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# An investigation into the potential of BUG

## Origins

I have used (Bi-Value Universal Grave) BUG as a simple and effective technique to end a Sudoku board rapidly for quite a long time. By accident I coded a board which had two apparent BUGs, in different units, of course. In this case the BUG character that resulted was the same for both cells and I wondered whether I could validly treat these two cells as a pincer and make an elimination. This worked and after a large number of attempts in other games I have concluded that this is valid. Further I decided to try chaining from one or other pincer (as usually they were not each the same value) until I had a pincer pair that could eliminate. And from time time I found THREE apparent BUGs and the same deletions were possible, though with three “candidates”, this could be trickier.

Only recently I looked again at the description entry for BUG in Andrew Stuart’s excellent Solver. This has a standard description of BUG with some examples – but links to a blog which goes a bit further. Here I discover that BUG forms BUG+1, BUG+2 and BUG+3 have been worked on and can resolve the board, though using chaining and forcing techniques. BUG+n is used as a term and I supposed a BUG could be found in theory for all nine occurences of a unit – row, column or box. But the key point I took away is that ONE of the identified BUG characters will be correct. So I have treated these as BUG Candidates (BC)
I have developed a second difference in that I learnt long ago (from Henk Westhuis’ excellent Into Sudoku site and solver) that states one should pick the candidate with three occurrences in the unit (this in reference to a standard BUG+1). I have extended this: to be the Most Commonly Occurring Candidate so the BC can be identified as the one which occurs most often in the unit. I can find no proof of this, but this approach clearly works and is much simpler than considering Deadly Patterns. Though I realise that this is taking the technique away from its origins! I had also assumed that BCs could be found in rows, columns, or boxes, but now dissent from this as far as the techniques I describe below are concerned. I do not know why, but equally I do not really understand why the technique I describe works at all!

There is also an important condition that the board is clean, with no deletions un-done. Probably this means a clean board as in Andrew Stuart’s excellent solver after the first six steps, and I have adopted this in the examples that follow.

## Development

Of course, as currently understood, the technique may be interesting, but it resolves only quite simple situations, and those at the end of the solution of a game.

However, I appear to have progressed the technique somewhat – and this by utilising two things:
(a) That BCs can be manipulated like any other pincer value; and
(b) by identifying BCs using my “most commonly occurring” principle. I refer to BCs occurring in boxes only.

The simple BC is to the right – the first example in Stuart’s description:

He identifies the BC as “2” in D8. And according to my technique the most commonly occurring candidate value in the box is also 2 and it relates to the cell with the most values (in this case three – 1 2 3). In the examples in the PDF it is simpler to show just a single box and ignore the rest of the board for the purpose of explanation.

Now, identifying BCs as I have done thus gives rise to some other permutations, and involving more unresolved candidates on a board.

The rest of the discussion and examples continues in the PDF document below.

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