JC’s Sudoku

Notation and Terminology

A grid or puzzle is made of cells grouped in rows, columns and nonets (or blocks).

Rows are numbered from 1 = top to 9 = bottom. They are abbreviated as R#
Columns are numbered from 1 = left to 9 = right. They are abbreviated as C#
Nonets are numbered from 1 = top-left-hand to 9 = bottom-right-hand, nonet 2 is the top central nonet, nonet 3 = top-right-hand… They are abbreviated as N#
Cells are referenced by their row and column. R1C1 is the upper left most cell.

Multiple cells or groups are referenced using multiple digits. Since we only deal with numbers with one digit, there is no need to use separators.
R1C12 designates the two cells at R1C1 and R1C2
R1C146 designates the three disjoined cells at R1C1, R1C4 and R1C6
R13C12 designates the fours cells at R1C1, R1C2, R3C1 and R3C2
R15 designates the two disjoined rows R1 and R5
N36 designates the two nonets N3 and N6

Ranges are designated by two dots “..” (or ellipsis “…” like in math notation) It should be prefered over the hyphen which may be confused with minus.
R1..4 designates the fours rows R1, R2, R3 and R4
R23C1..3 designates the six cells at R2C1, R2C2, R2C3, R3C1, R3C2 and R3C3

Buddies of a cell are all cells constrained where there may be no duplicate. This includes also the cells of the cage in addition to the row, column and nonet. For instance, buddies of R3C3 in the grid here above includes all cells in N1, R3, C3 and also the two cells R1C45 since they belong to the same cage.

Cages are designated by their sum and number of cells.
5/2 means cage with two cells adding up to 5.
18/3 means cage with three cells adding up to 18.
I use a slash to separate the sum and number of cells. Some authors will put the number of cells within parenthesis or square brackets : 18(3) or 18[3] is the same as 18/3.

The location of cages may be given as some group their belong to. To avoid ambiguity, one may give the location of their top-left-hand cell or their exact range.
5/2 in N4 designates the cage 5/2 within nonet 4 (assuming there is only one cage 5/2 in nonet 4)
18/3 in N58 designates the cage 18/3 across nonet 5 and 8
8/2 at R45C2 designates the cage with the two cells R4C2 and R5C2, adding up to 8
21/4 at R3C2 designates the cage 21/4 with top-left cell at R3C2

Values are designated by a single digit.
An equal sign designates an assignment, setting a value, like in programming languages.
R1C1 = 3 means cell at R1C1 is set to 3

Unordered candidates or values are denoted using curly brackets { and }, a usual math notation for sets. Since we only deal with numbers with one digit, we may ommit separators.
{12} means 1 or 2
{249} means 2 or 4 or 9
R1C1 = {46} means cell at R1C1 has the two candidates values 4 and 6
R1C12 = {4589} means the two cells at R1C12 both have the four candidates 4, 5, 8 and 9
5/2 = {23} means the two cells of the cage 5/2 both have the two candidates 2 and 3.
{1..5} means any value from 1 to 5 inclusive : {12345}

Ordered values are denoted using square brackets [ and ], a usual notation for ordered lists. The order goes from left to right and then from top to bottom.
R1C12 = [53] means R1C1 = 5 and R1C2 = 3

Alternatives or mutually exclusive candidates for several cells may be emphasized using the vertical bar symbol ’|’, the “or” symbol of some programming languages like C and Java.
R1C12 = {14|23} means the two cells at R1C12 can be either {14} or {23}.
9/3 = {126|135|234} means the cage 9/3 can be either {126}, {135} or {234}.

Mandatory values often play an important role in the reasoning. They may be designated by grouping alternatives within parenthesis.
8/3 = {1(25|34)} means the cage 8/3 must have a 1. It can be either {125} or {134} but whatever the case, the 1 is mandatory.

Grouping may also be used to emphasize some properties like a mandatory alternative pair of values. We can rearrange alternatives somehow like commutativity, associativity and distributivity of math operators.
5/2 = {(1|2)(3|4)} emphasizes the fact that a cage 5/2 must have either 1 or 2 (and also 3 or 4). For instance this may clarify a reasoning which excludes candidates with both 1 and 2 for some other cage.
9/3 = {(1|2)(26|35|34)} emphasizes the fact that the cage 9/3 must have 1 or 2 (or both).
R123C1 = {(8|9)…} emphasizes the fact that R123C1 must have 8 or 9, and “something we don’t mind”.

Deductions may be designated by a right pointing arrow “->” meaning “therefore”, “implies”, “thus”, “as a consequence”… I would avoid the math symbol “=>” which may be confused with “greater than or equal to”. Similarly, a left pointing arrow “<-" may designate “because”.
R1C12 = {12} -> R1C56 = [35] means R1C12 is {12} implies R1C56 is [35]
R12C4 = {34} <- R1C56 = {15} means R12C4 is {34} because R1C56 is {15}

Some other symbols may be used : “&” for “and”, “¬” for “not”…
R12C4 = {35} <- R1C56 = {14} & R2C56 = {27} means R12C4 is {35} because R1C56 is {14} and R2C56 is {27}
R12C4 = {35} -> R1C56 = ¬{36|45} means since R12C4 is {35}, R1C56 cannot be {36} nor {45}

Abbreviations of techniques :
45 on N1 -> R1C3 = 5 means the “45 rule” innies on nonet 1 implies that R1C3 is 5
45 on R1..4 -> R5C38 = 6/2 means the “45 rule” outies on rows 1 to 4 implies that the two cells R5C3 + R5C8 add up to 6
LOL means “Law Of Leftovers”
{R67C3} means the set of values in R67C3, here a pair of values

Some techniques like disjoined cage splitting let us restrict the sum of some parts or cell to a range. The two dots can be used for this. The symbols “<=" (less than or equal to) and “>=” (greater than or equal to) may also be useful.
Each in R12C1 <= 5 means each cell in R12C1 is at most 5.
R12C1 >= 15/2 means the sum of the two cells in R12C1 is at least 15.
R12C1 = (14..16)/2 means the sum of the two cells in R12C1 ranges from 14 to 16 inclusive.

Sometimes we can further restrict a range by exclusing some value, the vertical bar may be used.
R1C45 = (3|5)/2 = {12|14|23} = {(1|2)…} means the two cells R1C45 add up to either 3 or 5, which can be {12|14|23}, rearranged to emphasize the mandatory alternative {12}.

…

P.S There exists several adopted math symbols for such operators, but there are not commonly known and some browsers or systems may not support them. Those interested, may have a look at the Unicode table for range U2200 .. U22FF (PDF 136 Kb)