## Aligned Pair Exclusion (APE)

Happy New Year 2012!  I’m happy to be sharing my discoveries in this blog, and I can promise a more innovations in 2012. It’s about how we can best use our visual and mental abilities to solve difficult Sudoku puzzles.  Not a world changing enterprise, but dear to some hearts, and good mental exercise for all.  If you are just arriving, please glance at the beginner’s page, and start with the September posts to understand the blog at this point.

While you are looking for bv and other ALS in Sue de Coq box remainders and line remainders, it’s the ideal time to find candidate removals via a technique known as APE.  APE stands for Aligned Pair Exclusion, the “pair” being a pair of aligned cells seeing  several ALS that restrict the contents of the pair.

The APE method determines the pairs of numbers permitted in the solution of  two aligned cells by the aligned bv (or other ALS) they “see”.  Potential Sue de Coq chutes make ideal aligned pairs because all ALS in the remainder see both aligned cells.  In Sue de Coq, ALS that share numbers are less effective, but in APE, they are very welcome.

My favorite example comes from a doggedly difficult puzzle I encountered in the US Air Hemispheres magazine in July 2009.

The potential SdC Er4 sees many bv sharing 3. Here this is great. Neither 2, 4, nor 9 is permitted to pair with 3r4c7 in the solution, but cell r4c9 has to have something, don’t you think?  The only way out is to remove the 3-candidate.

That makes a good story, but it’s not a systematic basis for doing APE. When you suspect an APE, just list out all possible pairs and then scratch out all of those prohibited by the onlooking ALS. Here, that looks like

APE r4c79 = 12+14+19+32+34+39+42+49+92+94.

Now look to see if any of the left cell’s candidates are totally scratched out. In the above, it’s 3. Then do the same for the right cell.  Here r4c9 gets to keep 2, 4 and 9, which all appear unscratched on the right. With a little practice and good hand and eye coordination, you can simply leave the prohibited pairs off the list.  The 3 never shows up.

After the APE removal, the ALS 349r4c24 and 123 r56c89 create a two alternate SdC Er4 = 7(1+2)(4+9), removing 9r4c1 and 2r5c9.

APE r9c89 above shows how the ALS restrictions can be applied in another way. The 349 ALS r78c7 excludes 34 and 39, and commits the puzzle to either 8r9c7 or SEr9 = 537. Naturally, we try the latter, more specific, alternative. Follow the arrows to see how it fails, forcing two 8’s in r7, giving us the clue 8r9c8.

That’s not exactly the elimination anticipated in an Aligned Pair Elimination. I think of it as a trial generator, and call this application of APE an APT.

You might be thinking that it was hard to find the APT contradiction above, so I’d better come clean and admit that I don’t spend the time to find such things either. The sysudokie secret weapon is the trial trace described on the traces page.

Working from the trace, the trial can be  displayed  graphically, with arrows to show “implies”, solid lines for slinks and dotted for winks. You can show the inference path from SE3 to SW8 and S8, leaving out the rest.

The trial trace is the tool for verification of most single alternate Sue de Coq as well. As you see above, it is very much like the regular 2-D trace used in basic solving. But the 2-D trace is depth first, finding all the results of a list item before exploring the next. The trial trace is breadth first, exploring entire lists, but level by level, down the trace page.   When you reach a contradiction, you can then apply arrows to document the simplest path to the contradiction, if there is a contradiction.

Ready to try an SdC trial trace? OK, with the full grid above, apply the APE elimination and the APT clue. Then the APE chute becomes a classic Sue de Coq removing two candidates. Now do a trial trace on the Wr5 chute containing the naked pair np57, first identifying what Sue de Coq you are verifying.

Returning to APE, a weakness is that removals are not guaranteed.   You can expect many unsuccessful trials.   A strength is the ease of confirming a removal. Failed candidates for SdC are good prospects for the full APE, or at least an APT.

In the original order of posts, next week’s reveals more about this USair puzzle, and if you’re going through the posts by date, and want to do the basic solving, start with the straight up Calibri script numbers in the full grid above as the given clues.

Those who skipped over the bv map posts to follow the Sysudoku Order of Battle should now return to the November 15, 2011 post, The BV map and Regular XYZ-wings. I have a different assignment for you in that case. It’s another airlines puzzle, from Delta Sky magazine of June 2008. Do the basic solving and we’ll use it for several assignments. I’ll have checkpoint traces.

## About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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### 3 Responses to Aligned Pair Exclusion (APE)

1. ghfick says:

I decided to search out the puzzle from Hemispheres magazine. I found an online version. The puzzle is indeed on p124 in the July 2009 issue. The contentious puzzle is listed as ‘Hard’. It is described as from ‘Digital Underground’ by ‘Mitch Rose’.
I then entered the puzzle in Andrew Stuart’s site to check the ‘Solution Count’. Stuart’s counter displays 10 ‘solutions’. You can then follow the solver through to the ‘end’. The solver shows you all the remaining options after exhausting all techniques [except those based on uniqueness].
I then checked out the ‘solution’ given in the magazine. I then conducted a rather unstructured search for puzzles that would yield this solution and not be an ‘easy’ puzzle. There appears to be only one option. That option is that R9C6 would be 3. I can now speculate that the ‘3’ missing is a typo.
With this option added, the puzzle fits the description of ‘Hard’ in that all ‘solution paths’ with this puzzle need hard strategies. One path requires a Swordfish, a 2 String Kite, and a BUG+1. Of course there are many other paths. If indeed this typo is the issue, then this puzzle would be comparable with a puzzle that might appear in a newspaper.

• Sudent says:

Impressive work! I remember saying one of my sysudokie friends would set me straight. thanks for a fine contribution.
I think your typo theory is the most likely. At every opportunity I repeat for newbies the point that “solving” by baseless guessing tells the solver nothing. It doesn’t detect multiple solutions.

On multiple solutions, everyone: In the next post, “Trust But Verify” of 1/10/11, a Sue de Coq verification trial lifts the cover and reveals three solutions of Hemisphere 7/09. Later in the blog, after covering Medusa coloring, I had the means to track down multiple solutions in several collections by systematic coloring trials. Navigate to the collection reviews with “Find It”.

I think your typo theory is the most likely. At every opportunity I repeat for newbies the point that “solving” by baseless guessing tells the solver nothing. It doesn’t detect multiple solutions.

2. เพิ่มไลค์ says:

Increɗible points. Outstanding aгgumеnts. Keep up the great spirit.