Did Lagrange make any noteworthy contribution to the Knights Tour Problem?
It does not seem like there was much of a contribution, if any Lagrange is sometimes mentioned, without reference, e.g. by Borell, along with Taylor and de Moivre, as one who "worked" on it. But while Euler's work is called "serious" and "influential" not much else is said about the other three. A detailed history in Rediscovery of the Knight's Problem by Jelliss questions Taylor's involvement and does not even mention Lagrange. Neither does Rouse Ball who has a section on it in Mathematical Recreations and Essays. He does mention Legendre, and I wonder if there was some name conflation. From Jeliss:
"The modern study of the knight's problem appears to have begun in the 18th century without knowledge of the mediaeval work, save perhaps for the half-board tour in Guarini's work. The subject first reappeared in Jacques Ozanam's Récréations Mathématiques et Physiques... The first edition of Ozanam's work was published in 1694 but (according to one of the later editors, C. Hutton) Ozanam died in 1717.
The tours were supplied to the editor of that edition of Ozanam in 1722 by de Mairan, who was Director of the Académie Royale des Sciences, Paris. This date is printed in the margin of the book. Rouse Ball (1939) says the tours by de Montmort and de Moivre “were sent by their authors to Brook Taylor who seems to have previously suggested the problem”; unfortunately he gives no reference to where he learnt of the involvement of Taylor (another well known mathematician) and I have been unable to verify this. Similarly Kraitchik attaches the date 1708 to the de Montmort tour, but it does not appear in de Montmort's famous work l'Essai d'analyse sur les jeux de hasard, Paris 1708. A slight variation of the de Moivre tour in which the last three moves are reflected is mentioned in the text and is sometimes diagrammed in later accounts.
The first mathematical paper analysing knight's tours was presented by the most productive mathematician of the eighteenth century, Leonhard Euler (1707–1783), to the Academy of Sciences at Berlin in 1759 (but not printed until 1766). This paper has been of considerable influence, the starting point (but often alas the end also) of many later accounts of the subject."