This post describes a humanly practical way to identify ALS build an ALS map. The next considers how to use the map to build AIC and find ALS_XZ. The tools and procedures for ALS maps are demonstrated with the review solution of Stefan Heine’s ultrahardcore 311.
Here is an updated version of the ALS_65 shown in the post of November 3, 2020. It’s superimposed on the AIC segments left by the AIC building of last week. It confirmed N3, a step taken by a solver’s complex AIC in the November 10 post, but after the ALS_65. The confirmation replaced the ALS in r1 by a simpler one.
Let’s review the ALS_XZ and the problem of finding ALS.
An ALS is a set of n cells in a unit(house) containing candidates of n + 1 values.
ALS_XZ is based on the fact that when a value is removed from an ALS, there are n values for n cells. The remaining values are locked. Every remaining value group in the ALS has a true candidate. In this example, the single 6’s form the grouped weak link between ALS known as the restricted common. It guarantees that only one of the two ALS will have a X = 6 in the solution. One of the two value groups of Z = 5 will be locked. One 5-candidate outside the two ALS, 5r8c2, sees both 5 groups, and a true 5 candidate.
ALS_XZ is a consistent source of candidate removals, but spotting them is difficult. Finding them exhaustively seems overwhelmingly so, because ALS are so numerous, and because you have to see the X and Z connections between two ALS for each one. But like AIC building, there is a practical way for human solvers to do it. With an ALS map.
Here is the formidable looking Sysudoku ALS map of ultrahardcore 311, when the ALS_65 above is discovered.
It is the line marked grid, with ALS cells marked by curves. Bi-value cell ALS are already marked in green. Extra breakout panels and colors help make it barely possible to distinguish the ALS of UHC 311. A small circle is color matched to the encircling curve to mark the single candidate values. Trace how the ALS_65 above is spotted, and formulate your rules for finding more of the ALS_XZ.
You invest the time to build this map once, then bring updated copies along with you. ©Powerpoint is good for this. Anticipate using the map for spotting Beeby’s ALS-wings as well. But more generally, remember that after AIC building, we have segments of AIC on the grid that have not been extended by AIC nodes. An ALS node has strong links that exist between the value groups in every ALS. That can mean a second wave of AIC building. We will look into these possibilities in the next post.
So with a little exploration of the map, and with these prospects ahead of us, let’s take the time to walk through the construction of an ALS map. To even talk about it, we must have a way to describe particular ALS. We’ll use cell/value susets, a concept used many times in this blog. In the suset, two digit strings identify cell positions and values. The blue ALS is r1 27/256, with cells 2 and 7 of r1 holding 2, 5 and 6 value candidates.
For the map, we need to list all of the ALS in every row, column and box. Let’s start with r1 above, listing each cell by suset:
2/25, 5/56, 7/26
The bv are ALS, and there is an overlap of one value in each of 3 combinations, for
25/256, 27/256, and 57/256. A combination is a union of values and a union of cells.
All combinations now add a cell, reaching 257/256 which includes 3 cells and 3 values, and is not an ALS.
For a more typical case, let’s take the ALS in column 1 above. Here are the susets,
2/2358 4/23489 5/2459 7/589 8/58 9/259
Of course 8/58 is an ALS. Adding value 9 we include cell 7 for an ALS 78/589. Now looking for another such overlap of values, we combine cells 5 and 9 for 59/2459. And then we see 28/2358. In both cases cells are gaining on values. In our new list, we keep only the susets with added values in our revised list,
28/2358 4/23489 59/2459 78/589 8/58 9/259
Notice that the values in 4/23489 will pick up many cells if it has a 5, and no cells if it doesn’t. That gives us 245789/234589. Oops, 6 cells and all 6 values.
Now where is there more value overlap among values to pick up more cells per value? The last three. Add 2 to 589 gets 789/2589. Its an ALS adding one cell and one value added to an ALS. We now see that adding the 4 value to ALS 789/2589 is going to bring in cell 5, another ALS. Let’s list our ALS in order of increasing values:
8/58 78/589 789/2589 5789/24589, leaving behind 28/2358 and 59/2459 .
For homework, how about finding the other ALS in row 8, without checking on the map.
To prepare a map, record the susets for cells and ALS separately rows, columns, and boxes. Here are the ALS tables for UHC 311:
As you place ALS curves on the map, you can mark them for ALS_XZ partnering. To be an ALS_XZ partner, an ALS must have, for both X and for Z:
A single candidate value, or
An aligned value group with a possible Z victim, or
A box confined value group.
In ALS map, I mark single candidates for X and Z with small circles, matched in color with the ALS boundary curve. On the map, our r1 ALS has two single values 5 and 6.
Here, I use small pentagons to mark some potential Z victims of aligned value groups. Our ALS_65 r8 ALS has a matching 6 single value, and this map has the pentagon, but normally victims are not marked, because you can’t anticipate the partnering ALS.
This example of a box confined value group is from French Su-Doku Maestro magazine of July-Sept., 2009, and its L’attaque du cobra example 4. The green ALS , allows the grouped restricted common X = 5. The blue ALS qualifies for the map with the single 5 and the aligned victim 2.
The box confined value group is based on properties of both ALS. In this example, the 5r3c2 single value could be the starting signal. The spotting rule would be to check for an ALS group confined in the box of the single.
This post shows that exhaustive ALS map making can require a huge effort. At least you only have to make the map once, then update it, along with the bv map, X-panel and AIC segments, for an added dimension in solving very hard puzzles. Next week, we apply the map to the UHC 311 case to illustrate the several ways it can be used.