We will now be leaving methods relying on the bv map as the primary visual aid. With puzzles that survive these methods, we need to remember that the bv map file remains an important resource. With every clue or candidate removal, we should update the bv map along with the grid, and look for consequences of these updates.
The next weapon we grab, the X-panel, is a set of 9 tables, one for each number, showing the candidates, and only the candidates for that number. It will help us find X-chains and fish of all kinds, and will guide us in the invocation of Medusa coloring, the heavy artillery of Sudoku solving. Puzzles that succumb to the X-panel analysis we will honor as “challenging”. To show what the X-panel looks like, here is one from a Washington Post six star, of 6/4/10 encountered on a family visit.
The X-Panel template file is available by request to firstname.lastname@example.org , along with the puzzle template and bv map template files offered earlier. It is a ©Word file. There is an extra set of panels at the bottom (not shown here), which is handy for analysis and updating as solving continues.
I use X-panels for X-chains and fishing. We will consider these in separate posts, but actually both analyses are done on a panel as soon as it is completed. Fishing is the more extensive subject, so we’ll begin with the X-chain, a simple form of AIC, and therefore somewhat familiar territory for readers of this blog, right?
An X-chain is a series of linked candidates of a single number X. Winks and slinks play the same role as they do in other forms of Alternate Inference Chain (AIC). We saw that a weakly linked series of bv naturally forms the alternating slink/wink series of an AIC, the mandatory slinks being between the bv partner candidates. In an X-chain, the solver identifies the alternating links. The schematic picture we used to explain the inference propagation of an AIC is an exact representation of an X-chain.
Starting with a strong link, if the starting end candidate is false, then even candidates (outgoing wink) are true. Each even candidate forms a toxic set with the starting candidate, including the end candidate. Either the starting candidate is true, or the even candidate is. Yes, a slink is an X-chain all by itself. A candidate will need ER (or another dirty trick we’ll see later) to see both ends of a slink, but it does happen.
The odd links of an X-chain must be slinks, but the even links are not required to be winks. You may want to draw them as winks to show that they have the role of winks in the chain. You can restate the alternation requirement this way: any winks are even links.
That’s all the X-chain theory you need. They breed toxic sets like rabbits, if not exactly like XY-chains do. I guess you’re wondering why I waited this long. Its because explaining them is simple, but finding all of them is something else. From the Post X-panels above, its clear that the panels show you any slinks you may have missed. I find it helpful to sketch in the slinks on the panel, then look for connecting links of both kinds.
Don’t forget the ER winks. They are particularly easy to spot in an X-panel. Note the candidates seen by the candidate in r7c9 of the 3-panel. There are even more winks to be found when you desperately need one. For example, in the 2-panel, if the candidate in r3c1 is true, it forces the candidate in r6c2 to be false. That is the definition of a wink. This particular inference travels on two independent paths. There are enough of these chaining winks to clutter our panels, and most them are useless, so we only seek them out when we need them.
Fortunately with the MS ©Word and ©Powerpoint common drawing tools, you can place lines and dashed lines or curves on the panels to help you find X-chains, and to document those found.
There is an X-chain removal waiting for you in the Washington Post X-panels. You won’t need it, but I’ll put my marked up panel in the next post. The template of individual panel I use is in the panel file. It’s almost the same as the bv map.