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  X-Cycles
X-Cycles are the start of a large family of 'chaining strategies' which are fundemental to solving the harder sudoku puzzles. X-Cycles are strongly related to Simple Coloring.

A 'chain' is a series of links hopping from one candidate to another following very simple rules. A candidate can either be ON or OFF. That is, we either think it is a possible solution to that cell, or we do not. There are consequences to the rest of the board when we 'link' these two states.
In the X-Wing structure in Figure 1, as an example, we can consider B3 to be OFF. This forces the the 9 in B8 to be ON since it's the only remaining 9 in the row. If B8 is ON then that removes the 9 in H8. Again, the consequence is to turn ON the 9 in H3 - and that closes the loop by forcing the 9 back in B3.

In this X-Wing example three things are apparent. I went round clock-wise, but I could easily make the same logical chain going round from B3 to H3 to H8 to B8 and back to B3. Also, could decide that the start cell was ON and follow the loop round. And lasty, B3 is arbitary, I could start on any of those four cells.
 Figure 1: Nice Loop on 9
Figure 1: Nice Loop on 9
Another important way of thinking about this situation is from the point of view of the links. When a candidate is turned OFF AND there are only two candidates in the unit, then we can create a Strong link, as between B3 and B8 (marked in a thick blue line) and H8 and H3. Because there are just two 9s in each row, we know that if one is not a 9, the other must be. A strong link is where:
!A => B (if not A, then B)

Weak links are the opposite:

A => !B (if A, then not B)

Strong and Weak links alternate just as candidates are turned OFF and ON. When we turn a candidate X ON we effectively remove ALL other canidates of X in ALL other units. However, when we turn a candidate X OFF it has no effect unless the unit has only two of X in it.

A "Cycle", as the name implies, is a loop or joined-up chain of single digits with alternating strong and weak links, as the X-Wing in Figure 1 shows.

In Figure 2, we have a 2-2-2-formation Sword-Fish re-drawn to show the strong and weak links. The loop characterises the X-Cycle, and the strong/weak links alternate.
 Figure 2: Nice Loop on 4
Figure 2: Nice Loop on 4
Nice Loops have evolved a notation which is useful when accompanying a diagram or as part of an explanation. X is, of course, the digit, and we use the row letter and column number notation to identify cells (e.g., B2, F8). The cells in the loop are linked with a minus to indicated X has been turned OFF, and a + to indicated X has been turned ON - you will see these colours on the solver. An example:

+x[cell 1]–x[cell 2]+x[cell 3]–x[cell 4]

The 2-2-2 Sword-Fish above can be expressed as:

+4[B2]–4[B4]+4[H4]–4[H8]+4[F8]–4[F2]

The only thing not explicit in this notation is that the last cell joins back onto the first cell. Thus, 4[F2]+[B2]. The beginning of the loop is arbitrary, but it’s convenient to use the cell closest to the top left. Note that the loops can be of any length but they don't re-use any candidate.
Here is the notation for the 8-Cycle in Figure 3:

-8[B3]+8[B4]–8[C6] +8[D6]–8[F5]+8[F3]–8[B3]

The yellow cells are units where other 8s can be eliminated, which in this case correspond to the third column and boxes 2 and 5 – because that’s where the weak links are located. 		Figure 3: Nice Loop on 8
Figure 3: Nice Loop on 8
Nice Loops Rule 1

Nice Loops that alternate all the way round are said to be 'continuous', and they must have an even number of nodes. With a continuous X-Cycle, candidates are not removed from the loop since the loop does not have any flaws. Instead we are looking to eliminate on the units that can be seen by two or move cells that belong to the loop.
Figure 4 is a real-life example of an X-Cycle based on 8. The cells with links are in red and green. We can immediately see that C2/C7 is a weak link across the row because of the 8 in C3. G2/H3 is also a weak across the box because of the third 8 in G3. The last weak link is in box 9, J7/H9. Any other 8s in these units can be removed, which makes it a powerful technique. We end up with a loop containing only strong links – a result identical to a Colouring (Singles Chains) solution.

The output from the solver will contain the following information:

X-CYCLE (Alternating Inference Chain):
+8[C2]-8[C7]+8[J7]-8[H9]+8[H3]-8[G2]+8[C2]
- Off-chain candidate 8 taken off C3 - weak link: C2 to C7
- Off-chain candidate 8 taken off C9 - weak link: C2 to C7
- Off-chain candidate 8 taken off G9 - weak link: J7 to H9
- Off-chain candidate 8 taken off G3 - weak link: H3 to G2
 Figure 4: Nice Loop on 8
Figure 4: Nice Loop on 8: Load Example or : From the Start
In summary we can see that

  • X-Wing is a Continuous X-Cycle with the length of four.
  • Sword-Fish of the 2-2-2 formation is a Continuous X-Cycle with the length of six.

Although there are many other parallels as well.

This strategy is continued in X_Cycles Part 2 and the theme is continued in Grouped X-Cycles.



 
Comments

Friday 31-Dec-2010

... by: JimS

Laura,

Maybe Andrew answered your question privately or maybe it is posted somewhere else on this site but I thought that I would try to answer your question.

As you follow a continuous, alternating nice loop along the path some of the weak links can actually be strong links -- they just don't have any digits that can be eliminated. Imagine an X Wing which only has digits that can be eliminated in one unit not two.

Andrew might be more accurate if he said "weak or strong links" instead of just "weak links" but that would be cumbersome and wouldn't really help the understanding of the concepts.

Hope this helps.

Friday 12-Feb-2010

... by: Laura

I thought only strong links could have weak interference - not the other way around. I have come across several puzzles in which your solver showed a weak link and it should have been strong- there were only 2 of that number in the square. Is there a special circumstance where this could happen? Please advise thanks

Saturday 11-Apr-2009

... by: ECC

The missing detail in the explanation is that any odd length sequence of strong links counts as a single strong link, except that it cannot close the loop. That is, two consecutive strong links in a loop gives you an answer as described under discontinuous loops, but four does not.

Sunday 5-Apr-2009

... by: Elleda Katan

I'm finding again a problem I have encountered before : what seems like a contradiction between your explanation here and how x-cycles are 'used' in some of the daily puzzles. I'll use 4/5 [todays] as my example but I hit the same problem in 2/28 & elsewhere.
[1] In none of the x-cycle demos is the digit removed from the cells forming the loop. Instead it is eliminated " from cells that can be seen by two or more [loop] cells." In the 4/5 puzzle 7 is eliminated from E1, the beginning and end of the loop.
[2] The end result says your explanation is a loop containing strong links. However, in 4/5, eliminating 7 from E1 destroys the loop.
[3] In the documentation under the 4/5 puzzle, the loop is described as : 7[E1]-7[E4]=7[J4]-7[J3]=7[G1]-7[E1] "Discontinuity is two weak links joined……" However, 7[G1] to 7[E1] is a strong link, not a weak one, no? Shouldn't it read : 7[G1]=7[E1]?
Please I love love love your puzzles and am trying to get smarter at understanding the more advanced strategies, but this confusion has me X-ing out x-cycles from solutions because I am so baffled by them. Thank you for your time.

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Article created on 12-April-2008. Views: 64676
This page was last modified on 29-January-2011, at 11:11.
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Copyright Andrew Stuart @ Syndicated Puzzles Inc, 2011