The longest uncrossed knight's path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, an open path, which ends on a different field from where it begins; the longest open paths on an nxn board are known only for n ≤ 9. Their lengths for n = 1, 2... 9 are: 0, 0, 2, 5, 10, 17, 24, 35, 47 The longest closed paths are known only for n ≤ 10. Their lengths for n = 1, 2... 10 are: 0, 0, 0, 4, 8, 12, 24, 32, 42, 54 The problem can be further generalized to rectangular n×m boards, or to boards in the shape of any polyomino. Other standard chess pieces than the knight are less interesting, but fairy chess pieces like the camel and zebra lead to problems of comparable complexity. A knight's tour is a self-intersecting knight's path visiting all fields of the board.
TwixT, a board game based on uncrossed knight's paths. L. D. Yarbrough. "Uncrossed knight's tours". Journal of Recreational Mathematics. 1: 140–142. George Jelliss, Non-Intersecting Paths Non-crossing knight tours Uncrossed knight's tours
The Vosges are a range of low mountains in eastern France, near its border with Germany. Together with the Palatine Forest to the north on the German side of the border, they form a single geomorphological unit and low mountain range of around 8,000 km2 in area, it runs in a north-northeast direction from the Burgundian Gate to the Börrstadt Basin, forms the western boundary of the Upper Rhine Plain. The Grand Ballon is the highest peak at 1,424 m, followed by the Storkenkopf, the Hohneck. Geographically, the Vosges Mountains are wholly in France, far above the Col de Saverne separating them from the Palatinate Forest in Germany; the latter area logically continues the same Vosges geologic structure but traditionally receives this different name for historical and political reasons. From 1871 to 1918 the Vosges marked for the most part the border between Germany and France, due to the Franco-Prussian War; the elongated massif is divided south to north into three sections: The Higher Vosges or High Vosges, extending in the southern part of the range from Belfort to the river valley of the Bruche.
The rounded summits of the Hautes Vosges are called ballons in French "balloons". The sandstone Vosges or Middle Vosges, between the Permian Basin of Saint-Die including the Devonian-Dinantian volcanic massif of Schirmeck-Moyenmoutier and the Col de Saverne The Lower Vosges or Low Vosges, a sandstone plateau ranging from 1,000 feet to 1,850 feet high, between the Col de Saverne and the source of the Lauter. In addition, the term "Central Vosges" is used to designate the various lines of summits those above 1,000 m in elevation; the French department of Vosges is named after the range. From a geological point of view, a graben at the beginning of the Paleogene period caused the formation of Alsace and the uplift of the plates of the Vosges, in eastern France, those in the Black Forest, in Germany. From a scientific view, the Vosges Mountains are not mountains as such, but rather the western edge of the unfinished Alsatian graben, stretching continuously as part of the larger Tertiary formations.
Erosive glacial action was the primary catalyst for development of the representative highland massif feature. The Vosges in their southern and central parts are called the Hautes Vosges; these consist of a large Carboniferous mountain eroded just before the Permian Period with gneiss, porphyritic masses or other volcanic intrusions. In the north and west, there are places less eroded by glaciers, here Vosges Triassic and Permian red sandstone remains are found in large beds; the grès vosgien are embedded sometimes up to more than 500 m in thickness. The Lower Vosges in the north are dislocated plates of various sandstones, ranging from 300 to 600 m high; the Vosges is similar to the corresponding range of the Black Forest on the other side of the Rhine: both lie within the same degrees of latitude, have similar geological formations, are characterized by forests on their lower slopes, above which are open pastures and rounded summits of a rather uniform altitude. Both areas exhibit steeper slopes towards the Rhine River and a more gradual descent on the other side.
This occurs because both the Vosges and the Black Forest were formed by isostatic uplift, in a response to the opening of the Rhine Graben. The Rhine Graben is a major extensional basin; when such basins form, the thinning of the crust causes uplift adjacent to the basin. The amount of uplift decreases with distance from the basin, causing the highest range of peaks to be adjacent to the basin, the lower mountains to stretch away from the basin; the highest points are in the Hautes Vosges: the Grand Ballon, in ancient times called Ballon de Guebwiller or Ballon de Murbach, rises to 1,424 m. The Col de Saales, between the Higher and Central Vosges, reaches nearly 579 m, both lower and narrower than the Higher Vosges, with Mont Donon at 1,008 m being the highest point of this Nordic section; the highest mountains and peaks of the Vosges are: Grand Ballon 1,424 m Storkenkopf 1,366 m Hohneck 1,363 m Kastelberg 1,350 m Klintzkopf 1,330 m Rothenbachkopf 1,316 m Lauchenkopf 1,314 m Batteriekopf 1,311 m Haut de Falimont 1,306 m Gazon du Faing 1,306 m Rainkopf 1,305 m Gazon du Faîte 1,303 m Ringbuhl 1,302 m Soultzereneck 1,302 m Le Tanet 1,292 m Petit Ballon 1,272 m Ballon d'Alsace 1,247 m Brézouard 1,229 m Ballon de Servance 1,216 m Drumont 1,200 m Planche des Belles Filles 1,148 m Molkenrain 1,123 m Champ du Feu 1,099 m Baerenkopf 1,074 m Rocher de Mutzig 1,010 m Donon 1,009 m Taennchel 992 m Climont 965 m Hartmannswillerkopf 956 m Chatte Pendue 902 m Ungersberg 901 m Tête