REVIEW 2 major objections 6 minor 4 references
It's Common Knowledge
T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 5.1, simultaneous game formula] 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 7, Solution to Puzzle 13] 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.
minor comments (6)
- [Section 3.2] The text contains several typos: 'any knew information' should be 'any new information', and 'This conitnues' should be 'This continues'.
- [Section 6.1] 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 5.2] There is a typo 'the the set of numbers' in the discussion of Alice's turn when she has a 2.
- [Section 8] The author name 'Tanya Khovano va' should be 'Tanya Khovanova'.
- [References] Reference [1] lists 'USSR Moscow' as an affiliation; this appears to be a formatting artifact and should be cleaned up.
- [Section 2] 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.
Circularity Check
No circularity: all puzzle solutions are derived from explicitly stated game rules and the standard definition of common knowledge.
full rationale
The paper is a self-contained puzzle-analysis manuscript. Its solutions are derived from the stated game mechanics, the public-announcement semantics of YES/NO rounds, and the standard definition of common knowledge. There are no fitted parameters, no empirical predictions, and no load-bearing self-citations: references [1]-[4] are background sources, and the circular-hat analysis in Section 3 is re-derived inline rather than imported. The maximum-difference formulas in Section 5 and the sum-or-product case analyses in Section 6 follow from eliminating possible worlds after each public NO or YES; no equation is defined in terms of the quantity it is supposed to determine. The possible off-by-one issue in the Section 5.1 bound, if real, would be a correctness error in a derived interval, not a circular dependency. Accordingly, the paper's derivation chain is not circular.
Assumptions & free parameters
assumptions (6)
- domain assumption All players are infinitely intelligent and perfectly rational, and this is common knowledge.
- domain assumption Each player says YES if and only if they know their own value, and NO otherwise, and this rule is common knowledge.
- domain assumption All emperor announcements are truthful and become common knowledge.
- domain assumption Players have perfect memory and can perform arbitrarily long chains of nested knowledge reasoning.
- domain assumption Games proceed in synchronous rounds (simultaneous) or fixed-order turns (circular) with no interruptions or distractions.
- standard math Pigeonhole Principle
Cite this review
Pith. "Pith review of It's Common Knowledge." pith.science (2026). https://pith.science/paper/AX4C6GKL
@misc{pith2026190902415,
author = {Pith},
title = {Pith review of: It's Common Knowledge},
year = {2026},
howpublished = {\url{https://pith.science/paper/AX4C6GKL}},
note = {Machine review of arXiv:1909.02415}
}
read the original abstract
We discuss some old common knowledge puzzles and introduce a lot of new common knowledge puzzles.
Reference graph
Works this paper leans on
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[1]
J. H. Conway, M.S. Paterson, and USSR Moscow, A Headache Cau sing Problem, In: P. van Emde Boas, J. K. Lenstra, F. Oort, A.H.G. Rinnooy Kan, T.J. W ansbeek (eds.), Een Pak Met Korte Broek, A Book in Short Trousers , Amsterdam, May 18, 1997
work page 1997
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[2]
Morris Friedell, On the Structure of Shared Awareness, Behavioral Science 14 (1969): 28–39
work page 1969
- [3]
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[4]
Martin Gardner, Puzzles from Other Worlds. (1984), Vintage Books. ISBN 0-394-72140- 3. 24
work page 1984
Reviewed August 14, 2026 · model on record in the stance chip above.
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