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Asymptotics of the Tchoukaillon array and a conjecture of Beluhov

This paper proves Beluhov's conjecture regarding the asymptotic behavior of the Tchoukaillon array by establishing a precise uniform estimate for its entries, demonstrating that their square roots form a linear blend of the array's edge growth rates and enabling an efficient algorithm to locate any given integer within the array.

Original authors: Shisheng Li

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shisheng Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics, there is a quiet fascination with patterns that emerge from simple rules. Imagine a board game played with a single row of holes, where stones are moved according to a strict, repetitive logic. This is a form of Mancala, an ancient family of sowing games played across the globe. In a specific one-row version known as Tchoukaillon, the game is not about competition but about reaching a single, perfect starting position for any given number of stones. Mathematicians have long known that for every total number of stones, there is exactly one way to arrange them in the holes so that the game can be won. This unique arrangement creates a hidden map, a grid where every positive whole number appears exactly once, filling the space without gaps or overlaps.

For decades, researchers have studied the edges of this map. They knew how the numbers grew along the very first row and the very first column, finding that these boundaries followed predictable, curved paths. However, the vast interior of the grid remained a mystery. A mathematician named N. Beluhov, observing the numbers on a computer, guessed that the entire grid followed a single, elegant rule that blended the behavior of the two edges. He proposed that if you knew the position of any number in the grid, you could predict its value with a simple formula, but he could not prove it. The question was whether this beautiful pattern held true everywhere, deep inside the grid, or if it was just a coincidence near the edges.

A recent paper by Shisheng Li confirms that Beluhov's guess was correct. The author has proven that the entire infinite grid follows a precise mathematical law. The discovery reveals that the numbers in the grid are not scattered randomly; instead, they are arranged in a way that their square roots grow in a perfectly straight line. This means that if you were to draw a line connecting the positions of all numbers up to a certain size, that line would be straight, forming a clean triangular shape. The proof is significant because it does not rely on the known behavior of the edges to explain the middle. Instead, the author shows that the constants in the formula—the specific numbers that determine the shape of the grid—arise naturally from the game's own rules of movement. The number pi, which usually appears in circles, emerges here from the simple act of moving stones, generated by the game's internal logic rather than by any external geometric shape.

To reach this conclusion, the researcher did not look at the grid as a static picture. Instead, they followed a single number as it traveled through the game's different stages. Imagine a specific stone, say the number one hundred, starting in a specific hole. As the game's rules are applied step by step, that number moves to different positions. The author tracked this journey, watching how the number's row and column changed with every move. By reversing this process, they could start with a known position and work backward to find the number that belongs there. This method allowed them to strip away the complexity of the game and focus on the underlying rhythm of the movements.

The key to the proof was realizing that the movement of these numbers follows a pattern that can be described by two simple, opposing forces. One force is driven by the row position, and the other by the column position. As the number moves through the game, these two forces interact in a way that creates a smooth, predictable flow. The author showed that the distance a number travels is determined by a specific type of mathematical product, a sequence of fractions that, when multiplied together, naturally produce the number pi. This was a surprising finding, as it demonstrated that the constant pi is not just a feature of circles but is woven into the very fabric of this stone-moving game.

The result is a complete description of the Tchoukaillon array. The paper proves that for any large position in the grid, the value of the number can be calculated with high precision using a formula that blends the row and column coordinates. The error in this prediction is incredibly small, shrinking as the numbers get larger. This finding settles the long-standing question about the shape of the grid: the regions containing numbers up to a certain size are indeed triangles, with only a very thin, fuzzy edge where the perfect shape is slightly blurred. Furthermore, the research provides a fast and efficient way to locate any specific number in the grid. Instead of searching through millions of entries, one can use a simple set of steps to find exactly where a number like one million sits, a task that would otherwise take a long time.

The work also touches on a deeper question about the nature of these patterns. While the main formula is proven, the author notes that the actual numbers in the grid might fit the formula even better than the proof currently shows. The data suggests that the difference between the predicted value and the real value is tiny and constant, rather than growing slowly. This hints that there may be an even sharper, more precise rule waiting to be discovered, one that would eliminate the remaining small errors entirely. For now, however, the proof stands as a definitive answer to Beluhov's conjecture, revealing a hidden order in a game that has been played for centuries. It shows that even in a simple game of moving stones, the universe of mathematics can reveal profound and unexpected connections, turning a playful puzzle into a window on the fundamental structure of numbers.

Technical Summary: Asymptotics of the Tchoukaillon Array and a Conjecture of Beluhov

Problem Statement
The paper addresses the asymptotic behavior of the Tchoukaillon array, an infinite array of positive integers $\mathcal{C}_{i,j}$ arising from a one-row Mancala solitaire game. The array is defined such that every positive integer appears exactly once (a bijection $\mathbb{N}^2 \to \mathbb{Z}^+$ ). While the asymptotic behavior of the array's boundaries was previously known—Column 0 corresponds to the Flavius Josephus sieve ( $\mathcal{C}_{i,0} \sim \frac{\pi}{4}i^2$ ) and Row 0 corresponds to the Tchoukaillon numbers ( $\mathcal{C}_{0,j} \sim \frac{1}{\pi}(j+1)^2$ )—the behavior of the general interior entry $\mathcal{C}_{i,j}$ remained an open problem.

N. Beluhov conjectured, based on numerical evidence, that for large $i$ and $j$ , the entries satisfy the approximation:
$\mathcal{C}_{i,j} \approx \frac{(\pi i + 2j)^2}{4\pi}$
The paper aims to prove this conjecture, provide a rigorous error term, and determine the asymptotic shape of the level sets $\{(i,j) : \mathcal{C}_{i,j} \le V\}$ .

Methodology
The proof relies on a "trace" methodology, shifting the perspective from constructing the array to tracking the trajectory of a single integer value $M$ as the order of the array $k$ increases.

Trace Recurrence: The author defines coordinates $(X_k, Y_k)$ for the position of a value $M$ in the array of order $k$ . Starting from $M$ at position $(M-1, 0)$ in order $k=1$ , the value migrates to its final position $(i, j)$ at order $k = N = i+j+1$ . The motion obeys a coupled floor recurrence:
$X_k = X_{k-1} - \left\lfloor \frac{Y_{k-1}}{k} \right\rfloor, \quad Y_k = Y_{k-1} - \left\lfloor \frac{X_{k-1}}{k} \right\rfloor$
This recurrence is symmetric in $X$ and $Y$ , a structural feature crucial for the analysis.
Conjugate Widths: Instead of summing the drops in the recurrence directly, the author introduces "conjugate widths" $R_m$ and $S_m$ , representing the number of steps where the drop amounts exceed a level $m$ . This transforms the problem into summing a "staircase" of drops.
Linearization via Rounding: The recurrence for the widths involves floor functions. The author shows that these can be modeled as a lower-triangular system of linear equations perturbed by nearest-integer rounding errors. By removing the rounding terms, the system decouples into two scalar recurrences driven by the central binomial coefficients.
Wallis Modes: The linearized system yields two distinct modes (eigenvectors) involving the sequences $c_m = \binom{2m}{m}/4^m$ and $w_m = 1/(2m c_m)$ . These sequences are related to the Wallis product for $\pi$ . The constants $\pi$ and $2$ in the final asymptotic formula emerge naturally from these modes, derived solely from the array's recursion rather than from the boundary theorems.
Error Balancing: To recover the value $M$ from the widths, the author employs a "single-level reading" strategy. They balance two competing error sources: the "sandwich" error from reading $M$ at a specific level (which decreases as the level $m$ increases) and the "quantization" error from the rounding of the widths (which increases with $m$ ). The optimal balance occurs at $m \asymp N^{2/3}$ , yielding the final error bound.

Key Contributions and Results

Proof of Beluhov's Conjecture: The paper proves that as $N = i+j+1 \to \infty$ ,
$\mathcal{C}_{i,j} = \frac{(\pi i + 2j + 2)^2}{4\pi} + O(N^{4/3})$
uniformly for all $i, j \ge 0$ . This confirms Beluhov's conjecture (which corresponds to the leading term) and provides an explicit, centered main term.
Square-Root Law: Equivalently, the square root of the entry is asymptotically linear:
$\sqrt{\mathcal{C}_{i,j}} = \frac{\sqrt{\pi}}{2}i + \frac{1}{\sqrt{\pi}}(j+1) + O(N^{1/3})$
This demonstrates that the growth rate in the interior is a linear blend of the growth rates of the two edges (Column 0 and Row 0).
Geometric Shape of Level Sets: As a corollary, the paper establishes that the level regions $\{(i,j) : \mathcal{C}_{i,j} \le V\}$ are asymptotically triangular, bounded by the line $\pi i + 2j = 2\sqrt{\pi V} - 2$ , with a boundary layer of width $O(V^{1/6})$ .
Algorithm for Locating Integers: The trace recurrence leads to an $O(\sqrt{M})$ algorithm to determine the row and column indices $(i, j)$ for a given integer $M$ . This answers a question posed by Knuth regarding the rapid determination of an integer's position in the array.

Significance and Claims
The paper claims to resolve the asymptotic structure of the Tchoukaillon array, a problem that had remained open despite known results for its boundaries. The primary significance lies in:

Deriving Constants from Recursion: The constants $\pi$ and $2$ appearing in the asymptotic formula are shown to arise intrinsically from the array's recursive definition via the Wallis product, independent of the pre-existing edge theorems.
Uniformity: The result provides a uniform estimate across the entire array, bridging the gap between the known edge behaviors.
Methodological Insight: The technique of analyzing the "trace" of a single integer and converting the floor recurrence into a linear system with rounding errors offers a novel approach to similar combinatorial arrays.

The author notes modestly that while the proven error term is $O(N^{4/3})$ , numerical evidence suggests the true error may be $O(N)$ (or even $O(1)$ in the square-root scale). However, the current proof relies on pointwise estimates at a single level and does not yet control the correlations of rounding errors across levels required to improve this bound. The paper leaves the sharper error term as an open problem.

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