## The Complex 1-Way in AIC Building

This post is about an AIC building option offered by Philip Beeby’s PhilsFolly.com solver, and how and where it is best applied in DIY solving.  Here we refer to this solver as Beeby and this technique as complex 1-way

Complex 1-way is an extended form of what Beeby’s move notes call a Simple Discontinuous Loop. I call it a simple 1-way, or a 1-way, because it is based on an AIC applied in one direction. We start the AIC on a slink partner candidate that sees candidates of the same value, the potential victims.

Let’s look at simple 1-way from  Heine’s ultrahardcore 3. The AIC starts on 4r4c3. We’re in AIC building, just extending the AIC as far as it goes, by any means, while watching for any ending that would make any potential victim false. That happens here when a slink ends in 4r4c4’s cell on a different value, 9.

Why is 4r4c4 removed? The starting starting candidate is assumed false and every slink ending candidate is assumed true, 9r4c4 among them. The AIC makes 4r4c4 a victim. But if instead, the starting value 4r4c3 is true, 4r4c4 can’t be.

We put arrows on the slinks to mark that the AIC is only being applied in one direction. Actually, in the simple 1-way, the AIC is good in both directions. Starting at 9r4c4, this one is a boomerang, with 4r4c3 providing the closing wink. In the complex 1-way, however, we take advantage of the logical inference moving in one direction only.

Now let’s look at a simple case of the complex 1-way from UHC 3. The starting green 3 sees the victim 3r4c3. We can shortcut the chain and go directly to blue 7r6c9, where  a branch wink removes 7r4c8 under the 1-way assumption.

The branch wink limits the AIC to a single direction. Each one helps clear a slink for the  AIC  to continue.

In this case, it’s the 7 slink that allows the 1-way AIC to choose 7r4c3 over 3.

How do we spot this complex 1-way? We get to 7r6c9 and look for the next wink to slink combination to continue the chain. Then we see the branch wink creating a slink in r4, and that 3r4c3 will be removed.

In another example from UHC 3,  we may may see the c2 slink created by the two branch winks as an extension of the AIC before seeing the r1 slink  that removes 7r1c3.

The complex 1-way is not limited to a simple branch off. In this one from UHC 91, we are three slinks in, and maybe can see that last slink into r8c3. Then looking back from  2r5c9, we could see the slink off to 2r5c1 getting the fourth 1-way slink.

Now from UHC 135 we get to one of those examples that had me wanting to place the complex 1-way after ALS mapping in the Sysudoku order of battle.

In AIC building, however, we are only extending the AIC one step at a time. Starting at 1r4c7 we are on the 4th slink, and see the wink for the 5th in r9c7. From there we see 8 and 7 branches for the 6th, and an  easy one for the 7th slink. Either 1r4c7 or 1r4c3 is true, so 1 r4c4 is not.

Among Beeby’s ultrahardcore complex 1-ways, there were many that could be spotted in the first stage of AIC building, before ALS mapping, by watching for removals under the 1-way assumption that creates slinks.

However, examples such as this suggests a later of AIC building, perhaps along with trials, in which every branch wink is marked, one AIC branch at a time, for complex 1-ways.

Finally, there is one more human resource that I can illustrate with a Beeby complex 1-way on KrazyDad Insane v.4, book 7, #5.

Start on 6r7c3 and pick up a branch wink slinks 4 and 8. Slink 9 will take another branch wink from 4r9c3. We’re at 4r2c4 on the AIC.

Under the 1-way assumption, 4r2c4 is true, and the 9th slink disappears. But we are allowed to use it to continue the AIC.

But that last branch wink removes the last 4 in the NW box. That means the 1-way assumption must be wrong, and 6r7c3 is true!  We can abandon the AIC.

This example anticipates that last stage of AIC building in which complex 1-ways are exhaustively explored, without a specific destination. It is very much in the spirit of AIC building. The simple 1-way can stall indecisively. Exhaustively pursued, the complex 1-way can lead to a contradiction, confirming the starting candidate, and thereby removing all potential victims.

Beeby’s next complex 1-way in KDI 475 also raises eyebrows. If  6r2c1 is false, the 1-way confirms 6r1c8, making 6r2c8 false.

Since r2 must have a 6, the AIC is wrong. Unless there’s an error in the AIC, it’s the 1-way assumption that is wrong, making 6r261 true. Can  we just look at r2 and conclude that without the AIC? No, we have to show that the AIC does not fail to prove that it is wrong.

Beeby seems to start most complex 1-ways on bv, but any slink partner will do, as long as there are potential 1-way victims in the starter’s view. And any slink along the way could be a complex 1-way starter, until a branch is invoked.

The strategy for simple and complex 1-ways in Sysudoku then, is to include 1-ways in the first stage of AIC building, watching at first for complex branching opportunities only on last slinks, where the gain for the effort is reasonable. Then where earlier 1-way have failed, try the exhaustive  complex 1-way, perhaps, after giving ALS mapping a shot.

Next week, we’ll see how the above complex 1-ways  come up in KrazyDad Insane 475, and lead to a solution without trial. The puzzle is carried through Sysudoku Basic in the post of 9/3/13, but you can preview an update of the basic trace to include a 4-fill for the first clue. Trials were successful in updates of 9/17/13 and 1/22/19.