they are hard to utilize quickly but are applicable in some cases.

knowing how to use and apply these techniques can make more advanced puzzles very easy,

most of these do require candidate mark ups to identify and use correctly.

{or a very good retention for candidaes per cell}

understanding how these opperate requires a large amount of reading and a fair amount of time practicing.

i suggest testing the theory out for your self to fully comprehend it.

Definition:

A Bivalue Universal Grave (BUG) is any grid in which all the unsolved cells have two candidates, and if a candidate exists in a row, column, or box, it shows up exactly twice. example

A BUG-Lite is a partial BUG pattern that exhibits similar properties of a BUG where all nodes in the pattern are bivalue and if a candidate exists in a row, column, or box, it shows up exactly twice. example

A poly-valued cell for the purposes of this thread is a cell having more than two candidates.

A Local Bivalue Move or Localized BUG Move (LBM) is the selection of one or 2 candidates from a cell that causes each candidate in the 2-candidate selections in that row, column, or box show up exactly twice. example, example

A non-BUG candidate is a candidate that is excluded during a LBM from a cell. All non-BUG candidates are not part of a BUG.

A BUG+n is a BUG that has exactly n number of poly-valued cells. A BUG+1 is a BUG that has exactly one poly-valued cell left.

Theorem:

BUG grids can have either zero or more than one solution, and so are incompatible with a unique solution puzzle. Hence the puzzle solution must come from the non-BUG candidates. proof

Corollary 1: If a BUG can be formed out of the list of candidates by LBMs (without solving for any candidate), then the solution to the puzzle must make use of at least one non-BUG candidate. example,example

Corollary 2: Any deductions implied by all non-BUG candidates in the grid must be valid. example

Corollary 3: Any placement of a candidate which removes all non-BUG candidates is an invalid move. example,example

Corollary 4: Any placement of a candidate which forces a grid into a BUG+1 is a valid move. example

Corollary 5: Corollaries 1, 2 and 3 can be applied to a BUG-Lite.

extending A bug beyond the basics.

BUG LITE

Mutivalue universal graves (MUG)

Permeable Mug

to expand beyond these as well.

a person must first understand principles of an

Unavoidable Set:

Unavoidable sets exist in a solution grid. An unavoidable set is a group of cells which can be changed in such a way that another valid solution can be formed by only altering the cells inside this set.

The smallest unavoidable set has 4 cells. The largest unavoidable set has 81 cells. Any two rows or columns in a single chute form an unavoidable set of 18 cells. The combined cells for any pair of numbers also form an unavoidable set of 18 cells.

An unavoidable set can be a subset of one or more larger unavoidable sets. For example: A bivalue rectangle located in a single chute is both a subset of the unavoidable set for the 2 lines in that chute and of the unavoidable set for all 18 cells holding these two numbers.

Observation: Each unavoidable set has a complementary unavoidable set in each of its parents. This complementary set contains all the remaining cells in this parent.

Unavoidable sets can also intersect with other unavoidable sets. The union of these sets can be seen as unavoidable sets by themselves, but their constituent sets are not true subsets, since their complements are not unavoidable sets.

As stated above, in a valid Sudoku puzzle each unavoidable set is covered with at least one given value. As a result, a valid Sudoku puzzle does not contain unavoidable sets in which none of the cells contains a given value

with understanding the above some of these become more clear as concepts in solving.

Reverse BUG

Reverse BUG lite