This post details a fireworks analysis on “Hanabi”, shye’s fifth example. The first four examples used matching elbows alone, along with bi-value cells. This one uses forcing chains to identify the value winning placement in the neck cell shared by four elbows and a firework.

Hanabi allows the bypass a few clues, a naked pair, and two 3-fills.

Box marking adds box slinks (strong links), and line marking, line slinks and bv borders.

For elbows analysis, the line marked grid

is dissected by value in X-panels. X-Panels 1 – 4 display only show empty X-wings, but panels 5 – 8 reveal many elbows.

Fireworks proposer shye selects firework r9, SW, c1 on values 56789. There are no 9 – elbows. Here are the selected elbows and the 9-panel firework:

Returning to our “values’ version of the fireworks principle,

if the four 9 candidates outside the box are false, 9 is placed in r9c1. The four elbows compete with the 9 firework for placement in this common neck cell.  By including the 9 firework, we get five fireworks placing five values on r9c1 and four elbow cuffs.

In his forum post, shye designates r1c9 and r6c4 as base cells and r9c1 as a target cell and states that values missing from the base cells can be removed from the target cell, removing 56r9c1. That’s leaving out a lot.

The definition of shye’s firework base cell is unclear, but for an elbow, let’s say the base cell is the cell outside of the box that sees both cuffs. All the elbows above have the same neck cell. In every elbow above, if the base candidate is true, it is false in both cuffs and is therefore true in the neck of every elbow above.

The set of five cells, four cuff cells and the neck cell, must solve to the four elbow and 9 firework values.

Using a term introduced in Sysudoku to explain BARNs, this is a 5-set, an n-set with n = 5. An n-set is a naked set, but not a subset because more than one house is involved. The most common BARN is a 4-set in a bent area, the intersection of a box and a line.

Highlighting the cuff and neck cells, if the base cell candidate 8r1c9 is true, the two forcing chains shown here force the neck candidate 8r9c1 to be true. That means the base candidates of the other elbows must be false, because if true, they would force their value in the common neck cell r9c1.

Goodbye, 5r6c9, 6r1c4, and 7r6c4.

Starting with this, and working in trial tracing order, here is the trace to the point where we are resisted by

a locked rectangle of 59 bv cells.  However, with 7 already placed, and 9 in its cell, you can place 5r6c1 and break the rectangle to finish the solution.

Next week we continue with the full Sysudoku solver path, because it’s worth seeing, and to highlight the DIY benefit of examining your elbows.

Here is the solution, with some coloring from next week’s final wrap. In shye’s well chosen example, the elbows route is much easier.