**An international terminology for the Calcudoku-Kenken?.** (A proposal for an international terminology by the Cacudoku-Kenken solving strategies and tips).

I propose that, in order to make unnecessary the use of any particular language, so that any techniques or solution strategies proposed for the Calcudoku-Kenken puzzles can be understood by persons in the different countries, we build and agree some kind of mathematical terminology (like a common musical language or an “esperanto” for the Calcudoku), some kind of simple agreement easy to learn and easy to understand.

My initial suggestion or proposal is this:

**1.** If a process is made of various steps we will name them

**1.**,

**2.**, etc., with the use of the bolding and a dot though, in parenthesis, optionally, some comments, like the colours used, etc., may be included in the different languages, and this would be the only "free" part in the terminology, example:

**1.**(...).

**2.** The cages will be named “75x”, “2-“, “27+”, “4:”, “3mod”, “-4+”, “12|”, …, showing first the result of the operation followed by the type of the operation.

**3.** The position of the cages (if necessary to avoid uncertainties) will be referred to with parentheses inmediately after the name, in this way: “75x” (e1-f1-f2), with the cell positions separated with dashes and ordering the cells first by the column, then by the row, so (a1-a2-b1-b2) but not (a1-b1-a2-b2). With the same target, the reference to "any" cell of those occupied by the cage will considered enough, for instance: "75x" (e1) or "75x" (f2). Also we write "1280x" (ffff) if we want to refer to certain part of a cage, the part occupying the column f, or "1280x" (5555), in the case of row 5, for instance.

**4.** The “value” of a cage (the addition of all numbers inside the cage, or the multiplication, etc.) will be shown in this way: “75x” (+) = 13, ”17+” (x) = 300, “3mod” (+) = 8, “5|” (+) = 9, “1^” (+) = 13, etc. In case of including the position of the cage that would be “75x” (e1-f1-f2) (+) = 13 or, i.e., “17+” (a1-a2-b1-b2) (x) = 300.

**5.** When a cell must have a determined value we will use the equal sign, for instance, b1 = 7. If the cell must be different than a determined value we will use, for instance, c3 <> 5 (a little joke, please avoid the confusion you may create if you use “the mother of all formulas”: x < = > y).

**6.** If a cell may have several possibilities we write: b1 = 3, 5, 7, … If a cell must be different than several values we write: b1 <> 2, 6, 9, … If a group of cells must have a defined group of values we write: h3-h4 = [3, 5] (square brackets) this means that the 3 and the 5 must occupy the cells h3 and h4, the exact location to be defined later. If various independent cells (not a cage) may take several different values (from a pool) we write, i.e.: (a3, a5, a8) = 3, 4, 5, 6, 9. If various cells (independent or that may be part of a cage) must be different than one or more numbers we write, i.e.: (a3, a8, b3, b4) <> 7, 9 this means that the 7 and the 9 are forbidden for the cells a3, a8, b3 or b4.

**7.** If a cell must be even we write c3 = 2n. If odd, we write c3 = 2n + 1. If a cage must be even we write "75x" (c5-d5-d6) = 2n. If odd, we write, i.e.: "3:" (f2-g2-g3-h3) = 2n + 1 [The (+) after the cage is assumed in this case].

**8.** The rule of the addition, i.e., in an 7x7 puzzle, would be: (e+++) = 28; (5+++) = 28, where (e+++) represents the addition of all numbers in column e and (5+++) represents the addition of all numbers in row 5.

**9.** Similarly, for the rule of the multiplication, i.e., in a 6x6, we would write: (dxxx) = 6! = 720 (factorial of 6) for the column d, or (3xxx) = 6! = 720 for the row 3.

**10.** The maximum (///) or the minimum (\\\) of a cell, a cage, a sum, etc., would be represented in this way: ///[“0-“ (+)] = 14, etc. Or, for a sum of cages: ///[“0-“ (a4-a5-b5) + “2-“ (b2-b3)] = 16. Or, for an individual cell, \\\[g2] = 4.

**11.** If a cage must contain a already well defined numbers we say: “75x” (e1-f1-f2) = [3, 5, 5] (with the numbers inside the square brackets in ascending order). If a cage may have several possibilities we write: “120x” (a1-a2-b1-b2) = [2, 3, 4, 5], [2, 2, 5, 6], [1, 4, 5, 6], [1, 3, 5, 8], …; for instance, in a 6x6 puzzle: “300x” (a1-a2-b1-b2) = [3, 4, 5, 5], [2, 5, 5, 6]. If a cage must be different than some specific combination we say, i.e.: “13+” (a3-a4-b3) <> [4, 4, 5] or “13+” <> [4, 4, 5], [3, 5, 5] this last case meaning that both combinations are forbidden.

**12.** If some of the numbers inside a cage are unknown we write: “5880x” (a2-a3-b3-b4-c4) = [x, y, z, 7, 7]; in this case what we say is that the cage contains two 7’s, being x, y and z, three unknown numbers (in this moment, obviously). We will use for the variables the final letters of the alphabet (t, u, v, w, x, y, z).

**13.** If a cage can not contain a specific number, i.e, the 3, we write: “14+” (c4-c5-d4) <<>> 3; “14+” (c4-c5-d4) <<>> 3, 5 means that neither the 3 or the 5 can go inside that cage. If a number, i.e., the 7, is already present in a determined line we may use: (eeee) [] 7! or (5555) [] 5!, this meaning that the column e already contains a 7 (or that a 5 is already present in the row 5). Also, if what we want to say is that a certain number is required in a certain line, i.e., the 8 in column f, we would write: 8 >>> (ffff)! or 7 >>> (3333)!.

**14.** If an hypothesis or a conclusion implies another hypothesis or another conclusion we write: a3 = 4 ---> “3-“ (c3-d3) = [2, 5], [3, 6] or, in case of two or more simultaneous premises: (a3 = 4; b3 = 2) ---> “3-“ (c3-d3) = [3, 6]. A series of consequences would show several consecutive ---> among the statements. If an hypothesis drives to an impossibility we would write: (a3 = 4; b3 = 2) ---> “3-“ (c3-d3) = [3, 6] = [000] (this meaning that the pair [3, 6], i.e. in a 6x6 puzzle, is not allowed in those cells because of other reasons then arriving to the conclusion that the premises are wrong). For “because of” we will use *…* enclosing the explanation between asterisks (it is recommended to minimize the use of “because of”).

**15.** The first line of the text will always be, i.e.:

**C = 8** (unnumbered, C stands for "Calcudoku", 8 is the size of the puzzle). After this first line we will write the steps (in ascending order, without lack of numbers). To refer to certain part of a text we will use intermediate lines of ... ... ... (three groups of three dots) or when suppressing part of a text wich is not now significative. The statements will be separated by semicolons. Finally, to indicate the end of the process we will repeat the initial statement

**C = 8**.

An example, using this terminology to answer the jomapil’s question, for the puzzle Nov 07, 2011 (Puzzle id: 380447):

**C = 9****1.** “42x” (a8-b8) = [6, 7]; a7 = 7 ---> a8 = 6; b8 = 7

**2.** “9x” (a2-a3) = [1, 9]; “9:” (h9-i9) = [1, 9]; “18x” (f2-f3) = [2, 9]; e3-e4 = [4, 5] ---> e1-e2 = [3, 8] ---> e7-e8 = [2, 9]

**3.** “6x” (b2-b3) = [1, 6], [2, 3] ---> “6x” (+) = 2n + 1 ---> b9 = 2n

**4.** b9 <> 2, 4 ---> b9 = 6, 8; b9 = 6 ---> “6x” = [2, 3] ---> “0-“ (a6-b6-b7) (+) = 14 *(a+++) + (b+++) = 90* = [000] (Note * below) ---> b9 = 8 ---> c9 = 3 ---> e9 = 6 ---> e5-e6 = [1, 7] ---> f6 = 4

**5.** “6x” (+) + “0-“ (a6-b6-b7) (+) = 17; “6x” = [2, 3] ---> “0-“ (+) = 12 = [000] (Note ** below) ---> “6x” = [1, 6] ---> “0-“ (+) = 10 ---> “0-“ = [2, 3, 5]

**6.** “840x” (g1-g2-h1-h2) = [u, v, w, 7]; “588x” (g3-h3-h4-i4) = [x, y, 7, 7]; (“840x”; “588x”) ---> “8+” (i5-i6) <> [1, 7]; “8+” = [3, 5] ---> i6 = 3 *f5 = 3* ---> b7 = 3 ---> a6-b6 = [2, 5] ---> “6+” (c5-c6) <> [1, 5], [2, 4] = [000] ---> “8+” = [2, 6] ---> “18+” (i1-i2-i3) <<>> 7 ---> i4 = 7 ---> i8 + i9 = 4 *(i+++) - 18 - 7 - 8 - 8 = 4* ---> i8 = 3 ---> i9 = 1 ---> h9 = 9 ---> “18+” = [4, 5, 9]; i2-i3 <<>> 9 ---> i1 = 9 ---> i2-i3 = [4, 5]

**7.** “588x” = [2, 6, 7, 7], [3, 4, 7, 7]; (d3 = 3; e3-i3 = [4, 5]) ---> “588x” <> [3, 4, 7, 7] ---> “588x” = [2, 6, 7, 7]

… … …

**C = 9** The full solution would not extend for more than 18 or 20 steps and no more than half a page for a 9x9 puzzle.

I think then that the use of a terminology like this could have some advantages:

- An “international” agreement, a quick way of “speaking” with numbers and signs (implicitly we are already using part of this “language” in our Forum, intuitively).

- The solutions could be compiled by the programmers and converted to a representation with graphics, etc., with the significative numbers flashing, bolded, etc.

- Inversely, a computer´s presentation (which includes a representation with graphics) could automatically be translated generating the appropriate terminology (the “object” file) to be understood by any puzzlers regardless of the country/language.

- No voice is really required (but perhaps it may be introduced in the optional parenthesis of the steps).

Note *: I have exceptionally added this note to remind that (see previous posts) if a cage “n-“ has an “addition” value of V, the higher number inside the cage is (n + V) / 2 so, in this case, if the cage “0-“ had a value of 14, the higher number inside (that must be necessarily present) would be (0 + 14) / 2 = 7 and this is impossible due to the 7’s occupying the cells a7 and b8.

Note **: Neither of [1, 5, 6] or [2, 4, 6] or [3, 3, 6] are possible for “0-“ (a6-b6-b7)

Last edited Nov 11, 2011 to add graphics and correct some typo errors.

Reedited Nov 12, 2011 to correct some typo errors and make more clear some statements.

Reedited several times Nov 14, 2011 to improve the terminology.

Reedited several times Nov 15, 2011 to make more clear some statements.

Reedited several times Nov 22, 2011 to correct some typo errors and improve the terminology.

Reedited Nov 27, 2011 to improve the terminology.