## Hidden Logic Box/Line Interactions

This post looks at the brief and very late THLS treatment of box/line interactions in rc space. I say late, because box/lines (1/31/12) are essential in basic marking of candidates. But in The Hidden Logic of Sudoku,  box/lines are introduced after the application of symmetries to regular fish. Fish require a full set of candidates, being the expression of competition among candidates for positions in the solution.  This means Berthier has to use box/lines in THLS elaborations before explaining what they are.

Checkpoints for Akron Sudoku 16 participants are at the end of this post, along with a blog feature announcement.

Before we talk box/lines, however, let’s checkpoint the three hidden subsets which, after the mysteriously appearing 8r3c3, devastate 17-13727.  SudoRules first finds the hidden pair NWhp56, which you found in the bypass, and removes 3, 4 and 7 candidates.  Then it finds the hidden pair r2hp56, removing candidates 4, 7, 8, and 9 from r2c9. Without this help, it would still find the hidden triple NEht569.

As is usual, you probably saw the naked quad NE3478 first, but that would be a higher level of logical complexity for SudoRules. More normally, you would be reviewing boxes for subsets immediately after the third crossing line is marked. This example illustrates how they can jump out earlier. Either way, it’s much easier to deal with them when only the relevant candidates are present.

Now let’s turn to box/line interactions, and what THLS says about them.  On the positive side, Berthier covers the basics in Chapter IX with four rules, two for rows and two identical ones for columns.  I can best explain them with a chute schematic.

I call the intersection of a box and line a chute, and the remaining cells of each unit outside the chute, remainders. There is a line remainder and a box remainder for every chute.

SudoRules first two box/line rules RiB and CiB (Row and Column) state that if there are no candidates of a number in a line remainder, then candidates of the number can be removed from the box remainder. It’s obvious why, but we can miss this by failing to spot the empty line remainder. The follow up process of line marking catches this mistake, as we look for line slinks to mark.

The other two box/line rules BiR and BiC state the opposite. If there are no candidates of a number in the box remainder, then candidates of the number can be removed from the line remainder.  In Sysudoku basic, it is box marking where this kind of box/line is detected. We find the aligned slinks and triples that are only possibilities for candidates missing from the box remainder, and use them to sweep neighboring boxes along the line.

Reading through a SudoRules trace, we see this kind of step frequently:

On the Sysudoku grid, we see the 3-slink or aligned triple, and know that the elimination took place.

Clearly, what we mark as a box/line in a trace (bxl) is an instance where the last candidate of a number is removed from one of the remainders. This little comparison of traces reveals a serious deficiency in the wordy SudoRules trace of THLS. The pictured step is followed by others, but we have no idea which of the later steps are dependent on this one.

Just so you know, when these four box/line  interaction rules are added, rule set L1-0 becomes L1.

Berthier suggests one of his hidden symmetries “for detecting” conditions for each of his four box/line rules. For example, for the RiB box/line, in which the absence of a line remainder candidate removes candidates in the box, he suggests rn-space.

The chute contains a true candidate, in a slink or an aligned triple. In rn-space all candidates of a number will map into a single column. Cells of that column list the row locations of those candidates. In the chute row, the cell of the number column will contain only the numbers of columns crossing the box.  Although box divisions are preserved in a limited way, I don’t agree that scanning down a column looking for the exclusive presence of the “right”  range of numbers is anything like as easy as spotting the mandatory slink or aligned triple above in standard rc -space.

I have the identical objection to using rn-space for spotting CiB box/lines. And use of bn-space for spotting BiR and BiC box/lines, the deployment of slinks and aligned triples, is even more of a stretch.

With his box/line interaction rules, Berthier completes his coverage of the elements of Sysudoku basic solving, but without any reference to strong links, weak links, or bi-values cells.  He has leaped ahead into advanced solving with regular fish, in a hidden logic manner I review in the next post.  This blurring of any distinction between basic and advanced solving goes right along with the joyless practice of number scanning candidates before any logic is applied. From the human engineering point of view, a very bad idea.

Before taking leave of THLS basic solving, I would like for you to see what actually happens to many a hidden subset facing the systematic basic solving offered here. The interaction chapter traces two solutions of Royle 17-32227, one with and one without, box/line interaction. The sysudokie way is better than either.

See if you can get there first.

Now for the final bulletin from the Akron Sudoku Tournament of 2016:

Here is the grid of Akron Champion 2016 as the collapse is beginning. Line marking has reached the last line, row 4, where a naked triple 178 wipes out five candidates, and supplies candidates r178r5c5 for another naked triple in column 5, taking out three more. The solution follows immediately.

The line marking sequence by increasing free cells, and the beginning collapse, can be read from this trace:

Did our stage performers solve this puzzle?  Most likely, it was a matter of having the most cells right, but the Akron-Summit County Library will never tell.

FYI, the Round 3 puzzle is solved by the Sysudoku bypass as follows:

It takes a quick brain, and a quick hand, to do it in 20 minutes.  It could be very instructive to follow the above to a solution without a single guess. You just have to be willing to dig out the reason for each step. No excuses, please. Its all on the Traces page.  Enjoy!

Happy Thanksgiving!