Asymptotics of the Tchoukaillon array and a conjecture of Beluhov

Abstract: The Tchoukaillon array is an infinite array of the positive integers, arising from a one-row Mancala solitaire, in which each positive integer occurs exactly once. Its zeroth column is the Flavius Josephus sieve and its zeroth row is the sequence of Tchoukaillon numbers; the asymptotics of these two edges are classical results of Andersson and of Broline and Loeb. On the basis of numerical evidence, N. Beluhov conjectured (as relayed by Knuth) that the general entry $T_{i,j}$ satisfies $T_{i,j} \approx (πi+2j)<sup>2/(4π)$ as $i,j \to \infty$. We prove this conjecture. In fact we establish the stronger uniform estimate $$T_{i,j} = (πi+2j+2)^2/(4π) + O((i+j+1)<sup>{4/3})$$, in which both constants $π$ and $2$ are produced by the array's own recursion through a Wallis product, independently of the two edge theorems. Equivalently, the square root of the entry is asymptotically linear, $$\sqrt{T_{i,j}} = (\sqrtπ/2)\, i + (1/\sqrtπ)(j+1) + O((i+j+1)<sup>{1/3})$$, the linear blend of the two edge growth-rates. As corollaries we obtain that the level regions ${T_{i,j} \le V}$ are triangles up to a boundary of width $O(V<sup>{1/6})$, and an $O(\sqrt{M})$ algorithm that locates the row and column of a given integer $M$.

• The paper proves Beluhov's conjecture with explicit uniform error bounds, demonstrating that the Tchoukaillon array entries approximate the form $(rac{\pi i + 2j + 2}{2})^2\frac{1}{2} + O((i+j+1)^{4/3})$ through an exact coupled floor recurrence mechanism.
• A unique method is developed to LEARN about the floor recurrence system without relying on the bijection, monotonicity, or edge-theorem constants, establishing a self-contained analytic treatment of the Tchoukaillon game and array.
• The paper includes a detailed comparison of theoretical results with empirical findings, identifying a gap between numerically suggested O(N) errors and the proven O(N^4/3) that calls for further understanding of the floor-accumulated remainders across levels.

Background and the array

The Tchoukaillon solitaire is a one-row Mancala game in which a legal move at pit $k$ (holding exactly $k$ stones) sows those stones leftward into the store. For each total stone count $s$ there is a unique winning configuration, reachable greedily, so every statistic of "the winning game with $s$ stones" is a well-defined arithmetic function (2608.17517). Knuth organizes the resulting structure into a two-dimensional infinite array $\mathfrak X_{i,j}$ (the Cyrillic Che), defined as the limit of finite-order arrays $\mathfrak X^{(n)}$ governed by an explicit recursion; each entry stabilizes once $n \ge i+j+1$. The array is a bijection $\mathbb N^2 \to \mathbb Z^+$ with strictly increasing rows and columns. Its two edges are classical: column 0 is the Flavius Josephus sieve, with Andersson's estimate $\mathfrak X_{i,0} = \frac{\pi}{4}i^2 + O(i^{4/3})$, and row 0 is the sequence of Tchoukaillon numbers, with Broline and Loeb's $\mathfrak X_{0,j} = (j+1)^2/\pi + O(j+1)$. On numerical evidence, Beluhov conjectured (as recorded by Knuth) the interior law $k$0.

Main result

The paper proves this conjecture with an explicit uniform error term:

$k$1

uniformly over all $k$2, equivalently as the square-root law

$k$3

Two features distinguish this from a routine interpolation between the edge theorems. First, the proof does not use the bijection, monotonicity, or either edge theorem: both constants $k$4 and $k$5 emerge from the recursion itself via central binomial coefficients—a Wallis product. Second, the square-root of each entry is asymptotically linear in $k$6 and $k$7, with slopes equal to the growth rates of the two edges; the density is $k$8 for $k$9, whose square root is the straight-line interpolation between the endpoint rates. This linearity is precisely what forces straight level contours.

Method: tracing one value

Fixing $s$0, the value occupies cell $s$1 at order 1 and migrates to $s$2 at order $s$3. In the coordinates $s$4, $s$5, this migration obeys an exact coupled floor recurrence,

$s$6

whose $s$7 symmetry is structural: it produces conjugate widths, a telescoping two-sided estimate, and a Euclidean-type system amenable to analysis. A near-invariant $s$8 pinned to $s$9 caps $s$0 at $s$1; a matching quadratic lower bound is supplied later from the machinery itself.

The floor is the central obstacle. Removing it entirely collapses the system onto the diagonal $s$2 and predicts $s$3, against the true value $s$4: the accumulated remainders carry the dependence on $s$5. The remedy is a change of variables—summing the drop staircase by rows rather than columns. The conjugate widths $s$6 (last orders at which each coordinate's per-step drop reaches level $s$7) satisfy an exact lower-triangular nearest-integer recurrence,

$s$8

with sharply bounded corrections ($s$9, $\mathfrak X_{i,j}$0). Crucially, the non-local tail sums truncate exactly at $\mathfrak X_{i,j}$1 because the widths interlace ($\mathfrak X_{i,j}$2), so no smoothing hypothesis is needed here.

Dropping only the rounding diagonalizes the linear system into two scalar modes driven by the gap datum $\mathfrak X_{i,j}$3 and the row datum $\mathfrak X_{i,j}$4:

$\mathfrak X_{i,j}$5

An induction shows each true width stays within one cell of this profile at every level—the denominators $\mathfrak X_{i,j}$6 exactly wash out accumulated strays. Feeding the two-sided Wallis bounds on $\mathfrak X_{i,j}$7 gives $\mathfrak X_{i,j}$8, the linear form above.

The exponent and corollaries

Recovering $\mathfrak X_{i,j}$9 by summing all levels would accumulate a nonzero-mean one-cell error per level to $\mathfrak X^{(n)}$0. Instead the paper reads $\mathfrak X^{(n)}$1 off a single clean level using the no-skipping lemma (valid when $\mathfrak X^{(n)}$2, i.e. for $\mathfrak X^{(n)}$3): $\mathfrak X^{(n)}$4. Two errors then compete: the reading sandwich of width $\mathfrak X^{(n)}$5 (decreasing in $\mathfrak X^{(n)}$6) versus the amplified rounding error $\mathfrak X^{(n)}$7 (increasing). They balance at $\mathfrak X^{(n)}$8, both equal to $\mathfrak X^{(n)}$9, yielding the $n \ge i+j+1$0 term—or explicitly $n \ge i+j+1$1 for $n \ge i+j+1$2.

Two corollaries answer questions posed alongside Beluhov's conjecture in Knuth's text. The level regions $n \ge i+j+1$3 are triangles up to a boundary layer of width $n \ge i+j+1$4, extending the $n \ge i+j+1$5 strength of Andersson's counting result uniformly across all directions $n \ge i+j+1$6. And a trace-recurrence run forward locates any integer $n \ge i+j+1$7's cell in $n \ge i+j+1$8 arithmetic operations.

Limitations and open problems

The proven error is far from sharp. Exhaustive computation over the $n \ge i+j+1$9 cells with $\mathbb N^2 \to \mathbb Z^+$0 shows the square-root deviation lies in $\mathbb N^2 \to \mathbb Z^+$1 with mean $\mathbb N^2 \to \mathbb Z^+$2, nearly constant across interior deciles, suggesting the conjectural sharpening $\mathbb N^2 \to \mathbb Z^+$3—the interior analogue of the Erdős–Jabotinsky-to-Broline–Loeb improvement $\mathbb N^2 \to \mathbb Z^+$4 on the row edge. The current argument cannot close this gap: its pointwise single-level reading discards correlations among the rounding remainders $\mathbb N^2 \to \mathbb Z^+$5 across levels, which would need to cancel. Also open are local spacing laws such as $\mathbb N^2 \to \mathbb Z^+$6, which even the sharper edge estimates do not yield since their errors match the spacing scale.

On the edges themselves, the theorem's $\mathbb N^2 \to \mathbb Z^+$7 is weaker than Broline–Loeb's $\mathbb N^2 \to \mathbb Z^+$8 on row 0 and merely matches Andersson on column 0; the contribution is the uniform interior estimate, not an edge improvement.

Conclusion

This paper proves Beluhov's conjecture on the Tchoukaillon array in the strong uniform form $\mathbb N^2 \to \mathbb Z^+$9, deriving both constants from the recursion alone through a Wallis product. The reduction of the problem to a symmetric coupled floor recurrence, its exact reformulation as a nearest-integer width system, and the explicit balance-of-errors mechanism at $\mathfrak X_{i,0} = \frac{\pi}{4}i^2 + O(i^{4/3})$0 constitute a self-contained analytic treatment that simultaneously settles Knuth's questions on contour shape and integer location. The remaining gap between $\mathfrak X_{i,0} = \frac{\pi}{4}i^2 + O(i^{4/3})$1 and the numerically indicated $\mathfrak X_{i,0} = \frac{\pi}{4}i^2 + O(i^{4/3})$2 is the natural next target, requiring control of floor-residue cancellation not achieved here.