Equivalence Marks
People solving Sudokus with pencil and paper cannot use coloring techniques, unless they find an alternative to coloring cells and/or candidates. Equivalence Marks are such an alternative.
Since there is little room for elaborate markings in a pencilmark grid, the marks are very basic. Odd Parity is symbolized by an apostrophe ' and even parity by a quotation mark ".
Only coloring techniques that use a single cluster can be replicated, otherwise more marks are required.
The following techniques can be replicated by Equivalence Marks:
Marking Rules
Start with a bivalue cell or bilocal digit having other strong links which can be followed, and mark one candidate n' and the other n". These marks indicate that these candidates are strongly linked and are therefore mutually exclusive. From this seed link, an opposite parity mark can be given to any other candidates that are strongly linked to any of the marked ones. Additionally, when a candidate must be directly equivalent to a marked one, it can be marked with the same parity. This usually occurs because both see the same group of sister candidates, one of which must have the opposite parity.
It is critical that the links used are truly strong; for example assume 5 is bilocal to two cells holding (1359') and (2578). It is clear that at parity(i) 5 will be true in the second one, but it is NOT clear where it should be at parity(ii) so, without supporting evidence, it cannot be marked. However if one of the other candidates in cell 2 has the opposite parity mark, say (2"578) or if that cell can see a cell holding 5", then the matter is settled, as there is only possible parity for 5 in each cell.
Exclusion Rules
- A candidate that can see cell mates and/or sisters of opposite parities is false. e.g. (34'9") makes 3 false.
- When exclusions reduce a cell or house to a single marked survivor, that parity is true throughout. e.g. (
26"7) makes parity(ii) true. - If two sisters or cell mates carry the same parity mark, that parity is false throughout. e.g (35"8") makes parity(ii) false.
Marking Tips
Equivalence marks are persistent and can be extended progressively as other methods are applied. However, being limited to just one pair of marks, they will not show every available strong link until the clusters eventually merge. It is consequently wise to evaluate which strong link cluster will be most productive to mark (usually the biggest) at the start. It is therefore better, and quicker, to run simple colour checks first, even though equivalence marking will achieve the same reductions.
Should the deductions dry up, the parity marks can be extended to weakly linked candidates using Graded Equivalence Marks.
Example
Equivalence marking was started from (45)r9c5. In all but one case the marks were made through strong links at bivalue digits in the same cell or bilocal candidates in the same house. The exception is the marking of (8')r4c5 which could be marked because it is directly equivalent to (8')r5c7 (green). From the parity markings the bracketed digits in the orange cells must be false because at one parity another candidate in the cell must be true, and at the other, a sister candidate must be true in an underlined peer cell. This assigns (5)r2c3 and (8)r1c8 as they both become singles. After handling the further knock-on, basic exclusions, the marking can be resumed to find other deductions, when parity(i) will be found to be true.
