McKay and Wanless completed the exact enumeration through order 11. Their paper gives \(R_{11}=5363937773277371298119673540771840\) and says that the same graph-classification method was unlikely to reach \(R_{12}\) soon because there are more than \(10^{11}\) regular bipartite graphs of order 24 and degree 6.

McKay and Rogoyski had earlier reported a randomized estimate for order 12. Stones's 2010 survey explicitly says that \(R_{12}\) is unknown and lists the exact sequence only through order 11. A search on 2026-07-25 of the exact-count papers, the later order-11 class enumeration, the survey literature, and the current author-maintained count table found no subsequent exact value. The candidate therefore remains an open exact-enumeration problem.

For order \(n\), every reduced Latin square has exactly \(n!(n-1)!\) labelled forms, so \[ L_n=n!(n-1)!R_n. \] McKay and Wanless give \[ R_{11}=5363937773277371298119673540771840 \] and \[ L_{11}=776966836171770144107444346734230682311065600000. \] The companion replay checks the multiplication exactly. This calibration prevents the common error of substituting a count of labelled squares, isotopy classes, or main classes for the reduced count requested by the candidate.

The standard van der Waerden permanent bound quoted by McKay and Wanless gives \[ L_n\ge\frac{(n!)^{2n}}{n^{n^2}}. \] After dividing by \(12!11!\), the order-12 reduced count is at least the ceiling of the resulting rational number: \[ R_{12}\ge4410573106297854867267286926659737034. \] Their Theorem 2 also proves that \(R_{2m}\) is divisible by \(m!\). Thus \(R_{12}\) is divisible by \(6!=720\), and rounding the preceding lower bound to the next multiple gives \[ R_{12}\ge4410573106297854867267286926659737440. \] For an elementary upper bound, fix the first row and column. Each of the remaining 11 rows has at most \(11!\) possible fillings once its first entry is fixed. Ignoring the column restrictions can only increase the count, so \[ R_{12}\le(11!)^{11}=409933016554924328182440935903164918932547530146724293451448320000000000000000000000. \] The published estimate \(1.62\times10^{44}\) lies inside this broad rigorous interval.

McKay and Rogoyski generated a normalized Latin square row by row. At each step they chose uniformly among legal extensions with the required first-column entry. If \(e_i\) is the number of legal choices at step \(i\), then \(e_1e_2\cdots e_{n-1}\) is an unbiased estimator of \(R_n\). Their Table 2 reports 1,100,000 trials for \(n=12\) and the estimate \[ R_{12}\mathrel{\approx}1.62\times10^{44}. \] The authors described the displayed digits as probably accurate to one unit in the last shown digit. This value is statistical evidence. It is neither an exact integer nor a rigorous confidence interval.

The row-by-row backtracker fixes the natural first row and first column. Bit masks enforce that every symbol occurs once per row and once per column. A minimum-remaining-values choice selects the next column inside each row. The complete searches reproduce \(1,1,4,56,9408\) for orders 2 through 6.

The same program checks \(L_{11}=11!10!R_{11}\), evaluates the van der Waerden rational lower bound with `Fraction`, rounds it upward, applies the published factor \(6!\), and computes \((11!)^{11}\). Its complete standard output has SHA-256 digest `8c167454c650bc023b4ea8f3c237fa06fb595ede7def12744d31b4e89ebc8696`.