## X-panel Elbows for Fireworks

This post illustrates the use of the Sysudoku X-panel for fireworks analysis, as introduced  in forum corresponent shye’s first two examples.

Here is shye’s statement of the fireworks principle:

What does this actually mean? Given a row, column and intersecting cell, if a value in the cell cannot appear in the row or column outside the box containing the intersection, that value is placed in the intersection cell. This holds because no other candidate of that value in the box can provide that value to both row and column. Shye’s formulation of his fireworks principle uses the word “candidate”, when he means “value”.

A practical way to apply this fact is to find row/box/column combinations with only single candidates of an intersecting cell value in the row and column outside the box. Then by the fireworks principle above, the two single candidates and the intersection cell mark three cells, one of which contains the true candidate. For a reason soon to be apparent, we’ll refer to this combination of row, column and intersection cells, an elbow. We’ll call the single row and column cells containing the value of the elbow,  the cuff cells. To go along with that, the intersection cell will be the neck of the elbow.

In an elbow the value candidate is true, either in the neck, or in at least one cuff. If the neck candidate is false, at least one of the cuff candidates is true, because the true candidate in the box cannot be in both row and column of the elbow, so a cuff candidate is alone, in the row or the column. Another candidate of interest is one that sees both cuffs. If one of these is true, the neck candidate is also true, because both cuff candidates are false.

Commentator shye refers to elbows as fireworks, also uses that term for other, less well defined forms restricting intersection box values.

Shye’s first example in http://forum.enjoysudoku.com/fireworks-t39513.html illustrates the idea that, if you find three elbows matching three values at the neck and cuffs, you have a type of hidden triple.  The three cells must be reserved for these three values, so candidates of other values are removed.

Here is the Sysudoku line marked grid for shye’s first example, with matching  neck and cuffs for values 1, 2. There is no 3-candidate in c1 outside of the NE box, so value 3 is nevertheless  limited to the three cells of the two elbows, and candidates other than 1, 2, and 3  are removed.

We depict elbows on grids by freeforms because their “legs” are not necessarily strong links.

How hard is it to spot matching elbows? It’s much easier once you have all elbows of a value on an X-panel, with all 9 X-panels side by side. For that, you mark outside singles having outside singles from the same box in the crossing direction. After finding two matching elbows, you could look for a third one – armed elbow to match them. Here’s an elbow map for shye’s first example.

Values 1, 2 and 3 matching elbows, with three cells in common, a hidden triple. Other candidates are removed from these three cells. The 9 x-panels make it easy to verify that these and no more elbows of different values match. In this case, the removals bring an immediate collapse.

For the second example in shye’s fireworks introduction post, the homework of last week, here is a basic trace. This example shows how four elbows on a four cell rectangle,  sharing two pairs of values, makes a hidden quad.

Now in place of the usual line marked grid, here is the corresponding set of X-panels. Find and mark the elbows, including four matching on two values, that allow 1 and 2 in r9c6, SW r1c1 and r4c1; and 3 and 4 in r4c1, C r4c6 and r8c6, removing, 5r4c1, 89r4c6, 951c6 and 56r9c1.

On the line marked grid, the “the matching elbows form a 4-set quad.

Values 1 and 2 must appear in the 3 cuff and neck cells r4c1, r9c6, and r9c1. Similarly, values 3 and 4 must appear in cells r4c1, r9c6 and r4c6. That’s enough to remove 5,6, 8 and 9 candidates from the corner cells.

But also in this 4-set, a candidate of the other pair’s values cannot be true in its neck cell. If 1r4c6 were true, one cuff would have to solve to 3 and the other cuff to 4, so that both values could appear in that elbow pair. But that leaves the other pair missing 1 or 2 in all three cells. Basically, the row/column intersect cells can solve only to one of the pair’s two values.

The “quad” brings an immediate, but long, collapse.

Without the fireworks, this example keeps the ultrahardcore solvers at bay for 30 slides. My general conclusion from these first two examples is that, with X-panels, DIY exhaustive fireworks elbow detection is reasonable to do. A plausible  test of the frequency of results might be to see how many, if any, fireworks matches occur in the 22 ultrahardcore linemarked grids of the right and left page reviews. Volunteers, anybody?

Before you do that, let’s look at shye’s further examples of what to do with them. Next week, we start the new year with shye’s third example.