The New Sudoku Players' Forum

[Pong]

Line-Box Intersection

The argument below was given by Sue de Coq.

Consider the intersection of a line (row or column) and a box. Suppose there are n unsolved cells there, and these have together m candidates, and you can find m-n other cells in the same line or the same box that have no candidates other than these m, and such that for these m-n other cells there is no candidate common to a cell in the same box not on the line and a cell on the same line not in the box. Then matching applies: the m values go into the m cells, and the values can be removed elsewhere.

An example given by ronk:



The three yellow squares have together the five possibilities 2,3,4,5,9. The two green squares also have their possibilities inside the set 23459, and since no digit can be seen twice in the yellow and green squares together, these five squares contains these five digits. But that means that 345 can be eliminated elsewhere in the same box, and 259 elsewhere in the same column.

What I don't understand is the text highlighted in RED above. In the partial example below, the "6's" in yellow cannot both be removed...why not?

[Myth Jellies]

The 4 cells contain candidates 1246 with respective multiplicities of 1221. That is to say that you could possibly put two 2's or two 4's in those four cells at the same time. When you sum up those multiplicities you get a total of 6 in 4 cells. The only invalid move would be one that makes your total multiplicities less than the number of cells. Since no digit has a multiplicity of 3, no invalid moves can be noted with this set.

Compare that with the Sue de Coq example. There you have 5 cells with 5 candidates, 23459, with respective multiplicities of 11111, totaling 5. Thus any candidate which would prevent any of the five candidates from being placed in those 5 cells would reduce the total by one and would thus cause a crash.

For more information on this general way of seeing Sue de Coq's, ALS's, and other similar critters, check out this post on Subset Counting.