Solution of the 15 puzzle problem

Robert Hough; Yang Chu

REVIEW 4 major objections 6 minor 21 references

Solution of the 15 puzzle problem

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the generalized 15 puzzle on an $n\times n$ torus mixes in order $n^4$ steps, with a single tile's position converging to Brownian motion on the torus at time $c_{\mathrm{puz}} n^4 t$ for…

desk verdict A serious, novel attack on Diaconis's 15-puzzle problem that likely gets the n^4 order right, but the sharp Brownian constant rests on a two-sentence characteristic-function bound (Lemma 22) that needs a real proof. read the letter →

arxiv 1908.07106 v1 pith:LQGV2LNF submitted 2019-08-19 math.PR

Yang Chu , Robert Hough This is my paper · ORCID

classification math.PR MSC 60B1560J3520B20

keywords 15puzzlemixingtimerandomwalkonagroupMarkovchainDirichletformPoissonapproximationcut-offphenomenonBrownianmotiontorus

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper solves the 15-puzzle problem for an $n\times n$ board with periodic boundary conditions and random moves: how many slides does it take to randomize the board? The answer is order $n^4$ steps. Tracking a single marked tile, the proof shows that its position after $c_{\mathrm{puz}}n^4 t$ steps has total variation distance to uniformity converging to that of Brownian motion on the flat torus run for time $t$, with $c_{\mathrm{puz}}=\frac52(\pi-1)$. It also shows that after $n^4 f(n)$ steps with $f(n)\to\infty$, the number of tiles still in their original positions converges to a $\mathrm{Poisson}(1)$ law, matching a uniformly random permutation. A corollary is that this fixed-point statistic has no cut-off, while the full board mixes in at most $O(n^4\log n)$ steps.

What carries the argument

The load-bearing object is the renewal process of a marked tile: stopping times $t_i$ alternate between the empty square swapping with the tile from left or right and from above or below, with $H_i$ and $V_i$ the tile's horizontal and vertical displacements in those intervals and $r_i$, $s_i$ the elapsed times. The key identity is Lemma 21, which computes the limiting variance $s^2$ and mean time $\mu$ from the four directional return probabilities of Lemma 14, giving $c_{\mathrm{puz}}=2\mu/s^2=\frac52(\pi-1)$. The characteristic function of a single burst is expressed exactly as a resolvent formula (Lemma 23), then expanded at low frequencies; the resulting local limit theorem (Theorem 27) turns the renewal process into the Brownian $\theta$ kernel $\theta_t$. For several pieces, the mechanism shifts to a comparison of Dirichlet forms between the original puzzle, a symmetrized chain with overlapping pieces, and an abelian random walk with independent coordinates (Theorem 9), which yields the $d^2$ mixing bound $O(n^4)$.

What would settle it

Compute the limiting return probabilities in Lemma 14 by evaluating the $\mathbb{Z}^2$ lattice Green's functions $G_{1,0}$, $G_{1,1}$, and $G_{2,0}$ symbolically or to high precision; if the limits are not $\frac12$, $\frac12-\frac1\pi$, and $\frac2\pi-\frac12$, then $c_{\mathrm{puz}}$ changes, although the $n^4$ order would remain.

Watch

Extended reading notes

Core claim

The central discovery is that the mixing of a single tile in the $n^2-1$ puzzle reduces to a renewal process with a Brownian scaling limit. The tile moves only when the empty square is adjacent to it, and between such moves the empty square performs lazy simple random walk on the torus; the tile's displacement accumulates in alternating horizontal and vertical bursts. A local limit theorem for the triple (horizontal displacement, vertical displacement, elapsed time) of these bursts shows that the elapsed-time fluctuations are absorbed into the Brownian clock. Consequently the total variation distance of the tile at time $c_{\mathrm{puz}} n^4 t$ tends to $d_{\mathrm{Br}}(t)$, the total variation distance of Brownian motion on $(\mathbb{R}/\mathbb{Z})^2$, uniformly for $t$ in compact sets. For the whole board, the same renewal structure yields Poisson(1) fixed points after $n^4 f(n)$ steps and, via comparison with a 3-cycle walk, an upper bound of $O(n^4\log n)$.

Load-bearing premise

The sharp constant $\frac52(\pi-1)$ rests on four directional return probabilities in Lemma 14—stated as $\frac12$, $\frac12-\frac1\pi$, $\frac2\pi-\frac12$ with only a computer-algebra calculation as support—being exactly correct.

Editorial extensions

If this is right

A single tile in an $n\times n$ puzzle needs order $n^4$ moves to mix: at time $c_{\mathrm{puz}} n^4 t$ its distance to uniformity approaches that of torus Brownian motion at time $t$.
After $n^4 f(n)$ random moves with $f(n)\to\infty$, the number of tiles in their original positions converges to $\mathrm{Poisson}(1)$; if $f(n)$ stays bounded, this convergence fails.
The fixed-point statistic shows no cut-off phenomenon: mixing happens gradually over order $n^4$ steps.
The full board's total-variation and $\epsilon/|G|$-$\ell^\infty$ mixing time is $O(n^4\log n)$.
For each fixed $d$, the empty square and any $d$ labeled pieces can be coupled with their stationary copies in expected time $O(n^4\log n)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The renewal-and-local-limit structure should transfer to other 'moving-hole' walks, such as higher-dimensional grids or rectangular boards, where only the return-probability calculation changes.
The Poisson(1) fixed-point result suggests the induced permutation is locally uniform on low-complexity representations already at order $n^4$; a further test would be whether the full board's mixing time can be sharpened from $O(n^4\log n)$ to $O(n^4)$.
One could probe the sharp constant empirically by simulating small boards and comparing the single-tile distance to $d_{\mathrm{Br}}(c_{\mathrm{puz}}^{-1} t)$ at times $c_{\mathrm{puz}} n^4 t$; a mismatch would pinpoint the return-probability calculation as the fragile step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bob,

This paper claims to solve Diaconis's 15-puzzle problem on the torus: a single tile mixes at time c_puz n^4 with c_puz = (5/2)(π−1), the whole board mixes in at least order n^4 and at most O(n^4 log n), and the fixed-point statistic is Poisson(1) after n^4 f(n) steps. If the proof holds, it is a big deal: the renewal-process decomposition and the local limit theorem are genuinely new, and the constant is derived from first principles, not fitted.

The overall architecture is credible. The reduction to a random walk on S_{n^2−1} ⋉ (Z/nZ)^2 is clean; the single-tile renewal construction is natural; the comparison to the 3-cycle walk for the upper bound is standard. The Poisson fixed-point theorem is a nice bonus, and the proof via mixed moments and comparison to a symmetrized chain is inventive.

Now the soft spots. The reader's worry about Lemma 14 is misplaced: the Mathematica values are independently checkable against the potential kernel (a(1,1)=4/π, a(2,0)=4−8/π give exactly the stated return probabilities), so that is just an omitted calculation. The real load-bearing gap is Lemma 22, the uniform quadratic lower bound for the characteristic function of the renewal increments. The proof is two sentences about 'paths of bounded length' and 'variation in their phase.' That does not establish the required cancellation, especially in the delicate regimes ξ2 near 1/2 and ξ1 near 0. Theorem 27's truncation relies on this bound, so without a real proof of Lemma 22, Theorem 1 lacks support. This is fixable, but it is exactly the step a referee must verify before accepting the sharp constant.

The upper bound also depends on a sketch (Lemma 42) for the d2 mixing time of the 3-cycle plus torus walk, but that step is less central and the argument is plausible.

Bottom line: if you work in Markov chain mixing, this is worth reading and worth refereeing. I would not cite the sharp constant yet, but I would send it to a strong journal and ask for a rigorous proof of Lemma 22 and a fuller write-up of Lemma 42. It deserves a serious referee, not a desk reject.

Referee Report

4 major / 6 minor

Summary. The paper claims to solve Diaconis' '15 puzzle problem' for the n x n puzzle on the torus with lazy random moves. Theorem 1 asserts that a single labeled piece converges in total variation, at time c_puz n^4 t with c_puz = (5/2)(π - 1), to the corresponding Brownian motion limit, uniformly on compact time intervals. Theorem 2 states that after n^4 f(n) steps with f(n) -> infinity, the number of fixed points converges to Poisson(1), while this convergence fails if f is bounded; Corollary 3 concludes there is no total variation cutoff for this statistic. Theorem 4 gives an O(n^4 log n) total variation mixing time upper bound for the full puzzle, and Theorem 5 states an O(n^4 log n) expected coupling time for the empty square and any fixed number of labeled pieces. The main technical ingredients are a renewal description of a tracked piece's motion, a local limit theorem for renewal sums (Theorem 27), spectral comparison arguments, and a comparison with the 3-cycle walk for the upper bound.

Significance. If the proofs can be completed, the paper resolves a long-standing problem of Diaconis and establishes a sharp, parameter-free Brownian constant c_puz for the single-piece mixing. The Poisson fixed-point result and the absence of cutoff for that statistic are natural and interesting consequences, and the coupling theorem and spectral comparison framework provide tools for further study. The derivation of the constant is not circular: it is obtained from limiting return probabilities of the empty square rather than fitted to the target Brownian distance. The paper also presents a new renewal/local-limit method for this type of constrained random walk on the symmetric group. These strengths make the paper a potentially significant contribution to the probability literature, provided the under-derived technical estimates are supplied.

major comments (4)

[§5, Lemma 14 (Eq. (53))] The uniform bound |chi(xi1, xi2)| <= 1 - c max(xi1^2/n^2, xi2^2) is the load-bearing estimate for the truncation step in Theorem 27, and hence for the exact Brownian constant in Theorem 1. The proof is only two sentences: it invokes paths of bounded length or fixed displacement with positive probability and asserts that their phase variation has the given magnitude. That argument establishes, at best, that a single path has phase close to 1 in some regimes; it does not prove a uniform quadratic lower bound for 1 - |chi|, which requires controlling cancellation among many paths. The claim is especially delicate for xi2 near 1/2 and for xi1 near 0, where the phase scale is small relative to the stated quadratic bound. The authors should provide a rigorous derivation of Lemma 22, or replace it with a proved estimate that supports the frequency truncation in Theorem 27.
[§9, Lemma 42] The limiting return probabilities p_(1,0) = 1/2, p_(0,±1) = 1/2 - 1/pi, and p_(-1,0) = 2/pi - 1/2 are stated after 'The exact values were calculated in Mathematica,' with no derivation shown. These values enter Lemma 21 and determine the constant c_puz = (5/2)(pi - 1) in Theorem 1. The values are checkable from the potential kernel and harmonic relations, but the reader should not be expected to trust a black-box computation at a load-bearing point of the proof. Please include an analytic derivation, or at least a complete and reproducible calculation, of the Green-function limits G_(1,0), G_(1,1), and G_(2,0) used in equations (60).
[§6, Theorem 27] The proof of Lemma 42, which claims the d2 mixing time of the measure mu_S is O(n^2 log n), is only a sketch. In the one-dimensional representation case, the assertion that the spectrum is 'mixed in O(n^2 log n) steps' is stated without a quantitative argument covering the O(n^2) such representations. For higher-dimensional representations, the claim that an arbitrary factor n^2 in multiplicity can be saved by increasing the constant in the n^2 log n mixing time requires a justification, since the Plancherel sum includes multiplicities and the spectral gaps are only known up to constants. This lemma supports Theorem 4, so a complete proof should be supplied.
[§6, Theorem 27] In the contour-shift step between equations (117) and (118), the horizontal integral is dismissed as bounded by O_A(n^{-A}) without a demonstration. The shift moves the integration contour by i(t - 2N mu_n)/(2 N^{1/2} v_n), and the resulting horizontal contribution is controlled only if the Gaussian factor decays uniformly in the summation range and in the error region; this decay is not shown. Since Theorem 27 is the main local limit theorem used in the proof of Theorem 1, this estimate should be stated and proved explicitly.

minor comments (6)

[§8, Lemma 39] The notation in the lemma statement is garbled: 'let py,n,y P tp 1, 0q, p´1, 0q, p0, 1q, p0, ´1qu' should read 'let p_{y,n}, y in {...},' and the subscript typesetting throughout Section 5 should be cleaned up.
[§8, Lemma 40] The proof of Lemma 39 is very brief and uses an 'auxilliary step' that is not precisely defined. Since the lemma is used to justify the Gaussian approximation in Lemmas 40 and 41, please spell out the construction and the induction more carefully.
[§1, Abstract] In the proof of Lemma 40, the notation 'D d_m' in display (172) is unclear; it should be D^d_m or a similar indexed family. The same issue appears in the induction argument in Lemma 41. This is a presentation issue, but it makes already dense arguments harder to check.
[§1, Introduction] The abstract contains the typo 'annˆ n numbered grid'; please correct to 'an n x n numbered grid'.
[§6, Proof of Theorem 1] The related-work paragraph cites the authors' earlier paper [15], but that paper does not appear to be used in the proofs. If it is not needed, this is fine; if it is intended to motivate the renewal technique, a sentence clarifying the connection would help.
[§8, Deduction of Theorem 2] The upper-bound part of the proof states that 'convolution with the remaining distributions can only decrease the total variation distance, as can removing the conditioning.' This is not immediate for the conditional distribution described, and a brief justification or reference would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Brownian constant is derived from independent return probabilities, not fitted to the target mixing time.

full rationale

The derivation is self-contained against external benchmarks. Theorem 1's Brownian constant is computed from the return probabilities of simple random walk on Z^2 (Lemma 14), via the renewal-theoretic variance and mean in Lemma 21; no term in c_puz = 2mu/s^2 is fitted to the Brownian distance d_Br(t). The local limit theorem (Theorem 27) supplies the asymptotic independence of the renewal sum and is proved by Fourier inversion with explicit error bounds rather than by importing the target result. Theorem 2's Poisson(1) limit follows from moment estimates and the O(n^4) d2 mixing bound of Theorem 30, which is established by comparison with symmetrized chains whose spectral estimates are proven directly in Lemmas 40-41. The only author self-citation is [15], which appears in a related-work survey and is not used as an input to any theorem. Lemma 14's Mathematica computation is an unshown calculation, not a circular step, and Lemma 22's brevity is a completeness concern rather than a circularity. No derivation step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard Markov chain and Fourier techniques plus two external results: the comparison theorems of Diaconis-Saloff-Coste and the 3-cycle walk spectral estimate of Helfgott-Seress-Zuk. No new unproved entities are introduced; the main constants are derived from first principles, with one unshown Mathematica computation flagged separately.

assumptions (4)

standard math Comparison theorem for Dirichlet forms of reversible Markov chains (Diaconis-Saloff-Coste, Theorem 8 here).
Used repeatedly to compare the puzzle walk to simpler random walks on the same or enlarged state space; it is an external theorem cited from [8].
domain assumption Spectral gap 3/(n−1) and mixing time of order n log n for the random walk generated by 3-cycles on Alt(n) (Lemma 11, from Helfgott-Seress-Zuk [14]).
Provides the spectral input for the upper bound in Theorem 4; the paper quotes it from [14] without proof.
standard math Standard recurrence and Green's function facts for simple random walk on Z^2 (e.g., return probabilities and hitting time asymptotics).
Used in Sections 5 and 6 to compute return probabilities and hitting times; these are classical results, though some are quoted from [18].
standard math Plancherel formula and Fourier inversion on finite groups.
Used in Sections 8 and 9 to express distances to uniformity in terms of eigenvalues and irreducible characters; this is standard representation theory.

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Pith. "Pith review of Solution of the 15 puzzle problem." pith.science (2026). https://pith.science/paper/LQGV2LNF

@misc{pith2026190807106,
  author       = {Pith},
  title        = {Pith review of: Solution of the 15 puzzle problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LQGV2LNF}},
  note         = {Machine review of arXiv:1908.07106}
}

abstract

A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order $ n^4 $ when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any $f(n) \to \infty$ with $n$, the number of fixed points after $n^4 f(n)$ moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is $n^4$ without cut-off. We also prove an upper bound of order $n^{4 }\log n$ for the total variation mixing time.

Figures

Figures reproduced from arXiv: 1908.07106 by the authors.

**Figure 1.** A 15 puzzle. A move in the puzzle slides a numbered tile into the empty space. Theorem 1. Let dBrptq be the total variation distance to uniformity at time t ą 0 of standard Brownian motion started from p0, 0q on pR{Zq 2 . Let cpuz “ 5 2 pπ ´ 1q. As n Ñ 8, the total variation distance to uniformity of a single piece in the n 2 ´ 1 puzzle at time cpuzn 4 t converges to dBrptq uniformly for t in compact subsets of p0,… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Cut-oﬀ for conjugacy-invariant random walks on the permutation group

Berestycki, Nathana¨ el, and B. S ¸eng¨ ul. “Cut-oﬀ for conjugacy-invariant random walks on the permutation group.” Probability Theory and Related Fields 173.3-4 (2019): 1197-1241

work page 2019

[2] [2]

Cut-oﬀ for random to random c ard shuﬄe

Bernstein, M. and E. Nestoridi. “Cut-oﬀ for random to random c ard shuﬄe.” Annals of Probability , to appear

work page

[3] [3]

On the spectral gap for ﬁn itely-generated subgroups of SU p2q

Bourgain, Jean, and Alex Gamburd. “On the spectral gap for ﬁn itely-generated subgroups of SU p2q.” Inventiones mathematicae 171.1 (2008): 83-121

work page 2008

[4] [4]

Uniform expansion bounds f or Cayley graphs of SL 2pFpq

Bourgain, Jean, and Alex Gamburd. “Uniform expansion bounds f or Cayley graphs of SL 2pFpq.” Annals of Mathematics (2008): 625-642

work page 2008

[5] [5]

Group representations in probability and statistics

Diaconis, Persi. Group representations in probability and statistics. Lecture notes-monograph series 11 (1988): i-192

work page 1988

[6] [6]

Some things we’ve learned (about Markov chain M onte Carlo)

Diaconis, Persi. “Some things we’ve learned (about Markov chain M onte Carlo).” Bernoulli 19.4 (2013): 1294-1305

work page 2013

[7] [7]

Comparison techniq ues for random walk on ﬁnite groups

Diaconis, Persi, and Laurent Saloﬀ-Coste. “Comparison techniq ues for random walk on ﬁnite groups.” The Annals of Probability (1993): 2131-2156. 56 YANG CHU AND ROBERT HOUGH

work page 1993

[8] [8]

Comparison theore ms for reversible Markov chains

Diaconis, Persi, and Laurent Saloﬀ-Coste. “Comparison theore ms for reversible Markov chains.” The Annals of Applied Probability (1993): 696-730

work page 1993