LPO, a Limited Form of POM

The Pattern Overlay Method, or POM is frequently cited as a computer solving method, but seldom proposed as a human solving technique, and for good reason. A pattern is a subset of X-candidates that fit together to provide an X-clue in every unit.

POM is attributed to the colorfully pseudonymned Myth Jellies. It completely  solves a puzzle in two stages. Sudoku writers have described POM as a type of jigsaw puzzle.  The solver finds the pieces, the possible patterns for each number, then selects the single pattern of each number that fits with the other eight selected pieces .  Although there are only 9 correct pieces to this puzzle,  there are generally a very large number of pieces that don’t fit with other numbers, and a gigantic number of combinations to try.  In its pure form, this mathematically appealing concept is impractical as a human solving method.  It is also unappealing for most computer solver enthusiasts.  A computer can enumerate all patterns and stitch them together in a backtracking search, generating an answer.  But you already have that answer in the back of your puzzle book, and the fitting process tells you nothing more about the puzzle.

The legend of POM is a tale worth telling, however, because there is a limited version of it that stays within humanly practical limits, while offering a powerful weapon against particularly tough puzzles.  To avoid confusion with brute force POM, I call this variation Limited Pattern Overlay, or LPO.   Limiting patterns makes it a practical task to superimpose patterns with an expectation that the agreements and conflicts among them will lead to confirmations and removals of candidates and patterns.

In the Systematic Sudoku Order of Battle,  LPO can be invoked as early as X-panel analysis. The X-panels aid in the selection of numbers with sufficiently limited numbers of patterns.  An expandable form of the X-panel is used for enumerating the patterns of selected numbers. Coloring before LPO makes sense because coloring  limits patterns further.  All candidates of a pattern must have the same cluster color.

I place LPO in the SSOB after AIC and ALS toxic pairs for several reasons.  First, it works better there, where every constraint we have found limits patterns.  LPO still takes significant effort.  Its results occur in an extra dimension of number interaction largely unrelated to the techniques already tried. LPO fits the “if all else fails” scene.

I use the term LPO enumeration to describe the identification of patterns among the candidates of a number.  A systematic technique for complete LPO enumeration is the subject of my next post.  Following that, we will walk through the LPO process to solve a Maestro Fall 2009 puzzle. Diligent sysudokies  may be way ahead of me by now with the Maestro 22 that gave us a swordfish in line marking in the post of April 3. But It has turned out to be an excellent example of LPO.  Let’s get updated on it and ready to do some pattern mashing and bashing.

We can start by checking out the nightmare of a candidate grid that Maestro 22 line marking left us.  The line marking trace leading to this grid, picking up after the swordfish, was

5f: c1, c4, c6, c9.  7f: r2, r5, r8. 9f: r5.  Close: c2, c3, c5, c7, c8.

If you solve it without LPO, tell me later.  Good hunting.