]^[4] Assembly adheres to specific constraints: the pieces interlock conceptually through precise edge alignment on an underlying triangular grid, rather than physical tabs, and each can be rotated in 12 discrete orientations corresponding to 30-degree increments, though border pieces face additional orientation limits due to their perimeter role. The pieces are double-sided and identical on both faces, rendering flipping unnecessary and effectively prohibited as it yields no change. Despite its finite scope, the puzzle's design evokes an "eternal" sense of endless complexity through the combinatorial explosion of possible arrangements.^[4]^[2] A verified complete solution results in a seamless dodecagon covering, free of protrusions or voids, where all internal edge colors align flawlessly across the entire structure, confirming the tiling's integrity.^[16]

Solving Process

Rules and Assembly Challenges

The official rules for assembling the Eternity puzzle required using all 209 unique pieces exactly once to form a complete dodecagon without gaps or overlaps, utilizing the provided frame and a specified starter position for one piece on the included grid.^[17] Participants were prohibited from cutting, tracing, or employing any external aids beyond the physical pieces themselves, with submissions necessitating photographic proof of the assembly along with a traced outline on the solution grid for piece-by-piece verification by judges.^[17] Entries had to be mailed via registered post to a designated address in London by the deadline of noon GMT on September 30, 2003, and were held unopened until annual review periods, with the judges' decision deemed final.^[17] No purchase was necessary to participate, and details could be requested from the manufacturer, Ertl Europe Ltd.^[17] Manual solving presented immense challenges due to the puzzle's high combinatorial complexity, estimated at approximately $10^{500} possible arrangements, rendering exhaustive trial-and-error approaches utterly impractical for human solvers.^[4] The irregular, asymmetric shapes of the pieces—composed of 30-60-90 degree triangles—further compounded difficulties, as initial placements often appeared viable but led to frequent dead ends from geometric mismatches on adjoining edges that had to align precisely with neighbors.^[1] Parity constraints, such as the orientation and directional alignment of pieces (e.g., up/down or left/right), added another layer of intricacy, often grouping incompatible pieces early in the process and stalling progress.^[2] Human attempts typically resulted in prolonged frustration, with many enthusiasts reporting being unable to complete the puzzle after months of effort and often getting stuck around 150 pieces, at which point remaining pieces failed to fit despite extensive reconfiguration.^[7] Border placement proved particularly problematic, as committing to the outer frame early frequently locked in internal configurations that proved incompatible later, exacerbating the inefficiency of manual trial-and-error.^[18] Common pitfalls included over-relying on shape compatibility while neglecting precise edge matching, leading to invalid partial assemblies that required complete disassembly, and mishandling piece orientations by attempting alignments outside the allowed 12 rotational positions dictated by the triangular grid, which wasted significant time without advancing the solution.^[4]

Methods and Discovery of Solution

The Eternity puzzle, released in June 1999, was solved on May 15, 2000, approximately 11 months later, by Alex Selby, a software engineer, and Oliver Riordan, a mathematician at the University of Cambridge.^[19]^[5] Their breakthrough came amid a £1 million prize competition that incentivized computational efforts to crack what was marketed as a near-impossible tiling challenge. Selby and Riordan employed a computer-assisted approach using two personal computers, developing custom programs that prioritized the most difficult pieces first to establish a "favorable state" in a reduced region of the puzzle.^[14] This two-phase algorithm began with preprocessing to position awkward border and internal pieces, minimizing mismatches early, followed by an exhaustive brute-force search on the simpler remaining tiles, which proved feasible on desktop hardware after weeks of runtime.^[14]^[5] The method incorporated combinatorial insights and probability theory to guide the search, avoiding the infeasible full enumeration of configurations estimated at up to 10^{95} possibilities.^[5]^[14] The pair collaborated as former colleagues, exchanging ideas and code refinements, with Selby handling much of the programming while Riordan contributed mathematical analysis.^[21] Their solution was independently verified through multiple checks, including by the puzzle's creator and official adjudicators, confirming its validity. A second, independent manual solution was achieved by German puzzler Guenter Stertenbrink and submitted on July 1, 2000, though it did not qualify for the prize as it followed the first review period.^[19]^[12]^[1] This underscored the puzzle's design, which resisted human intuition alone but yielded to targeted algorithmic and persistent manual efforts.

Prize Competition

Prize Details and Terms

The Eternity puzzle, released in June 1999, offered a £1 million prize—equivalent to approximately $1.6 million USD at the time—to the first individual or group to fully assemble its 209 pieces into the specified dodecagonal frame without gaps or overlaps.^[7] This substantial incentive was announced by the puzzle's creator, Christopher Monckton, as a key marketing element to underscore the challenge's difficulty, with the funds primarily derived from puzzle sales projected to exceed 500,000 units.^[22] However, under the terms of a 50/50 indemnity insurance policy with Lloyd's of London, Monckton personally guaranteed half the prize amount, anticipating the puzzle would remain unsolved long enough for sales to cover costs.^[13]^[22] The competition rules stipulated a submission deadline of 30 September 2003, approximately four years after the launch, with annual verification batches closing on 30 September each year, during which claimants needed to provide verifiable proof of the complete solution, such as photographs or physical assembly verification.^[13]^[22]^[17] Eligibility was open to anyone worldwide, including children as young as eight, with no limits on team size or restrictions beyond purchasing the puzzle, though submissions from individuals were given priority in case of simultaneous claims.^[11] Intellectual property rights to the solution were to remain with the successful claimants, while Monckton retained limited reproduction rights for promotional or commercial purposes related to the puzzle. The puzzle remained unsolved for over a year after release, aligning with Monckton's expectation of a multi-year challenge.^[13] To encourage engagement, additional incentives included three smaller promotional puzzles—Meteor, Delta Wing, and Heart—each offering clues to substitute up to two Eternity pieces upon completion, though these did not carry monetary rewards. These terms emphasized manual solving over computational methods, as Monckton designed the puzzle to resist easy algorithmic resolution.^[11]

Verification and Payout

Following the discovery of the solution, Alex Selby and Oliver Riordan submitted it on May 15, 2000, providing both digital files and physical assemblies to the puzzle manufacturer and their solicitor to ensure secure handling under the competition rules.^[19] The rules stipulated that solutions could not be examined until the end of September, after which an independent verification process commenced. The verification was conducted by appointed independent scrutineers, who performed multiple checks, including manual reassembly of the pieces to confirm the tiling matched the puzzle's objective without gaps or overlaps. This audit, which also cross-verified the digital submission against physical components, was completed efficiently within a few weeks, culminating in official confirmation of the solution's validity in late October 2000.^[19] An independent solution was also submitted by Günter Stertenbrink on July 1, 2000, and similarly verified, resulting in two verified solutions at that time.^[19] The full £1 million prize was disbursed in October 2000, divided equally between Selby and Riordan at £500,000 each.^[12] The non-disclosure period on solution details, enforced until payout to maintain the puzzle's commercial appeal, was upheld through prior agreements signed by claimants. Post-verification, puzzle creator Christopher Monckton personally endorsed the solution, noting its correctness after review.^[12] No legal disputes emerged regarding the verification or payout, and the public announcement prompted a brief resurgence in sales as renewed interest drew buyers seeking to attempt the now-solved challenge.

Mathematical Analysis

Tiling Properties

The Eternity puzzle constitutes an edge-to-edge tiling of an irregular dodecagon using 209 distinct irregular pieces, each formed by combining 12 right-angled 30-60-90 triangles, analogous to polyiamonds in a triangular grid with between 7 and 11 sides per piece.^[1]^[2]^[23] This geometric foundation ensures total area conservation, as the combined area of all pieces precisely equals the dodecagon's area of 2,508 elemental triangles, guaranteeing a perfect fit without gaps or overlaps upon correct assembly.^[1]^[2]^[14] A key constraint arises from parity matching on the pieces' edges, where edges are classified by orientation parities (such as up/down and left/right directions relative to the triangular grid), distributed such that the total imbalances across all pieces are balanced overall, for example, 116 pieces balanced in up/down parity and 93 with specific excesses. This balanced distribution imposes a combinatorial restriction, requiring that every internal edge pairing matches both shape and parity, thereby limiting invalid configurations and promoting unique connectivity.^[2] The puzzle was engineered to possess exactly one valid solution, a property confirmed post-design through exhaustive computational enumeration of feasible arrangements, though subsequent independent discoveries revealed at least two solutions. Border pieces, comprising a subset with predefined perimeter-compatible edges, anchor the outer boundary of the dodecagon, substantially contracting the effective search space by constraining rotational and positional freedoms early in the assembly process.^[1] Fundamentally, the Eternity puzzle extends principles from polyomino tiling theory, wherein the irregular, asymmetric shapes of the pieces elevate combinatorial entropy by permitting a vast array of potential placements—estimated at up to 10^{95} without constraints—while the rigorous edge and parity matching rules counteract this complexity to enforce solvability within a narrow set of valid tilings.^[24]

Computational Approaches

The core computational approach to solving the Eternity puzzle involved a backtracking algorithm enhanced with sophisticated pruning techniques, developed by Alex Selby and Oliver Riordan.^[25] Their method utilized a two-phase strategy: an initial phase to establish a "favorable state" by placing the most constrained "awkward" pieces along the border, followed by an exhaustive search for the remaining interior tiles.^[14] Starting from the irregular dodecagonal border and expanding inward via a beam search (a breadth-first variant of backtracking), the algorithm prioritized positions with the fewest viable piece options to detect dead ends early.^[25] Parity adjacency considerations played a crucial role in pruning, representing edge orientation compatibilities to preemptively exclude invalid placements and reduce the search space by scoring partial tilings based on estimated tilability probabilities, maintaining a beam width of 10,000 to 1,000,000 promising states.^[25] This setup required substantial resources for the era, implemented in custom C++ code running on two Pentium III processors.^[25] The algorithm evaluated a large number of possible configurations, but aggressive pruning limited the effective search to a computationally feasible scale, yielding the solution after several weeks of continuous operation, with the discovery on May 15, 2000.^[25]^[1] Post-solution analyses have classified the Eternity puzzle as NP-hard, akin to exact cover problems due to its edge-matching constraints requiring precise combinatorial satisfaction across all tiles.^[24] With advancements in parallel computing, the puzzle could now be solved in days or less on modern hardware clusters, leveraging multi-core processing to distribute the beam search across nodes.^[16] However, manual human computation remains infeasible given the enormous branching factor—estimated at over 10^95 potential tilings beyond 70-80 pieces—highlighting the puzzle's reliance on automated depth-first exploration.^[25] The Eternity puzzle's structure also relates to subsequent optimizations in constraint satisfaction algorithms, such as Donald Knuth's dancing links technique for efficiently solving exact cover variants with edge-matching elements like parity adjacencies.^[26]

Legacy and Influence

Sequels and Variants

Eternity II, developed by Christopher Monckton in collaboration with mathematicians Oliver Riordan and Alex Selby, was released on July 28, 2007, by TOMY UK Ltd as a direct sequel to the original puzzle. Unlike its predecessor, which featured 209 irregularly shaped pieces forming a dodecagon, Eternity II consists of 256 square pieces that tile a 16×16 grid, with each piece divided diagonally into four colored quadrants and allowing 90-degree rotations but no reflections. The design incorporates 22 distinct colors—five for the frame and 17 for the interior—along with matching symbols on edges, significantly increasing the complexity and search space compared to the original's geometric edge matching and irregular geometry.^[27]^[28]^[2] The puzzle offered a $2 million prize for the first complete solution, with submissions required by December 31, 2010; no solution was found, and the prize expired unsolved, though a $10,000 award was given to Swedish solver Louis Verhaard for a partial configuration achieving 467 matching edges out of 480, covering all but 12 pieces (over 95% of the board). This heightened difficulty stems from the square format, rotational variability, and more intricate edge constraints, making brute-force computation infeasible without advanced algorithms. The official solution remains unpublished by the creators.^[27]^[29] Beyond Eternity II, no official major sequels have been released, but the puzzle inspired smaller physical variants for educational use, such as a 72-piece edition that simplifies assembly while retaining core edge-matching mechanics. Digital adaptations and fan recreations include open-source software tools like the Eternity II Editor for grid creation and analysis, as well as solver programs implemented in languages like Julia and Golang for research and experimentation. These tools have facilitated studies on NP-complete tiling problems, generating Eternity II-style puzzles with adjustable parameters.^[30]^[31]^[32] Eternity II continues to be commercially available through online retailers like Amazon and eBay, without the associated prize, sustaining interest among puzzle enthusiasts and academics despite the expired competition. Third-party efforts have produced algorithmic variants and partial tilings, but no large-scale commercial follow-ups have emerged post-2010.^[30]

Academic and Cultural Impact

The Eternity puzzle has significantly influenced academic research in combinatorial optimization and tiling problems, serving as a benchmark for studying complex constraint satisfaction challenges. Since its solution in 2000, researchers have developed various algorithmic approaches to analyze its structure, including integer linear programming formulations that model the 209-piece edge-matching task as an exact cover problem.^[33] A linear algebraic method, treating the puzzle as a matrix equation over finite fields, has been proposed to enumerate solutions efficiently, highlighting connections to polyomino tiling and graph theory.^[14] These studies underscore the puzzle's NP-complete nature, akin to general edge-matching problems, and have informed broader investigations into the computational hardness of commercial puzzles. In educational contexts, the puzzle has been integrated into combinatorics and discrete mathematics curricula to illustrate real-world applications of search algorithms and probability. For instance, it features in university lectures and online resources as a case study for brute-force enumeration and heuristic optimization, with mathematicians like Oliver Riordan using it to demonstrate how probabilistic modeling can crack seemingly intractable problems.^[15] The puzzle's irregular dodecagonal board and rotational symmetries provide practical examples for teaching concepts in constraint programming and backtracking, fostering student engagement with NP-hard problems.^[4] Culturally, the Eternity puzzle captured public imagination in the late 1990s, rapidly achieving commercial success by securing 20% of the British board game market within its first month of release, surpassing established titles like Trivial Pursuit.^[23] Its £1 million prize drew widespread media coverage, positioning it as a symbol of intellectual challenge and inspiring discussions on the intersection of games and mathematics in outlets like BBC News and The Guardian.^[12] The puzzle's reputation for "eternal frustration" has endured in popular narratives, with retrospective analyses in books such as Mysteries of the Equilateral Triangle portraying it as a modern geometric enigma.^[34] Popular science videos, including a 2023 documentary-style exploration by Cardboard Mountain, have further amplified its legacy, emphasizing the human-computational triumph in solving it ahead of schedule.^[35] As of 2025, the puzzle maintains relevance through digital adaptations and its role in advancing AI techniques for constraint satisfaction. Online platforms like NRICH provide articles and resources for educational exploration of the puzzle.^[4] It has inspired evolutionary computing and hybrid heuristics in AI research, where the puzzle's structure tests algorithms for large-scale optimization, as seen in studies adapting it to model similar tiling scenarios.^[36] The Eternity puzzle's prize model also contributed to the trend of high-stakes challenges in recreational mathematics, echoing formats like the Beal Conjecture competition by incentivizing collaborative problem-solving.^[37] Christopher Monckton's design innovations from the project influenced subsequent puzzle developments, though his later career shifted toward public policy.

References

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Eternity -- from Wolfram MathWorld
Eternity is a puzzle devised by Christopher Monckton and consisting of 209 pieces, each of which is a 12-polydrafter (ie, a compound of 30-60-90 triangles).
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eternity - MathPuzzle.com
Each piece in the Eternity puzzle is made from twelve 30-60-90 triangles, also known as a Drafter's Triangle (or Draughter's Triangle). It seems that ...
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Monckton saves the day! | Climate science scepticism and denial
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eternity
### Summary of the Eternity Puzzle Invention and Development
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Former Thatcher aide puts £1m profit motive into jigsaw puzzle
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May 14, 2015 · In 1999 Christopher Monckton launched a new type of puzzle, similar to a jigsaw but with 209 plain green plastic pieces with geometric ...
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This NP-hard problem [17] sparked a lot of interest following the launch of the Eternity puzzle challenges. ... edge colors are drawn at random; see Figures 1 and ...
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Jun 1, 2014 · Unlike the first version, the Eternity II is very much unsolved. The only people that know of a solution are its creators. Also, unlike the ...Missing: publisher Elysium Ertl<|control11|><|separator|>
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Description
### Summary of Color Matching, 19 Colors, Balanced Distribution, Tiling Constraints, Uniqueness, Exhaustive Search, Border Pieces
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### Summary of Key Information from https://www.archduke.org/eternity/talk/notes.html
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Recreation 24 (Eternity Puzzle [242]). The Eternity Puzzle [242] is a jigsaw puzzle comprised of 209 pieces constructed from 12 hemi-equilateral. (30◦ − 60 ...
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Prize specimens
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