Non-Consecutive

A Non-Consecutive Sudoku is a Sudoku Variant such that no two adjacent cells have consecutive values.

In the example below: because r4c8 = 5, then none of r3c7, r4c7, r4c9 and r5c8 can have the digits 4 or 6.

Because of these limitations, fever givens are needed for a valid puzzle. (10 in the example)

Techniques for Non Consecutive Sudoku

The following text is mirrored from the now defunct url jcbonsai.free.fr/sudoku/?p=139

The rules:

In this variant no cell may hold a value consecutive to any of its adjacent cells. If R5C5 = 5 -> R46C5 & R5C46 may not hold {46} (nor 5 of course). The rules are not clear on this, but usually this holds true for cells across a block/nonet : if R4C4 = 5 -> R35C4 & R4C35 <> {46} My non consecutive grids are designed with this restriction. Some other authors may drop this restriction. Unless expressed, this does not “wrap around” the limits of the puzzle : if R1C1 = 5 -> R1C2 & R2C1 <> {46}, but R9C1 & R1C9 may hold {46} This does not “wrap around” 1 & 9 : if R5C5 = 1 -> R46C5 & R5C46 <> 2, but may hold 9.

Unless for the easy ones, non consecutive usually require pencil marks and candidate elimination.

The basic technique is to eliminate consecutive candidates from adjacent cells whenever a cell is solved.

Single cell techniques:

A single cell with only two consecutive candidates {X,X+1} -> eliminate {X,X+1} from the adjacent cells eg R5C5 = {45} -> R46C5 & R5C46 <> {45}

A single cell with only two “doubly” consecutive candidates {X-1,X+1} -> eliminate X from the adjacent cells eg R5C5 = {46} -> R46C5 & R5C46 <> 5

A single cell with only three consecutive candidates {X-1,X,X+1} -> eliminate X from the adjacent cells eg R5C5 = {456} -> R46C5 & R5C46 <> 5

Dual cells techniques:

If two adjacent cells have all their candidates among {X,X+1,Y} -> Y is locked in one of them eg R45C5 = {569} -> 9 is locked in R45C5 -> not elsewhere in C5, N5 eg R4C5 = {59}, R5C5 = {569} -> 9 is locked in R45C5

If a cell = {X,Y} and an adjacent cell = {X-1,X,X+1,Y} -> Y is locked in one of them eg R4C5 = {59}, R5C5 = {4569} -> 9 is locked in R45C5 -> not elsewhere in C5, N5

The other way:

If some candidate X for some “house” is locked in two adjacent cells -> eliminate {X-1,X+1} from these two cells eg 5 of R5 locked in R5C45 -> R5C45 <> {46}

If some candidate X for some “house” is locked in two “nearby” non adjacent cells -> eliminate {X-1,X+1} from the cells adjacents to both of them The cells may be in a row or column with one cell in between, we can eliminate candidates from the cell in between eg 5 of C5 locked in R24C5 -> R3C5 <> {46} The cells may be in the same block/nonet touching at their corners, we can eliminate candidates from the 2 cells adjacent to both of them. eg 5 of N5 locked in R4C4,R5C5 -> R4C5 & R5C4 <> {46} This works also for diagonal non consecutive when some candidate is locked in 2 nearby cells. eg 5 of D\ locked in R3C3,R4C4 -> R3C4 & R4C3 <> {46}

Triple cells techniques:

Some dual cells techniques may be extended to 3 cells For the following techniques to work, the adjacent cells must be “buddies” of each other : either in the same row, column or block with an I shape or in the same block/nonet with an L shape.

If three adjacent cells have candidates among {X-1,X,X+1,Y} -> Y locked in one of them eg R5C456 = {1238} -> 8 is locked in R5C456 -> not elsewhere in R5, N5 As for the corresponding dual technique, some cells may miss one of the candidate from {X-1,X,X+1} eg R5C4 = {138}, R5C5 = {238}, R5C6 = {128} -> 8 is locked in R5C456 One cell may miss the candidate Y eg R5C4 = {138}, R5C5 = {123}, R5C6 = {138} -> 8 is locked in R5C46 eg R5C4 = {123}, R5C5 = {138}, R5C6 = {1238} -> 8 is locked in R5C56 NB If two cells miss the candidate Y, then simpler techniques may be used eg R5C4 = {123}, R5C5 = {123}, R5C6 = {1238} R5C4 = {123} -> R5C5 <> 2 = {13} R5C5 = {13} -> R5C46 <> 2 -> R5C4 = {13}, R5C6 = {138} R5C45 = {13} = naked pair -> R5C6 = 8

If three adjacent cells : {X,Y,Z}, middle cell = {X-1,X,X+1,Y,Z}, {X,Y,Z} -> both {Y,Z} locked in two of them eg R5C4 = {379}, R5C5 = {23479}, R5C6 = {379} -> both {79} locked in R5C456 As for regular subset techniques, this holds true if some cell miss some candidate eg R5C4 = {39}, R5C5 = {3479}, R5C6 = {37} -> both {79} locked in R5C456

If some candidate X for some “house” is locked in three adjacent cells -> eliminate {X-1,X+1} from the middle cell eg 5 of R5 locked in R5C345 -> R5C4 <> {46} eg 5 of N5 locked in R5C45,R6C4 -> R5C4 <> {46}

If some candidate X for some block/nonet is locked in three “nearby” non adjacent cells -> eliminate {X-1,X+1} from the cells adjacents to all of them For this to be useful, the cells should be in the same block/nonet in a check pattern, in a V shape. eg 5 of N5 locked in R4C46,R5C5 -> R4C5 <> {46} This check pattern is frequent, but seldom useful

If both {X,X+1} for some house are locked in 3 adjacent cells -> the two end cells = {X,X+1}, the middle cell <> {X-1,X,X+1,X+2} eg {45} of R5 locked in R5C456 -> R5C46 = {45}, R5C5 <> {3456} NB Some cell may miss some candidate

If both {X-1,X+1} for some house are locked in 3 adjacent cells -> eliminate X from these three cells eg {46} of R5 locked in R5C456 -> R5C456 = <> 5

Quad cells techniques

Some of the techniques may be extended to more cells, for instance a candidate locked in a house in 4 cells in a check pattern, but it is even less useful than the 3 cells version.

Complex naked pair

Two adjacent cells with all candidates restricted to a range {X..X+4} must include some alternative pairs. eg R1C12 = {1234} The only 3 possible combinations are {13|14|24}. Whichever the combination it must include either {1|2} and also {1|4} (or both) and also {3|4} This group may then be considered just like a single cell with two candidates. eg R1C12 = {1234}, R1C5 = {14} forms a complex naked pair on {14} -> {14} is locked two of these 3 cells and nowhere else in R1

Conflicting combination

From the example above, we can deduce R1C12 = {1234} cannot be {14} since it would cancel all possibilities from R1C5 = {14} eg R1C5 = {14} -> R1C12 <> {14} = {13|24} We have now limited the combinations for R1C12 which now must also include either {2|3}

Complex naked pair may involve two such groups of adjacent cells. The groups must be disjoint, cannot share even one cell. eg R1C12 = {1234}, R2C23 = {3456}. Possible combinations for R2C23 = {35|36|46} which must include either {3|4} Since both R1C12 and R2C23 must include either {3|4}, both {34} are locked in two of these four cells and nowhere else in N1

These groups of adjacent cells may also play a role in XY-Wing, XY chains…

Compiled with the help of Udosuk and Pyrrhon

Filed under: Solving techniques by Jean-Christophe, August 17th, 2006