This closing post adds two examples of a fierce looking line marking being de-fanged by hidden dublex on the over abundant X values
A double line exclusion, or dublex, is one of the primary moves of Basic sudoku, in which the intersection of two parallel lines through a box cannot contain a candidate of the dublex value, forcing a third line chute to contain a true candidate of the value.
When each of the intersecting lines only contain candidates of the value, then each contains a true candidate. We just don’t know which line remainder contains which true candidate. We call that a hidden dublex, or in trace language, an hdx.
The hidden dublex is a frequent move in the bypass and follow up, but is especially useful when an over abundant value needs weeding. Scroll down for the givens of Extreme Hard 1503, with one 7-candidate and no sign of the 8 value.
The bypass leaves 4 3-fills, each with an 8 candidate to be placed.
On the 7 value of box marking, columns 4 and 6 contain true 7 candidates outside of the C box, for a 7-slink, and soon after SW7m generates the second slink W7m for a 7-wing, another form of hidden dublex.
The 7-wing’s E7m slink is trimmed to clue E7 by the second hdx in c79, squeezing c8 to the naked pair 18, for a 1-wing keeping 1 out of r5c9. But the real damage is done in the 8 camp as C8m starts a collapse.
The collapse trace:
I have one more Extreme Hard example of the hidden dublex avoiding a tough line marking. It’s 1506.
At the 9 value in box marking, note the hdx on r8 which creates slink W9m. The hdx chain continues with a c2hdx forcing NW9!
The immediate collapse is typical of unbalanced puzzles tripped up when headed for a cluttered line marking.
To illustrate, here is the line marked grid when the hidden dublex chain is ignored. The 8 and 9 candidates are everywhere.
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