This post develops a property of a 2 value elbow match, and uses it with matching bv. Before we start on that, here are the elbow panels for the last week’s second example, just in case you want to compare it to your homework. 

The number of elbows is easy to count, and matches are easy to make, just looking at the horizontal or the vertical lines.

Here is the basic trace for the third fireworks example, leading to the line marked grid below.

We may want to do immediate X-panel only for very difficult puzzles, but in addition to a quick fireworks analysis, the panels will access X-chains, regular fish, XYZ, XY railway, freeform pattern analysis, and coloring assessment from the start.

On Fireworks 3 we have one elbow match, values 1 and 9. We could anticipate more as removals come in

Looking back at the line marked grid, there is something to note about the matching values 1 and 9, and their matching elbows. The the elbow cells outside the intersection box, the cuff cells,both see bv of values 1 and 9! These are on r1 and c9. Each elbow cell outside the box sees one of these bv. At least one of these, r1c1 or r9c9 or both, must have a 1 or 9 in the solution. The bv on the same line must have the opposite 1 or 9 in the solution.

That means 9r1c9 sees a true 9, either in an  elbow cell or a bv, hence its removal.

For future reference, let’s encapsulate the critical fireworks property at work here in the matched elbow rule:

The rule doesn’t mention bv. That’s because the rule is a general property of matched elbows that we might apply without matching bv.  Why is it true? It’s because two of the three cells of the elbow, the two cuff cells and the intersection cell of the fireworks rule, solve to the two matched elbow values. If one of the cuffs doesn’t, then the other cuff does.

By the way the removal of 9r1c9 collapses fireworks 3.

The fourth example in shye’s introductory post uses bv in a similar way, but instead of a single pair of matching elbows and two bv, there’s two pairs of elbows and one bv. The firework is easily spotted, but the logic applies the matched elbow rule in a hypothetical way. Next time.