# SUDOKU QUICK COURSE

(from the SUDOKU PERSONAL TRAINER v 2.2 program help file)

## Sudoku is a puzzle, to be solved logically.

Its most common Scheme consists in 81 Cells laid out on a grid (or table) with  9 Rows and  9 Columns; the Scheme shows also, by more evidenced borders, 9 Boxes of 3 x 3 Cells each (also called Blocks or Regions).

The puzzle starts with the numbers (1 to 9) in some Cells only; to solve the puzzle the player must fill the empty Cells with those numbers, respecting the Base Rule that “each Row, Column and Box  must contain each number 1 to 9 exactly once”.

## TERMINOLOGY

-  Row, Column, Box, Cell: as defined above;

-  Zone: means a Row, a Column or a Box indifferently;

-  Value: is a number (1-9) present in a Cell, or from the beginning or played during the game  (the initial Values are black, the played ones are blue);

-  Candidates: are all the allowed Values you can play in a Cell, in respect of the Sudoku Base Rule. Since every new move adds a new Value it should force the player to do new reasoning, but that is done by the program (the Candidates that are automatically calculated in this way are red, while those adjusted directly by the player are blue);

-  Exclusions: are all not admitted Values in a Cell,  because conflicting with the Sudoku Base Rule;

-  Double: if a Cell admits only two Candidates that is a Double;

-  Conjugated Pairs: when, during the game, a Candidate appears (alone or with some others) only in one Cell of a Zone (for instance a Column), then it is certainly true and it is the Value to be assigned to that Cell. But if  it is present in two Cells of a Zone, it realises a Conjugated Pairs that have the propriety that the above Candidate will be true in a Cell and false in the other;

-  Twins: (not to be confused neither with Double nor with Conjugated Pairs)  you have Twins when in a Zone two Cells accept the same two Candidates. Twins can be “naked” (pure, and you have two equal Doubles), or “hidden” when some other Candidate appears in one or both those Cells;

-  Triple: as above when there are three Cells which share in any way the same 3 Candidates;

-  Quad: as above when there are four Cells which share in any way the same 4 Candidates.

At large, a given set of numbers on a Sudoku Scheme can:

-     have no Solution;

-     accept only one Solution: that is the case of those you find on papers and magazines;

-          accept more than one Solution: in this case the Sudoku program follows the one it finds first.

Some Schemes can be solved without Candidates' evidence (I call this way “at a glance”). Very often, anyway, after a first set of Moves, it becomes necessary to write the Candidates into their Cells for appropriate reasoning. The program let you work in both ways, passing from one to the other at any time you want: in fact a Move you discover with Candidates put in evidence may take back the opportunity to play new simple Moves "at a glance".

There is a set of “logical Methods” (reasoning or techniques) to play the Moves.  Those rational constructions are sufficient to solve the Sudoku Schemes you find on general daily and magazine game pages.  However, on Sudoku specialised magazines or Internet pages you can find Schemes where, after having used all available rational methods, you must proceed “by Trials”. The SUDOKUPTI program offers the environment that helps you to play that way, as well.

METHODS  (OR  TECHNIQUES)

METHOD 1 - At a glance: by crossing Rows/Columns/Boxes with the same Value (1-9), you find that some Zone remains with  just one  not-excluded Cell (the Method may be known as “slice and dice”). Easy Sudokus might be entirely solved with this Method.

Sample 1a - Crossing Rows and Columns having the 5 Value, happens that in the marked Box only the azure Cell can accept that Value.

Sample 1b - It is the same case when the target Zone to cross is a Row. The central grey Box, with its pink 4 Value, forbids 4 in the empty Cells shared with the grey Row. The grey Column, with its pink 4 Value, forbids 4 in the Cell shared with the grey Row: if the grey Row (here not entirely shown) does not have other free Cells, only the blue Cell can receive 4 as Value.

METHOD 2 - Only one missing Cell to complete a Zone (Single Cell).

Sample 2 - in the marked Row, only the azure Cell has to be done so you can confirm 2 there.

METHOD 3 - Only one Candidate in a Cell (Single Number).

Sample 3 - The azure Cell admits only one Candidate,  so you can confirm it.

METHOD 4 - A Zone shows a certain Candidate in one Cell only, but it is accompanied by some other Candidate (Hidden Single or hidden loner).

Sample 4 - In Row H the Candidate  9 exists only in the azure Cell (even if accompanied by others Candidates) so you can confirm 9 there.

METHOD 5 - A Zone shows exact Twins Candidates,  i.e. the same pure Double appears twice (Naked Twins or pure twins).

Sample 5 - In the marked Zone there are two Twins (the two Cells with the Double 12  in pale grey). Using only two Candidates,  here 1 and 2, you cannot satisfy any other Cell,  so the 1 appearing in the azure Cell must be cancelled from there.

METHOD 6 - A Zone has two Cells with the same double of Candidates, but  one double or both are accompanied by other Candidates (Hidden Twins).

Sample 6 - The Row A shows only two Cells with the Candidates  1 and 2,  but accompanied by others Candidates. On two Cells you cannot play more than two Candidates, and since they appear only there you must cancel every different one. In this example the PERSONAL TRAINER says to liberate the azure Cell, but obviously you should clean up also the second Cell interested by these Twins (marked in pale grey).

METHOD 7 - A Box locks a Candidate on a Row or Columns of  its, but that Candidate appears on the same Row or Column also outside that Box (Box Locks Candidate).

Sample 7 - The Box ABCabc foresees the Candidate 4 (pale grey Cells) only on its part of Row A; it will surely go there, but, since the same A Row shows the Candidate 4 also (azure Cell) outside that Box,  the 4 must be cleaned up from there.

METHOD 8 - A Row or Column locks a Candidate in one Box only, but the Candidate appears also in other Cells of that Box (Row/Col. Locks Candidate).

Sample 8 - The Row F foresees the Candidate 1 only within the marked Box; the Candidate 1 will surely be true in one of the pair of those pale grey Cells,  as a consequence it cannot be accepted in any other Cell of the same Box (not in the azure one; note that there is another Cell to clean but the program is suggesting the first now).

METHOD 9 and 10 - 3 (o 4) Cells of a Zone admit  just  all or some of the same 3 (or 4) Candidates, but those Candidates appear even in one or more other Cells of that Zone (Candidates Sharing).

Sample 9 - Using only three Candidates you cannot satisfy more than three Cells of the same Zone. It happens here for the marked Box and its three pale grey Cells,  showing 4, 7 and 9; as a consequence you can clean up 4 from the azure Cell (note that are other Cells to clean but the program is suggesting the first now).

Sample 10 - Using only four Candidates you cannot satisfy more than four Cells of the same Zone. It happens here for the marked Row and its four pale grey Cells showing 2, 3, 8 and 9; as a consequence you can clean up 8 and 9 from the azure Cell.

METHOD 11 - A Zone shows 3 Cells with exactly the same three Candidates, that appear also in some other Cell of  its (Naked Triples or pure triples).

Sample 11 - In the marked Zone there are three pure Triples (the three Cells  in pale grey with the Triple 679 ). Those Candidates cannot be true in any other Cell of the same Zone, so you can clean up 7 from the azure Cell.

METHOD 12 - A Zone shows 3 Cells with the same three Candidates, ma some other Candidate accompany them in those Cells (Hidden Triples).

Sample 12 - In the A Row, the three marked Cells (azure and pale grey) hide the Triple 168; you can start cleaning up 2 from the azure Cell, and then clean up conveniently the other two pale grey Cells to leave only 168 in each of them.

METHOD 13 - The Scheme shows an X-Wing Rectangle. When two Rows shows the same Conjugated Pair and those Cells belong to the same two Columns (those four Cells form a rectangle), then every other presence of that Candidate on those Columns is certainly false.

The same can be said switching Rows and Columns, when the above Conjugated Pairs lay on Columns.

Sample 13 - The Candidate 8 appears on the corners of a X-Wing Rectangle, based on D and G Rows having just those  8 as Conjugated Pairs; accordingly the theory the 8 Candidate is admitted only on the rectangle’s corners,  but not in other points of its sides or of  its sides’ continuations; as a consequence you can clean up the azure Cell of the second Column.

METHOD 14 - The Scheme shows a SwordFish  Butterfly. When 3 Rows shows 2 or 3 Cells with the same Candidate and those Cells lay on the same  3 Columns (so that you can see two partially overlapping rectangles, just about two butterfly’s wings), then every other presence of that Candidate on those Columns is certainly false.   The same can be said switching Rows and Columns.

Sample 14 - The Candidate 5 realises a Swordfish Butterfly lying on three Rows (the horizontal sides of two Rectangles with an overlapped corner showing 79,  as the Butterfly centre); accordingly the theory you can cancel the Candidate 5 outside the corners,  if it appears somewhere else on the Columns of the three vertical rectangle sides; that is the case of the azure Cell.

METHOD 15 - A Zone shows 4 Cells with exactly the same four Candidates, that appear also in some other Cell of  its (Naked Quads or pure quads).

Sample 15 - In the Row D the Quad 2689 appears exactly four times, in the pale grey Cells. But the azure Cell  shows the Candidates 89 already necessary for the Quad itself,  as a consequence you can clean them up from the azure Cell.

METHOD 16 - A Zone shows 4 Cells with the same four Candidates, ma some other Candidate accompany them in those Cells (Hidden Quads).

Sample 16 - In the Row D there are four Cells (in pale grey and azure) having the Quad 2789, but somewhere those Quads are not pure. You can clean them up from any unwanted Candidate, maybe starting taking away 1 in the azure Cell and then the cleaning up the others as necessary.

METHOD 17 – When none of the previous methods can be used, the player must proceed by Trial, that is  Try and error”: assume that a certain Candidate is the true Value and see what the hypothesis generates.

Two remarks:

- “Trying” with pencil, paper (and rubber) might be very complex;

- Playing with “Trials” isn’t cheating, because it requires co-ordination and cleverness.

When in “Trial Mode” the program temporarily hides all the Values and switch to an environment where the player may choose a Cell, a starting Candidate and trigger a step by step try process. The program offers two ways: trying seeing All the current Candidates or only those achieving some Conjugated Pairs (see TERMINOLOGY)..

## HOW TO USE THE PERSONAL TRAINER

When the player does not find a right Move to continue the game or when he wants to go faster, he can appeal to the PERSONAL TRAINER (or Teacher). The PERSONAL TRAINER gives his Advice accordingly to the standard Priority or the Priority the player can modify from the “Tools” menu.

Who really wants to learn, should ask him respecting the following gradual stages:

- “Method” button: it prompts on the blackboard the most convenient Method in that moment and this information may be enough for the player to proceed autonomously;

-  Value“ button  (to be used if  the “Method” button does not solve enough): it appends on the blackboard another line saying what to write or cancel on the Scheme; even if the interested Cell is not mentioned yet, the information may light the way to the right Move;

-  Where” button (to be used if  it’s still too dark!): keeping it pressed you can identify on the Scheme the Cell to write into (azure background), other Cells that are interested by the suggested Method (grey background) and other possible Cells to consider (maybe with pink background o a pale-grey one).

When releasing the “Where” button the coloured backgrounds disappear and only the Cell to play into keeps the evidence with its yellow background and can be manually played, but for the laziest …

-  Do it” button (or F2): can make the program itself to write the Move.

## THE NETS

A Candidate’s Net is the drawing obtained joining with horizontal and vertical lines all the Cells that at that moment allow the Candidate. Reasoning on the Nets’ shapes  may help to play some correct Move.

The SUDOKUPT program, as said before, offers among the “Advanced Functions” the Nets viewing.

Here you find some Net patterns: they are real cases, the program has produced.

Normally the Candidates are represented with an empty circle, but if from the “View” menu the choice “Show Solution” has been selected, a filled circle is used for “true” Candidates; this is the case of the samples below, but don’t overrate empty or filled, because those depended from the specific underlying game. Pay attention to the comments, instead.

FREE BRANCHES – The “ends” are surely true, the “nodes” false.

GRAFTED BRANCHES – Starting from a free “end” (that is true, as said before) you can assume sequences of false/true  “nodes”.

SIMPLE RECTANGLE – One couple of opposite corners is true, the other is false: it depends on the specific game. The Simple Rectangle a the bare figure that can complicate itself in many ways (to become, maybe, the X-WING following here).

X-WING – It occurs when two Rows (or two Columns) contain the subject Candidate only twice and in such a way that those Cells are the corners of a Rectangle. If that occurs, then all other instances of the subject Candidate outside its corners are surely false. As for true Candidates: it depends on the game.

TWO CORNER-JOINED RECTANGLES – One of three combination is true: aef, bcg, bdf.  When the d Candidate is missing, you must consider only aef, bcg combinations.  (This is the base shape that can origin a SWORDFISH pattern that follows here).

SWORDFISH -  It occurs when three Rows (or three Columns) contain the subject Candidate only twice in such a way that its Cells lie on the same three Columns (or Rows) and those Cells are the angles of Two Corner-Joined Rectangles. If that occurs, then all other instances of the subject Candidate outside its corners are surely false. As for true Candidates: it depends on the game.

Note – The same if  the Candidate exists in the joined corner Cell, as well. (three instances on that Row/Column).

MORE CORNER-JOINED RECTANGLES -  (with or not the subject Candidate in the Corner-Join Cells) – Every combination of  opposite corners might be true (one is shown here).

NESTED RECTANGLES – There are three possible true combinations (one is shown here).

A SQUARE OF RECTANGLES – No exclusion a priori. Every combination: ach, aef, bdf (here shown), bce might be true. Some Candidate (like d e g h) may be missing.

SQUARES SEQUENCE – You can consider any opposite corners combination, beside the one shown here.