## Counting Cases

Everything about Sudoku that doesn't fit in one of the other sections

### Counting Cases

I was going to post this in the backdoor thread, but as I can't understand the jargon there, I could be corrupting it, so I'm startng a new thread just in case it promotes any discussion.

My view is that strategically we may say we don't make assumptions when choosing our point of attack, but tactically they must be made in order to follow the various implications and I'm far more interested in the tactical ones.

As Ruud pointed out 7 or 8 years ago we consider the reduced candidate lists in a Sudoku grid to be established 'Sudoku truths' which can be freely used without being accused of making an assumption. We do this without thinking, but just try solving puzzles without the pencil marks to discover the implications.

Taking an individual candidate in a candidate list as being true is splitting off a case for examination, commonly called making an assumption or guessing.

The process of following the implications of that case must now alternately consider the candidates that would be made false an odd number of inferences away and true an even number of inferences away. This involves having to use alternating strong and weak inferences, which automatically makes the implication chain segments stemming from the start node bi-directional.

In tracking an implication chain as a new end case is found the previous case is forgotten. At any time therefore there are only two active cases, one at each end, available for comparison in an un-branched chain. Otherwise each branch followed adds an active case which is dropped when the branches converge. Similarly each remembered case also increments the active case count until it is used.

For chain and net methods active case counts provide a relatively simple means of 'costing' them. As I'm only interested in methods that are within the scope of a human solver (perhaps with a helper program) this is the basis I intuitively use to decide but how tactically assumptive, and therefore complex, it is.

There is one rider however. A derived inference exists when a relationship between two remote candidates can be shown. Once one has been proved, it shouldn't be necessary to re-prove it every time it's used. Consequently it should be possible to optionally use a solution step to register a derived inference as a Sudoku truth together with its active case cost.

Using known Cover Sets and Uniqueness patterns presents problems for working out the active case counts, but my rough and ready rule is to use the minimum number of houses needed to contain the pattern. When a pattern is embedded in a chain, 2 should be deducted to allow for the overlaps.
David P Bird
2010 Supporter

Posts: 1040
Joined: 16 September 2008
Location: Middle England

### Re: Counting Cases

Hi David,

AFAIK champagne uses some kind of magic which sounds similar to yours.
dobrichev
2016 Supporter

Posts: 1622
Joined: 24 May 2010