魔法骑士在更高维度的旅行
Magic Knight's Tours in Higher Dimensions
摘要：
棋盘上的骑士之旅是一系列的骑士移动，每一个方块只走一次。如果每一行和每一列的数字之和相同（魔术常数），则在正方形棋盘上的骑士之旅称为魔术骑士之旅。高维（n > 3）骑士之旅是古老的骑士之旅领域中的一个新课题。本文证明了奇阶n维超立方体中不可能存在幻骑士环和闭骑士环。3 \times 4 \times 2n-2是最小的立方体（n \geq 2），4 \times 4 \times 4 n-2是n维中可以进行骑士之旅的最小立方体（n \geq 3）。魔法骑士的游览可以在4 \times 4 \times 4 \times 4和4 \times 4 \times 4 \times 4 \times 4超立方体中进行。
英文摘要：
A knight's tour on a board is a sequence of knight moves that visits each square exactly once. A knight's tour on a square board is called magic knight's tour if the sum of the numbers in each row and column is the same (magic constant). Knight's tour in higher dimensions (n > 3) is a new topic in the age-old world of knight's tours. In this paper, it has been proved that there can't be magic knight's tour or closed knight's tour in an odd order n-dimensional hypercube. 3 \times 4 \times 2n-2 is the smallest cuboid (n \geq 2) and 4 \times 4 \times 4n-2 is the smallest cube in which knight's tour is possible in n-dimensions (n \geq 3). Magic knight's tours are possible in 4 \times 4 \times 4 \times 4 and 4 \times 4 \times 4 \times 4 \times 4 hypercube.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
