The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center squ...The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or when square (1, 3), (2, 4), (3, 3), (1, n – 2), (2, n – 3), or (3, n – 2) is removed.展开更多
通过分析欧拉所给出Knight’s Tour Problem的解法,结合哈密尔顿路和哈密尔顿圈的相关知识,得出其解法对应着二部图中的一条哈密尔顿圈.由此再充分利用8×8棋盘所对应的8×8表格的对称性及同格图的特性,对欧拉所给出的Knight’s...通过分析欧拉所给出Knight’s Tour Problem的解法,结合哈密尔顿路和哈密尔顿圈的相关知识,得出其解法对应着二部图中的一条哈密尔顿圈.由此再充分利用8×8棋盘所对应的8×8表格的对称性及同格图的特性,对欧拉所给出的Knight’s Tour Problem的解法作了进一步的探讨,得出了以欧拉的解法为基础的以任一棋格为骑士周游起点的另外一系列解法.最后,把Knight’sTour Problem推广到m×n棋盘上,考虑到移动规则的特殊性,利用图论的相关知识,得到3×4,8×16和16×16棋盘上的Knight’s Tour Problem的解法,同时给出8m×8n(m>2,n>2)棋盘上Knight’s Tour Problem的猜想.展开更多
Using geometric techniques, formulas for the number of squares that require k moves in order to be reached by a sole knight from its initial position on an infinite chessboard are derived. The number of squares reacha...Using geometric techniques, formulas for the number of squares that require k moves in order to be reached by a sole knight from its initial position on an infinite chessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and 28k – 20 for k ≥ 5. The cumulative number of squares reachable in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k2 – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more mathematically accessible then preceding proofs.展开更多
COTTON USA’s Supply Chain Marketing Buyers Tour will be held in China from 7th to 11th June.Cotton Council International’s("CCI")representative offices in the U.S.,Europe and North East Asia have recruited...COTTON USA’s Supply Chain Marketing Buyers Tour will be held in China from 7th to 11th June.Cotton Council International’s("CCI")representative offices in the U.S.,Europe and North East Asia have recruited nearly展开更多
■The world’s most prestigious art shows teamed up in Europe this spring to create a European art tour that offered an unparalleled opportunity to examine contemporary art from around the world.It was an endless expe...■The world’s most prestigious art shows teamed up in Europe this spring to create a European art tour that offered an unparalleled opportunity to examine contemporary art from around the world.It was an endless experience of展开更多
The security of critical data is an important issue for distributed storage system design,especially for long-term storage.ESSA (An Efficient and Secure Splitting Algorithm for Distributed Storage Systems) is presente...The security of critical data is an important issue for distributed storage system design,especially for long-term storage.ESSA (An Efficient and Secure Splitting Algorithm for Distributed Storage Systems) is presented,which takes advantage of a two level information dispersal scheme to strengthen the security of data.In ESSA,the approach of knight’s tour problem,which is NP-Complete,is introduced to scramble data at the first level,and a split cube is used to split scrambled data at the second level.Thus,it is very difficult for the malicious user to get information because the task of reconstructing the original data needs more computation than they can tolerate.We prove that the security of ESSA is better than encryption algorithm and not inferior to erasure codes and secret sharing.Experimental results show that distributed storage systems exploiting ESSA has greater efficiency than that exploiting keyed encryption,erasure codes and secret sharing.展开更多
文摘The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or when square (1, 3), (2, 4), (3, 3), (1, n – 2), (2, n – 3), or (3, n – 2) is removed.
文摘通过分析欧拉所给出Knight’s Tour Problem的解法,结合哈密尔顿路和哈密尔顿圈的相关知识,得出其解法对应着二部图中的一条哈密尔顿圈.由此再充分利用8×8棋盘所对应的8×8表格的对称性及同格图的特性,对欧拉所给出的Knight’s Tour Problem的解法作了进一步的探讨,得出了以欧拉的解法为基础的以任一棋格为骑士周游起点的另外一系列解法.最后,把Knight’sTour Problem推广到m×n棋盘上,考虑到移动规则的特殊性,利用图论的相关知识,得到3×4,8×16和16×16棋盘上的Knight’s Tour Problem的解法,同时给出8m×8n(m>2,n>2)棋盘上Knight’s Tour Problem的猜想.
文摘Using geometric techniques, formulas for the number of squares that require k moves in order to be reached by a sole knight from its initial position on an infinite chessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and 28k – 20 for k ≥ 5. The cumulative number of squares reachable in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k2 – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more mathematically accessible then preceding proofs.
文摘COTTON USA’s Supply Chain Marketing Buyers Tour will be held in China from 7th to 11th June.Cotton Council International’s("CCI")representative offices in the U.S.,Europe and North East Asia have recruited nearly
文摘■The world’s most prestigious art shows teamed up in Europe this spring to create a European art tour that offered an unparalleled opportunity to examine contemporary art from around the world.It was an endless experience of
基金This study is supported by National Natural Science Foundation of China （No.60973146） National Natur al Science Foundation of Beijing （No.4092029） The Fundamental Research Funds for the Central Universities （No.2009RC0217）. We also thank the anonymous reviewers for their constructive comments.
文摘The security of critical data is an important issue for distributed storage system design,especially for long-term storage.ESSA (An Efficient and Secure Splitting Algorithm for Distributed Storage Systems) is presented,which takes advantage of a two level information dispersal scheme to strengthen the security of data.In ESSA,the approach of knight’s tour problem,which is NP-Complete,is introduced to scramble data at the first level,and a split cube is used to split scrambled data at the second level.Thus,it is very difficult for the malicious user to get information because the task of reconstructing the original data needs more computation than they can tolerate.We prove that the security of ESSA is better than encryption algorithm and not inferior to erasure codes and secret sharing.Experimental results show that distributed storage systems exploiting ESSA has greater efficiency than that exploiting keyed encryption,erasure codes and secret sharing.
