Alternating Inference Chains
Chaining strategies now take a new leap with Alternating Inference Chains. These extend X-Cycle into a new dimension - where X-Cycles stuck to a single number, AICs use any candidate number.
AICs encapsulate all the discussion of chaining strategies so far. It’s very useful to split out chain-like strategies into X-Wings, XY-Chains, Forcing Chains, XYZ-Wings, X-Cycles, Nice Loops and so on, since they have special characteristics which make them spottable. But it turns out they are all part of a larger extended family.
As we saw in the previous chapter, alternation is just what X-Cycles are about. However, you’ll remember that X-Cycles are applied only to a single candidate number. AICs, on the other hand, take everything from an X-Cycle and extend the logic to as many different candidate numbers as necessary.
AICs ask the question “How many ways are there to make a strong or a weak link?” If there is more than one way, we can join them up in an alternating manner and make deductions leading to eliminations. Let’s look back on the previous chain-like strategies and note the following:
There are also other ways (see later articles), but for now let’s keep it simple and stick to these two dimensions – links between cells and within cells.
Nice Loops Rule 1
Nice Loops that alternate all the way round are said to be 'continuous', and they must have an even number of nodes. With a continuous AIC, candidates are not removed from the loop since the loop does not have any flaws. Instead we are looking to eliminate on the units that can be seen by two or more cells that belong to the loop.
Starting a B7 we turn 4 ON.
This removes the 4 in B2 turning on 7 in that cell.
That turns OFF 7 in B5 turning on 6.
6 in B5 turns off 6 in H5.
That turns on 4 in H5 removing 4 in H7
Which confirms 4 must be ON in B7
Thus...there is no contradiction in the loop. The nice thing about Nice Loops is they can be reversed. Try starting with 4 ON in B7 and turning 4 OFF in H7 - it will come back round with the same conclusion. In fact, the loop is especially "Nice" because you can start with any candidate in the loop and work your way round, provided it is the same On/Off state as described in the example.
So having proved the loop we can look for extra candidates on any unit linked by the chain - or indeed, extra candidates in the same cell where an ON/OFF has occurred. (There are none in this case, only bi-value cells have been used).
Now we turn to flawed loops - ones that show a discontinuity. In terms of strong/weak links, there are two types, as described in the article on X-Cycles. Those where two strong links join up and those where two weak links join up.
If the adjacent links are links with strong inference (solid line), a candidate can be fixed in the cell at the discontinuity. It removes all other candidates as is the solution to that cell. This type is unfortunately much rarer than the Nice Loop Rule 3, two weak links.
Our third rule dictates what happens when two weak links form a discontinuity in a loop:
If the adjacent links are links with weak inference (broken line), a candidate can be eliminated from the cell at the discontinuity. In terms of ON/OFF this is where you try and set a candidate to be ON but the loop comes round and shows that doing so forces that candiate to be turned OFF.
AICs are chains of links going across a unit and within a cell.
If a candidate is turned ON you create a weak link turning OFF any other candidates in that cell and across the units it can see.
If you turn a candidate OFF you can turn ON other candidates in that cell (if there are only two candidates in the cell - bi-value) or across the unit if candidate X occurs just twice (bi-location).
However, bi-value and bi-location are just two ways of making chain links. There are other interesting ways of making chain links: Grouped Cells and Almost Locked Sets are documented here and other patterns as well can be made into links, even Unique Rectangles.