{"id":"bce567b4-0900-4211-be6f-57f2155c771a","arxiv_id":"1909.02415","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A collection of new common knowledge puzzles with solutions, covering hat colors, sight impairments, maximum difference, and sum-or-product games.","lead":"This paper collects many new puzzles about common knowledge, where people deduce hidden facts from what others know or do not know. A smart generalist might read it to see how a simple idea, everyone knows that everyone knows, generates surprising reasoning chains.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5.1's central simultaneous-game formula is off by one: the interval M ∈ [(k−1)D, kD−1] misstates the first YES round for the maximum-difference game.","rationale":"I read the paper in good faith as a puzzle collection whose central advertised claim is that every puzzle has a correct solution, including the systematic characterizations highlighted in the strongest claim. I looked for the least secure condition for that claim to hold. The synchronization assumption identified by the reader is real, but the authors explicitly acknowledge it and it is a standard idealization for this genre. A more concrete and checkable problem is that Section 5.1's flagship formula is internally off by one under the paper's own stated update rule. This is an internal inconsistency, not a dispute with external consensus, so it directly bears on correctness. The error is localized and likely fixable: the correct interval appears to be M ∈ [kD, (k+1)D−1] for the first YES round, and the circular formulas in the same subsection warrant rechecking for the same shift. I therefore do not propose changing the reader's CONDITIONAL verdict; I would keep it conditional, but the required revision should include correcting this formula and retesting the cases that rely on it. My concern differs from the reader's stated weakest assumption, hence the disagreement flag.","tokens_in":16954,"tokens_out":22236,"duration_ms":206685,"concrete_test":"Run the two-player simultaneous maximum-difference game for D=1 with number pairs (2,1) and (1,0) under the paper's own round definitions, and record the first round in which the player with the maximum says YES. The protocol gives round 2 for M=2 and round 1 for M=1; the printed interval M ∈ [(k−1)D, kD−1] assigns M=2 to round 3 and M=1 to no round. Independently re-derive the threshold from the paper's own statement that after k rounds of mutual NO the numbers are known to be more than kD−1; the resulting condition is the least k with M < (k+1)D, i.e. M ∈ [kD, (k+1)D−1]. If this simulation and derivation confirm the shift, Section 5.1's characterization and any downstream use of it must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is not the synchronization idealization but an internal off-by-one error in a claimed systematic result. Section 5.1 states: 'the person with the largest number M, will say YES on round k, where M ∈ [(k−1)D, kD−1].' This is inconsistent with the paper's own update rule. In the simultaneous two-player game, after one round of mutual NO it is common knowledge that every number is at least D. Thus on the second round the max player, who sees M−D, can rule out the alternative M−2D as soon as M−2D < D, i.e. M < 3D. In general the first YES round is the least k with M < (k+1)D, equivalently k = floor(M/D), i.e. M ∈ [kD, (k+1)D−1]. The printed interval is shifted by one block. Concrete consequences: with D=1 and numbers (2,1), the max player says NO in round 1 and YES in round 2, because the round-1 NO has ruled out 0; the formula instead places M=2 in round 3. With numbers (1,0), the max player sees 0 < D and says YES in round 1; the formula gives no round for M=1. Since this formula is the main characterization of Section 5.1, the advertised claim that all systematic solutions are correct is not currently supported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a recreational-mathematics collection of common knowledge puzzles. It begins with the classic hats puzzle, then introduces variations with blind, near-sighted, and far-sighted players. It then replaces hats with integers on foreheads and analyzes two families of games: maximum-difference games, where the emperor announces the largest difference D between numbers, and sum-or-product games, where an announced integer M is either the sum or the product of the players' numbers. For each family the authors provide solutions, including closed-form round counts and complete case analyses for the two-player sum-or-product game. The paper also leaves several puzzles for the reader, with solutions deferred to Section 7.","tokens_in":17232,"tokens_out":50433,"duration_ms":420314,"significance":"If the results are correct, the paper offers a useful and enjoyable collection of puzzle variants with systematic solutions. The most valuable parts are the two-person sum-or-product case analysis (Section 6.1), which appears thorough and is presented as a complete classification, and the maximum-difference analyses (Section 5), which give explicit formulas for when players first know their numbers. The paper is clearly written for an informal audience and does not claim formal theorem-proving status. However, the correctness of the advertised systematic characterizations is load-bearing for the paper's value; a single off-by-one error in Section 5.1 currently undermines that part of the paper.","major_comments":[{"comment":"The claimed characterization 'the person with the largest number M, will say YES on round k, where M ∈ [(k−1)D, kD−1]' is inconsistent with the paper's own update rule. In the simultaneous two-player game, after r rounds of mutual NO it is common knowledge that both numbers are at least rD. Before round k there have been k−1 NOs, so the max player, seeing M−D, can rule out the alternative M−2D as soon as M−2D < (k−1)D, i.e., M < (k+1)D. Since the player must not have known in earlier rounds, the first YES round is the least k with M ≥ kD and M < (k+1)D, i.e., M ∈ [kD, (k+1)D−1]. The printed interval is shifted by one block. For example, with D=1 and numbers (2,1), the max player says NO in round 1 and YES in round 2, but the printed formula places M=2 in round 3; with (1,0), the max player says YES in round 1 but the formula gives no round for M=1. This is the main systematic result of Section 5.1 and must be corrected.","section":"Section 5.1, simultaneous game formula"},{"comment":"Puzzle 13 asks 'What is Bob's number?' but the solution concludes that Bob's number is either 2 or 25. The puzzle as stated is therefore underdetermined. The authors should either rephrase the question (e.g., ask what can be concluded about Bob's number, or ask to identify all possible pairs), or add an additional condition that selects a unique number. As written, the claim that the puzzle has a unique answer is not supported.","section":"Section 7, Solution to Puzzle 13"}],"minor_comments":[{"comment":"The text contains several typos: 'any knew information' should be 'any new information', and 'This conitnues' should be 'This continues'.","section":"Section 3.2"},{"comment":"In the circular-game summary, the first 'YES, YES' line reads 'or or A = 1 and B = M = 2; B is a non-divisor'—the doubled 'or' and the dangling clause make the classification hard to read.","section":"Section 6.1"},{"comment":"There is a typo 'the the set of numbers' in the discussion of Alice's turn when she has a 2.","section":"Section 5.2"},{"comment":"The author name 'Tanya Khovano va' should be 'Tanya Khovanova'.","section":"Section 8"},{"comment":"Reference [1] lists 'USSR Moscow' as an affiliation; this appears to be a formatting artifact and should be cleaned up.","section":"References"},{"comment":"The paper assumes perfectly synchronized, equally fast thinkers but does not formally state that the timing protocol (e.g., fixed-length rounds) is common knowledge; spelling this out would strengthen the rigor.","section":"Section 2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The off-by-one error in Section 5.1 is localized but serious because the paper advertises systematic solutions. I would ask the authors to re-derive all closed-form expressions in Section 5 (especially 5.3) as part of the revision, since similar indexing mistakes may lurk. The Puzzle 13 issue is a design flaw in the puzzle statement, not a deep mathematical problem, and should be easy to fix. The paper's scope is recreational, and the informal style is appropriate to the venue; I would not require formal proofs. The novelty is modest, but the collection is coherent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a likeable PRIMES STEP paper, not a major research claim. It is a collection of common-knowledge puzzles, several genuinely new in design, with informal solutions. Most of the logic is checkable and right, but the paper's showcase systematic formula in Section 5.1 is off by one, and Puzzle 13 asks for a unique answer it doesn't have.\n\nWhat's actually new: the blind/near-sighted/far-sighted hat variants, the max-difference games for N players, and the two-person sum-or-product case analysis. The Section 6.1 case analysis is the most substantial piece; I spot-checked several of the summarized patterns and they hold. The paper is honest about being informal, and the writing has a nice expository voice for showing how common knowledge works.\n\nThe soft spots, in order of importance. First, Section 5.1's simultaneous-game interval M ∈ [(k−1)D, kD−1] is shifted. In the two-player game with D=1 and numbers (2,1), the max player sees 1, and after the first round's mutual NO it is common knowledge that both numbers are at least 1, so the only remaining possibility is 2; the max player says YES in round 2, not round 3. With (1,0), the max player says YES in round 1, not round 2. The correct condition is M ∈ [kD, (k+1)D−1]. Since this interval is the advertised systematic result, it needs to be fixed before the paper can be trusted as a reference.\n\nSecond, Puzzle 13 is underdetermined. The dialogue rules out everything except Alice=25, but Bob's number is either 2 or 25, and the solution itself says Bob does not know his number. The puzzle asks 'What is Bob's number?', so either the question or the setup needs to change.\n\nThird, some of the case analyses (e.g., Section 3.2 summaries) are asserted rather than derived. For a recreational paper that is acceptable when the claims are true, and they mostly are, but a couple of typos like 'knew information' make it look sloppier than the mathematics warrants.\n\nThe synchronization assumption is not a real flaw here; it is the standard idealization for common-knowledge puzzles, and the Alex/Betty/Claire anecdote explicitly acknowledges that real humans break it.\n\nWho this is for: teachers and puzzle people who want worked examples of common knowledge, maybe a fun reading-group session. I would not cite it in my own research. But it deserves a serious referee for a recreational-math or math-education venue, and with a revision fixing the off-by-one and being honest about Puzzle 13, I'd be happy to see it published.","headline":"A fun, mostly sound collection of common-knowledge puzzle variants; the main systematic claim in Section 5.1 has an off-by-one error and Puzzle 13 is underdetermined.","tokens_in":17771,"tokens_out":7667,"would_cite":false,"duration_ms":72971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["00A08","91A80","03B42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper presents and solves a family of common knowledge puzzles, turning vague waiting times into exact round-by-round schedules for both simultaneous and circular play.","keywords":["common knowledge","hat puzzles","simultaneous games","circular games","maximum difference","sum or product","forehead numbers","public announcements"],"falsifier":"Check the transcript of any simultaneous maximum-difference game: if the largest number $M$ first triggers YES on a round $k$ with $M \\notin [(k-1)D, kD-1]$, the characterization is wrong; none such configuration should exist. The synchronization assumption can be tested directly by rerunning the three-person hat experiment with one player forced to wait an extra beat, which should shift or destroy the predicted first-YES round.","tokens_in":16772,"feed_emoji":"🎩","tokens_out":6987,"duration_ms":69010,"temperature":0.7,"pith_summary":"The paper's aim is to turn common knowledge puzzles from isolated riddles into a systematic body of solved problems. It formalizes two modes of play, simultaneous rounds and circular turn-taking, and shows how each public \"I don't know\" narrows the possible configurations until someone can say \"I know.\" New variations add blind, near-sighted, and far-sighted players, and replace hat colors with integers whose values matter. For the maximum-difference and sum-or-product families, the paper gives explicit round-by-round characterizations, including a complete two-person analysis of the sum-or-product game. A sympathetic reader should take the paper as claiming that all of these puzzles have determinate solutions under synchronized, perfectly rational play.","feed_headline":"Common knowledge puzzles get exact round-by-round answers","feed_subtitle":"Classic hat riddles and new number variants all reduce to counting the rounds of silence before \"yes.\"","key_machinery":"The load-bearing object is the public announcement round: a fixed interval in which every player says YES or NO simultaneously, so that a NO heard by everyone upgrades the announced fact from mutual knowledge to common knowledge one level deeper. In the hat puzzle, each all-NO round proves that the number of red hats is at least one larger than previously known; in the number puzzles, the same silence eliminates all numbers below a rising threshold. This round structure lets the paper derive exact announcement times, so the strategy is not \"wait a while\" but a precise schedule.","core_discovery":"On its own terms, the paper claims that every puzzle it introduces has a definite solution, and that the solutions share a common inductive mechanism: once a round passes with everyone saying NO, that silence becomes common knowledge and raises the minimum number of red hats or the lower bound on the hidden number. In the simultaneous maximum-difference game with announced difference $D$, the person holding the largest number $M$ says YES on round $k$ exactly when $M \\in [(k-1)D, kD-1]$; the holder of the minimum then follows in the next round. For the two-person sum-or-product game with announced integer $M$, the paper gives a complete case analysis: depending on whether the visible number divides $M$, whether one player sees $1$, and whether the two numbers are equal, the game ends in a predictable pattern of YES and NO answers over at most three rounds. In circular games the order of speakers determines who learns their number, and the paper shows that in many cases only the first informed player, or the last speaker, can ever know their own number.","pith_inferences":["The same \"silence is a signal\" mechanism should transfer to any public protocol with a fixed clock, so a testable extension is to replace verbal NOs with a button press or a light that players control; the predicted round numbers would remain unchanged.","The authors leave open the full sum-or-product analysis for $N>2$ with general composite $M$; the semiprime case solved here suggests the general answer will depend on the additive and multiplicative partitions of $M$, and could be derived by extending the divisor/non-divisor case split.","Because the puzzles assume common knowledge of rationality, they predict that adding a single irrational or forgetful player changes outcomes dramatically; a simple human experiment could measure how many rounds of silence real groups require before the first YES, and compare the distribution to the predicted round.","The near-sighted and far-sighted variants show that local visibility creates local knowledge; one could push this to a network version where players see only along a graph, with the red-hat rule replaced by a graph-distance threshold, to test whether the same round-counting logic survives on non-circular topologies."],"forward_implications":["In the standard simultaneous hat puzzle with $N$ players and $r$ red hats, all red-hatted players will first say YES in round $r$, and all blue-hatted players in round $r+1$.","In the circular version, the number of players who ever learn their color depends on the order of play and ranges from one to the number of blue hats plus one; players after the first red-hatted speaker to answer can learn immediately.","In the simultaneous maximum-difference game with difference $D$, the maximum holder's first YES occurs in round $k$ where $M \\in [(k-1)D, kD-1]$; the minimum holder then knows in the following round.","For the two-person sum-or-product game with announced number $M$, the entire YES/NO conversation lasts at most three rounds, and at least one player always learns their number, though in circular play one player may remain ignorant forever.","For prime or semiprime announced numbers with $N$ players, the paper classifies which distributions make everyone, some, or no one know their numbers in the first and second rounds."],"supporting_citations":[{"why":"Introduces the common knowledge concept that the announcements in the puzzles are meant to create.","marker":"[2]"},{"why":"Provides the formal account of knowledge and covers the circular hat game that the paper extends.","marker":"[3]"},{"why":"Collects the classic hat and muddy-children variants that serve as the paper's starting point.","marker":"[4]"},{"why":"Supplies the forehead-number format with an announced product or sum that Sections 4 through 6 develop.","marker":"[1]"}],"fun_headline_variants":["Silence reveals all: hat puzzles solved round by round","Common knowledge: counting NOs to say YES","Hat riddles crack when silence speaks","Round-by-round logic pins down every puzzle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All players must be perfectly synchronized, infinitely rational, and completely reliable in announcing NO whenever they do not know their number; if one player thinks at a different speed or is distracted, the chain of common knowledge breaks, as the paper's own opening experiment with three friends illustrates.","fun_headline_variants_meta":{"raw":{"variants":["Silence reveals all: hat puzzles solved round by round","Common knowledge: counting NOs to say YES","Hat riddles crack when silence speaks","Round-by-round logic pins down every puzzle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000871,"raw_usage":{"total_tokens":3667,"prompt_tokens":737,"completion_tokens":2930,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":353,"completion_tokens_details":{"reasoning_tokens":2872}},"tokens_in":353,"tokens_out":2930,"duration_ms":21450,"temperature":1.0,"reasoning_tokens":2872,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:21:58.682875+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the transcript of any simultaneous maximum-difference game: if the largest number $M$ first triggers YES on a round $k$ with $M \\notin [(k-1)D, kD-1]$, the characterization is wrong; none such configuration should exist. The synchronization assumption can be tested directly by rerunning the three-person hat experiment with one player forced to wait an extra beat, which should shift or destroy the predicted first-YES round.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the common knowledge concept that the announcements in the puzzles are meant to create."},{"cited_title":"Fagin, J","cited_arxiv_id":null,"evidence_quote":"Provides the formal account of knowledge and covers the circular hat game that the paper extends."},{"cited_title":"(1984), Vintage Books","cited_arxiv_id":null,"evidence_quote":"Collects the classic hat and muddy-children variants that serve as the paper's starting point."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the forehead-number format with an announced product or sum that Sections 4 through 6 develop."}],"review_version":1}