1. Finding Multi-fish

A working definition of a multi-fish is a combination of inconclusive fish patterns for different digits which together reveal a core set of cells that will be forced to hold one or other of the digits in the set, so allowing various exclusions to be made.

The digits are partitioned into two sets – a 'Home' set typically of 4 digits and the complementary 'Away' set of 5 digits (these names have been selected to avoid any confusion with base and cover sets.)

Individual houses can now be selected to be covered by one or other of these sets. As sets are assigned, any digits that occur as given or solved cells in the house, are removed, leaving only those that are still to be placed to remain as digit covers.

At any stage as the covering sets are being selected, there will be a potential Naked Set consisting of all the cells with every candidate covered, and a potential Hidden Set consisting of all the cells with at least one candidate covered.

Let

NS = the total number of cells with every candidate covered (the potential Naked Set).

HS = the total number of cells with any candidate covered (the potential Hidden Set).

DC = the total number of digit covers

We then get HS >= DC >= NS

When selecting covering sets to assign, the aim is to balance one of these inequalities while avoiding leaving a house or without a digit, or a cell with no candidates. Any candidates covered twice will affect the Potential Eliminations, so these also must also be watched.

Eliminations

When HS = NS => An established Locked Set already exists

- No eliminations

- Digits in the partially covered cells that are covered more than once are false.

- All covered digits in the partially covered cells are false.

Digits in the fully covered cells that are covered more than once are false

- If exactly one cell has digits covered more than once, the other digits in that cell are false

- If no digits are covered more than once, digits in the partially covered cells that are uncovered are false

If exactly one cell has digits that are covered twice, any uncovered digits in that cell are false

Note: in this treatment there is no need to be conscious of whether the sets being selected should be considered to be weak or strong. However for the so-minded, when naked sets are found the Home & Away sets should be considered to be weak and the digits in the NS cell to be the strong set. When hidden sets are found, the HS cells make the weak set and the digit cover sets become strong.

Tactics

In selecting the focus digit set concentrate on digits that appear as givens together in the same rows and columns most frequently and rarely in combination with the other digits. One way of doing this is to start with a digit which has two givens, and identify which others occur in the same rows or columns to progressively widen the set up to 4 members.

Choose either rows or columns to hold the Home covering sets and the other to hold the Away covering sets. Now select some banker rows where two of focus digits occur as givens for the covering set and some banker columns where none of them do for the complementary cover set. From this start point explore the options for including other rows and columns to be covered to see if a locked or almost locked set can be found.

If it happens that 5 rows and 5 columns are needed, then switching the direction to hold the Home and Away sets, and selecting the unused rows and columns instead, which will give a smaller 4x4 equivalent pattern.

If an AHS is found then this can be converted to its complementary ANS by keeping the same rows and columns but switching over the Home and Away sets. This is advisable as the NS elimination rules are stronger and the notations are shorter.

General Points

Because the user needs only to enter house cover sets without specifying if they are weak or strong, this approach is simple and fast in comparison to using truth and link sets. However it's less powerful as it will only handle rank 0 and rank 1 patterns.

Against this, a cocktail of truth and link sets should often be able to make the same eliminations with fewer sets, but at the expense of a more complicated procedure.

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[Edits 1-5 (Jul 2013) Extensively re-worked ]

[Edit 6 (Jul 2016) TAG Added ]