{"id":"5d99c260-5715-4fba-ac6a-3a2bf62d209d","arxiv_id":"1908.06003","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every permutation of the numbers 1 through 12 on the vertices of an icosahedron admits an all-different-tile solution to the Icosoku puzzle.","lead":"This paper reports that every way of placing the numbers 1 through 12 on the twelve pegs of an Icosoku puzzle can be completed into a solution that uses twenty distinct triangle tiles. The authors share a constraint-programming model, two affirmative answers, and four open questions for puzzle and constraint-satisfaction researchers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal ADTS claim depends on the unstated face-vertex incidence behind Constraint (4) and an unreleased implementation; if that labeling is not the true icosahedron, the exhaustive result does not apply to the puzzle.","rationale":"The reader's weakest assumption identifies the unstated vertex-face incidence, and I agree that this is the most load-bearing gap. However, the reader also worries that the rotation-symmetry reduction from 12! to (12−1)!/5 may be invalid; that part appears sound. Fixing the vertex carrying value 1 at v0 is best read as a coordinate choice using vertex transitivity of the icosahedral rotation group, and quotienting by the order-5 stabilizer of v0 gives 11!/5 = 12!/60, exactly the number of orbits under the orientation-preserving symmetry group. Since all values are distinct, no nontrivial rotation fixes an assignment, so the count is correct. The real weakness is that Constraint (4) is specified only as a template, and the authors explicitly decline to give Fsubset. The paper also provides no code, only a partial tuple table, and no per-instance output, so the 'proved' universal claim cannot be independently checked from the submitted text. This is a verification/reproducibility weakness rather than a demonstrated mathematical error, so the reader's CONDITIONAL verdict remains appropriate; no change is needed.","tokens_in":5428,"tokens_out":12450,"duration_ms":126892,"concrete_test":"Obtain from the authors the exact Fsubset incidence table (or the Choco source) used in Constraint (4). Verify that the 20 rows of F describe a regular icosahedron: each vertex index appears in exactly five faces, each row lists three distinct vertices, and the resulting graph has 12 vertices, 30 edges, and 20 triangular faces and is isomorphic to the standard icosahedron graph. If the supplied mapping passes this check, the geometric-concern is resolved; if it cannot be produced or fails, the universal claim remains unverified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central assertion—that every one of the 12! peg arrangements admits an ADTS—is a computational existence result, but the manuscript does not supply the incidence data that defines Constraint (4). Section 3 explicitly says: 'we do not explicitly specify the five elements in the Fsubset because they depend on how the triangular faces and the variables representing their vertices on the icosahedron are labelled in practice.' Thus the 20×4 matrix F and the scalar constraints are only templates; the actual Choco model is not contained in the paper. If the implementation used a labeling or incidence that is not the true face-vertex structure of a regular icosahedron, the reported SAT answers would not be about Icosoku. The exhaustive enumeration over (12−1)!/5 assignments is a sound symmetry reduction: the rotation group about the vertex fixed to 1 acts freely on distinct-value assignments, giving 12!/60 orbits. That part is not the main risk. The main gap is the unverifiable mapping: no code, no full tuple table, and no per-instance data accompany the 'proved' claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies what it calls 'all different triangular solutions' (ADTS) of the Icosoku puzzle: an assignment of the values 1..12 to the 12 vertices of the icosahedron such that each vertex value equals the sum of the five surrounding face-corner values and such that the 20 triangular face types are pairwise distinct. The authors formulate a constraint programming model (Constraints (1)-(5)) and report two computational results: a first ADTS was found, and an exhaustive test of (12-1)!/5 symmetry-reduced peg permutations showed that every such permutation admits at least one ADTS. The remaining questions Q3-Q6 are left open. The paper is short, explicitly framed as a preliminary result, and aimed at encouraging further work in group theory and constraint programming.","tokens_in":5549,"tokens_out":9314,"duration_ms":89842,"significance":"If the central claim is correct, the paper answers two natural mathematical questions about Icosoku: ADTS existence and universality over all 12! peg arrangements. The constraint model is a direct and natural encoding of the puzzle rules, and the symmetry-reduction argument from 12! to (12-1)!/5 cases is sound because the ADTS property is invariant under rotations fixing the vertex v0. The paper also formulates five further questions that are interesting open problems. However, the manuscript omits the concrete face-vertex incidence data needed to instantiate the scalar constraints (4), provides only a partial tuple table, and releases no code or per-instance data, so the computational results cannot currently be verified by a reader. The contribution is therefore an interesting preliminary report rather than a self-contained proof.","major_comments":[{"comment":"The scalar constraints (4) use a subset Fsubset of the matrix F that is never defined. The text explicitly says that the five elements are not specified 'because they depend on how the triangular faces and the variables representing their vertices on the icosahedron are labelled in practice.' Since the universal ADTS claim depends on the specific 20-face incidence structure of a regular icosahedron, the model as written is only a template. A reader cannot determine whether the implemented constraints correspond to the true puzzle geometry, and the claimed exhaustive result does not follow from the paper as written.","section":"Section 3, Eq. (4)"},{"comment":"The experimental results are not reproducible from the manuscript. Table 1 lists only a subset of the 64 tuples, with no full tuple table in an appendix or supplement. The paper reports overall runtime statistics but does not give the first ADTS solution, does not release code, and does not provide per-permutation data from the exhaustive run. Consequently, both the existence claim and the universal claim rest on an unverifiable implementation. The authors should supply at least an appendix with the explicit Fsubset incidence, the complete tuple table, and a link to the code, or a machine-checkable certificate for the claimed results.","section":"Section 4"},{"comment":"The conclusion states that 'we have proved the existence of the ADTS, and any permutation of {1...12} for the vertices of the icosahedron can produce at least one ADTS.' The term 'proved' is appropriate for a computational search only if the search is exhaustively specified and the artifacts are available. The symmetry reduction is described plausibly, but the missing incidence data, the incomplete tuple table, and the unreleased implementation prevent an independent check. The paper should either soften the claim to a computational report or provide the artifacts needed to verify the exhaustive result.","section":"Section 5"}],"minor_comments":[{"comment":"The text says 'all combinations of values that can be assigned to every row of the matrix F are 644-tuples'; this appears to be a typo and should read '64 4-tuples' or '64 tuples'.","section":"Section 3, near Eq. (3)"},{"comment":"The footnote says the scalar constraint is called the 'linear and LinearInt constraint in Geode and JaCoP'; 'Geode' should be 'Gecode'.","section":"Footnote 1"},{"comment":"The phrase 'the sum of values assigned to the vertices of the triangle surrounding this vertex' is imprecise; the authors mean the sum of values on the five triangular faces meeting at the vertex, and should phrase it accordingly to avoid confusion.","section":"Section 3, paragraph before Eq. (4)"},{"comment":"The caption contains 'T able' with an errant space, and the caption should also note that the full tuple list is available only from an external source or supplement, since the paper provides only a partial list.","section":"Table 1"},{"comment":"The statement that 'four-fifths of the symmetries are removed' is imprecise; after fixing v0=1, the remaining 11! value assignments are partitioned into orbits of size 5 under the rotations about v0, so (12-1)!/5 representative permutations are tested.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is not self-contained as a formal verification of the universal ADTS claim. The key missing pieces are the explicit face-vertex incidence underlying Constraint (4) and the code/data used in the exhaustive experiment. If the authors can supply these, the claim may be confirmable; if not, the wording of the conclusion should be weakened to a report of what their implementation found. Please consider asking for a supplementary appendix with the incidence mapping and a reproducibility package."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent, small computational paper that answers two new Icosoku questions, but the main universal claim is not independently checkable from the text because the actual face-vertex incidence is never given and no code or data ship with it. I read the model as probably correct, but \"proved\" is too strong for an unreleased search.\n\nWhat is actually new: the paper poses the all-different-tile question and the universality question, and reports an exhaustive computational answer to both. The CP encoding is standard: allDifferent on the twelve vertex values, table constraints linking each face's three corner values to its tile type, and scalar constraints enforcing that each vertex value equals the sum of the five surrounding face-corner values. The symmetry reduction is sound: fixing v0 to 1 and quotienting by the five rotations around v0 leaves 12!/60 representative assignments, and the paper says each was solvable. If you accept the enumeration, the conclusion follows.\n\nThe main soft spot is reproducibility. Section 3 explicitly declines to give Fsubset, the 20x4 incidence matrix connecting faces to vertices, saying it depends on the labeling. The tuple table is partial, and no source code or per-instance data is included. For a computational existence result, this is a load-bearing gap: a referee cannot verify that the solver was working on the actual icosahedron, nor re-run anything. This does not mean the result is wrong, but it does mean the paper is currently a black box. The CSPLib benchmark proposal is fine but speculative, and the paper honestly says the remaining four questions are open.\n\nThe citation pattern is clean: standard CP textbooks, solver documentation, and a prior paper of theirs on the Social Golfer Problem. The self-citation is benign and not load-bearing.\n\nBottom line: this is a useful, well-posed contribution for the constraint-programming and puzzle communities, not a broad AI paper. It deserves a serious referee, but only with a request for the Fsubset mapping, the full tuple table or code, and enough data to check the exhaustive run, or alternatively a weakened claim that the result was found by their implementation rather than proved. If the artifacts are not provided, I would not accept the universality claim as verified.","headline":"A plausible, well-scoped computational answer to two new Icosoku questions, undercut mainly by missing incidence data and unreleased code; referee it with a request for artifacts.","tokens_in":6139,"tokens_out":3247,"would_cite":false,"duration_ms":32502,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A constraint model proves every Icosoku peg arrangement admits a solution with 20 distinct tiles.","keywords":["Icosoku","constraint programming","constraint satisfaction problem","all-different constraint","table constraint","scalar constraint","symmetry breaking","icosahedron"],"falsifier":"Run the exhaustive check again with an independently generated face-vertex incidence table that does not come from the same labeling routine, or with a brute-force enumerator that does not use the constraint solver; if any of the $12!$ peg arrangements yields no ADTS in that independent implementation, the universality claim would be refuted.","tokens_in":5159,"feed_emoji":"🧩","tokens_out":8456,"duration_ms":69921,"temperature":0.7,"pith_summary":"This paper sets out to establish two properties of the Icosoku puzzle, a regular icosahedron with pegs numbered 1 through 12 and triangular tiles whose vertices carry 0 to 3 dots. It defines an all-different triangular solution (ADTS) as a solution in which the 20 face tiles are pairwise distinct, and asks whether an ADTS exists and whether every arrangement of the pegs admits one. The paper reports an affirmative answer to both: the constraint model finds an ADTS, and an exhaustive run over the peg arrangements, reduced by symmetry, finds at least one ADTS for every case. If correct, this settles the first two of six questions the paper poses and makes Icosoku a concrete bridge between constraint programming and group theory.","feed_headline":"Every Icosoku peg arrangement has a distinct-tile solution","feed_subtitle":"A constraint program checked all peg orders, cut by symmetry, and found a solution for each.","key_machinery":"The central object is the constraint satisfaction model of the Icosoku, a CSP with 92 variables and 35 constraints built around a $20\\times4$ matrix $F$. Each row of $F$ lists the three vertex dot counts of one triangular face and the corresponding tile type from the 24 rotationally distinct types; the table constraint restricts each row to the 64 allowed tuples, the allDifferent constraint forces the 20 tile types to be pairwise distinct, the scalar constraints enforce that each vertex value equals the sum of its five incident face-vertex values, and the arithm constraint fixes the first vertex to 1. The mechanism carries the argument by turning a geometric/combinatorial existence question into a finite CSP whose exhaustive solution, with value and rotation symmetry broken, certifies the universality claim.","core_discovery":"The paper's central claim is that the Icosoku puzzle has an all-different triangular solution (ADTS)—a feasible placement in which the 20 face tiles are pairwise distinct—and, more strongly, that every assignment of the values $\\{1,\\dots,12\\}$ to the 12 vertices of the icosahedron admits at least one such solution. The evidence is a constraint satisfaction model with 92 variables and 35 constraints: vertex variables are constrained by allDifferent, a $20\\times4$ face matrix is constrained by table constraints linking each face's three vertex dot counts to one of the 24 rotationally distinct tile types, scalar constraints require each vertex value to equal the sum of the five incident face-vertex values, and one arithm constraint fixes the first vertex's value to break value symmetry. Running this model in a constraint solver finds an ADTS for the first tested arrangement and, after reducing the $12!$ permutations to $(12-1)!/5$ by fixing one vertex and quotienting the fivefold rotational symmetry, the exhaustive run took $7.03\\times10^5$ seconds and found an ADTS for every remaining arrangement. On this evidence the paper concludes that Questions 1 and 2 are answered in the affirmative.","pith_inferences":["If the universality result extends to the remaining open questions, the six-question hierarchy suggests a structural conjecture: the set of all ADTSs may form a single orbit under the icosahedral rotation group, which would reduce Questions 4–6 to counting orbits of 20-tile sets.","The same constraint model could be re-run on the other Platonic solids to test whether 'every vertex-labeling admits an all-distinct-face solution' is a general property of polyhedral symmetry or special to the icosahedron's five-faces-per-vertex structure.","A constructive certificate that maps every peg arrangement explicitly to an ADTS, without search, would replace the 8-day computation and could yield a human-readable proof of the universality claim."],"forward_implications":["The first two questions are settled: an ADTS exists, and every permutation of $\\{1,\\dots,12\\}$ on the vertices admits at least one ADTS.","The original puzzle's 'always solvable' assertion holds in a stronger form: it remains solvable even when all 20 face tiles are required to be pairwise distinct.","The symmetry reduction shows that the exhaustive check covered $11!/5 \\approx 7.98$ million genuinely different peg arrangements after fixing one vertex and quotienting the fivefold rotational symmetry.","The total computation time of about 8.13 days on a laptop makes Icosoku a feasible but nontrivial benchmark for constraint solvers."],"supporting_citations":[{"why":"The constraint solver in which the model was implemented and both experiments were run; it supplies the allDifferent, table, and scalar constraint machinery.","marker":"[7]"},{"why":"The symmetry-breaking technique used to fix the first vertex value to 1 and prune value symmetries.","marker":"[8]"},{"why":"The handbook that supplies the formal CSP definitions and constraint machinery the model instantiates.","marker":"[9]"}],"fun_headline_variants":["Icosoku: every vertex setup has a distinct-tile solution","Constraint search proves Icosoku's all-different tile answer","Exhaustive CP run: all Icosoku puzzles solvable","Symmetry-breaking search finds Icosoku answers for all cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exhaustive result depends on the unstated mapping from the icosahedron's actual geometry to the five-tuple $F_{\\text{subset}}$ used in the scalar constraints, and on the rotation-symmetry reduction from $12!$ to $(12-1)!/5$ being exact; if either is wrong, the claimed universality may not apply to the physical puzzle.","fun_headline_variants_meta":{"raw":{"variants":["Icosoku: every vertex setup has a distinct-tile solution","Constraint search proves Icosoku's all-different tile answer","Exhaustive CP run: all Icosoku puzzles solvable","Symmetry-breaking search finds Icosoku answers for all cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001008,"raw_usage":{"total_tokens":4214,"prompt_tokens":855,"completion_tokens":3359,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":3286}},"tokens_in":471,"tokens_out":3359,"duration_ms":24847,"temperature":1.0,"reasoning_tokens":3286,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:58:18.840337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the exhaustive check again with an independently generated face-vertex incidence table that does not come from the same labeling routine, or with a brute-force enumerator that does not use the constraint solver; if any of the $12!$ peg arrangements yields no ADTS in that independent implementation, the universality claim would be refuted.","supporting_citations":[{"cited_title":"TASC - LS2N CNRS UMR 6241, COSLING S.A.S","cited_arxiv_id":null,"evidence_quote":"The constraint solver in which the model was implemented and both experiments were run; it supplies the allDifferent, table, and scalar constraint machinery."},{"cited_title":"Constraints 10(1), 23--46 (2005)","cited_arxiv_id":null,"evidence_quote":"The symmetry-breaking technique used to fix the first vertex value to 1 and prune value symmetries."}],"review_version":1}