数独高级技巧 X环(x-cycle)

数独高级技巧 X环(x-cycle)

原贴来源http://www.sudoku.com/forums/viewtopic.php?t=2752

翻译有误的地方，大家要热情的给我提出来哈，以我的英文水平，难免有错误。不过先自己给自己加个精，鼓励鼓励。

金山词霸伺候，先选自己能翻的先。

An x-cycle is a cycle in which all cells are linked by a single digit 'x'. It can be of any lengt，h. It can be continuous or discontinuous.

x-cycle is equivalent to all current techniques which operate on a filtered digit such as:

一个X环，由一个单一数字“X”连接而成，可以是任意长度的环，可以是连续和不连续的。

Due to its simplicity, like xy-chain, x-cycle is a subset of nice loops that can be identified without the need of a bilocation/bivalue plot.

In an x-cycle, nice loop propagation always follows alternate links with 'strong inference' and 'weak inference' (refer definitions of 'link', 'strong link', 'strong inference' and 'weak inference' here), eg.

在X环，环之间的连接由“强关联”和“弱关联”交替构成。<译者注：这个很关键，是交替构成>

......[cell 1]-x-[cell 2]=x=[cell 3]-x-[cell 4]=x=[cell 5]-x-[cell 6]=x=[cell 7]-x-[cell 8].........

where:
the notation '=x=' is a link with a "strong inference" (+ve label)
the notation '-x-' is a link with a "weak inference" (-ve label)

强关联用‘=X=’符号表示
弱关联用‘-x-’符号表

In an x-cycle nice loop, if the links propagate alternately in a cyclic manner (ie. no adjacent links are of same type), the loop is said to be 'continuous'.

在X环，环之间的连接由“强关联”和“弱关联”交替构成，不存在邻近的两个连接是相同类型的，我们称这是连续的环，反之，邻近的两个连接是同类型的，称为不连续的环。

With continuous x-cycle, candidates can be eliminated outside the loop but within the unit of the links with weak inference (broken line) as demonstrated below:Nice loop notation:

在连续的X环，可以直接排除在弱连接上（虚线表示）上，在这个循环之外的其它候选数。下面是三个例子，灰色部分都可以排除对应的候选数。<译者注：下面的的符号[r2c6]，r表示第2行，第6列，虚线表示弱关联，实线表示强关联>
example 1 - x-wing: -[r2c2]=5=[r2c6]-5-[r8c6]=5=[r8c2]-5-[r2c2]=
example 2 - swordfish of 222 formation: =[r2c2]-8-[r2c4]=8=[r9c4]-8-[r9c8]=8=[r5c8]-8-[r5c2]=8=[r2c2]-
example 3: =[r2c1]-3-[r9c1]=3=[r7c3]-3-[r7c8]=3=[r5c8]-3-[r5c6]=3=[r3c6]-3-[r2c4]=3=[r2c1]-

A 'discontinuous' x-cycle nice loop has exactly one discontinuity between 2 adjacent links of the same type (ie. both links with strong inference or both links with weak inference).
在X环，环之间的连接由“强关联”和“弱关联”交替构成，不存在邻近的两个连接是相同类型的，我们称这是连续的环，反之，邻近的两个连接是同类型的，称为不连续的环。不连续的连接只允许出现一次。

If the adjacent links are links with strong inference (solid line), a candidate can be fixed in the node at the discontinuity, as demonstrated below:

If the adjacent links are links with weak inference (broken line), a candidate can be eliminated from the node at the discontinuity, as demonstrated below:

不连续的X环分两种情况：

1：不连续的部分由强关联（实线）构成，节点（两个连接相交的点）上的数字可以确定就是对应的数字。

2：不连续的部分由弱关联（虚线）构成，节点（两个连接相交的点）上的数字可以排除对应的数字（与上相反）。

Nice loop notation always starts from the discontinuity:
example 4 - turbot fish: [r8c8]=7=[r3c8]-7-[r3c1]=7=[r1c3]-7-[r8c3]=7=[r8c8] => r8c8=7
example 5 - turbot fish: [r3c7]-2-[r3c2]=2=[r6c2]-2-[r6c9]=2=[r2c9]-2-[r3c7] => r3c7<>2
example 6: [r4c8]-4-[r7c8]=4=[r7c1]-4-[r3c1]=4=[r3c6]-4-[r1c4]=4=[r4c4]-4-[r4c8] => r4c8<>4

Demonstrated below is the identification process for x-cycles. From the filtered grid of candidate '6', strong links (shown as double lines, one solid and one broken representing strong and weak inferences respectively) and a link (shown as broken line representing weak inference) are drawn for the '*6'-cells as shown.
下面这个环不同与前面的概念，由4个强关联和1个弱关联构成，不属于X环。但可当成X环特例。在弱关联的两个点上排除数字"6"

下面举了一些例子，并对特殊的X环进行命名，不太符合中国习惯<还是我们自己来命名吧>。
An x-cycle has alternate links with strong and weak inferences. Since a strong link has both strong inference and weak inference, the following 5 cases of x-cycle can be identified. The weak inferences selected from strong links are shown in red for clarity.

x-cycle 1 & 2 are equivalent to simple colouring.

x-cycle 3, 4 & 5 are equivalent to turbot fish.

An x-cycle can be proven by double implications. Consider x-cycle No.5 above:

Nice loop notation: [r3c4]=6=[r3c7]-6-[r1c9]=6=[r6c9]-6-[r6c4]=6=[r3c4] => r3c4=6

Proof (Implications can start from any node in the nice loop):
r1c9=6 => r3c7<>6 => r3c4=6
r1c9<>6 => r6c9=6 => r6c4<>6 => r3c4=6
Therefore r3c4=6

这是怎么推导出我们对于X环结论的一个证明：

X环：[r3c4]=6=[r3c7]-6-[r1c9]=6=[r6c9]-6-[r6c4]=6=[r3c4] => r3c4=6
由 r1c9=6 => r3c7<>6 => r3c4=6
由 r1c9<>6 => r6c9=6 => r6c4<>6 => r3c4=6
所以 r3c4=6