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# How to solve to a pike perch?

the Latin square

In the 18th century the great Swiss mathematician Leonard Euler thought up the entertaining numerical structure called “ Latin square “ (Carr é latin) which became a prototype of new game - the numerical crossword puzzle which appeared in the USA in 70 - x years of the last century. This game those years had no special popularity since did not correspond to dynamic American mentality. However, having got to Japan a decade later, game found huge popularity and received the name “ To Sudok “ (in translation from Japanese literally means “ numbers - nearby “) . Already from Japan game promptly extended worldwide, having made the serious competition to usual crossword puzzles.

Rules of the game are simple

of the Rule of the game: in classical option the matrix of 9õ9 in size divided into 9 blocks of 3õ3 in size is set. The part of cages is in advance filled with figures from 1 to 9. The less cages are in advance filled, the complexity of game is higher. I will call the blocks of 3õ3 in size outlined by the fat line sectors, ranks of the cages located across - lines, and down - columns. I will call sectors, lines and columns in general clusters, and all initial table 9õ9 - a matrix. The only rule of this remarkable game is that figures in each cluster of a matrix should not repeat.

The game purpose

the Purpose of game consists in filling empty cages (cells) of a matrix by rules of the game. It is recommended to fill empty cells with a pencil that it was easy to correct errors and to clean incorrect options.

Decision to a pike perch

Rule No. 1. the simplest and obvious rule is that if in a cluster 1 cell is not filled, then it is filled with number 45 - S where S - the sum of numbers in a cluster.

Rule No. 2. we Will assume that we authentically know that number N can be only in two blank cells of a cluster. In this case these cells I will call semi-filled. If only in the same two cells of a cluster there can be also number M, then these cells should be considered filled. So, two cells semi-filled with two figures are filled. In other words, in them there can be no other figures.

From this the simple and obvious rule there is an interesting consequence: if in a cluster 3 cells, but two of them a poluzapolnenna are not filled with numbers N and M, then in the third cell there has to be number 45 - N - M - S where S - the sum of the filled figures in a cluster. This consequence extends to any number of couples of semi-filled cells.

Rule No. 2A. we Will assume that we authentically know that number N can be only in three blank cells of a cluster. In this case these cells I will call filled on 1/3. If only in the same three cells of a cluster there can be also numbers M and L, then these cells should be considered filled. So, three cells filled on 1/3 three figures are filled. In other words, in them there can be no other figures.

Continuing on induction, we receive the similar rule for four, five etc. cells.

Rule No. 3. Two clusters having the general cells I will call interfaced. If one of the interfaced clusters - sector, and another - the column or a line, then crossing makes three cells if a column and term, then one. If in crossing of the interfaced clusters number N meets, then it can be in the crossed parts of both clusters.

From this obvious rule there is a consequence: if in not crossed part of one of the interfaced clusters there is number N, then it also is in not crossed part of other cluster.

From this consequence we receive one more rule:

Rule No. 4. All matrix can be divided across and down into 3 equal parts (third) till three lines or three columns. In each of these parts each number meets three times: one time in every line (column), and in each sector. If number N in any third meets twice, then the third has to be on crossing of that line (column) and that sector in which this number does not meet.

And now the most interesting: this rule works also for semi-filled and on 1/3 filled cells if they are located in one sector and go along a line for horizontal (consisting of lines) thirds of a matrix, and along a column for vertical (consisting of columns) thirds.

Example: let one of horizontal thirds of a matrix be filled as follows:

2. x. x. x. x. x. x. x. x

X. x. x. x. 2?. 2?. x. x. x

X. x. x. x. x. x. z. z. z

Where x and z - the blank cells, and 2? - the cells semi-filled with number 2.

Then crossing of the lower line with the last sector (the cage designated by a letter z) have to on 1/3 be filled with number 2.

Combining the rule No. 4 for horizontal and vertical thirds of a matrix, we can unambiguously place numbers in sector of their crossing. It is also the basic rule of the decision to a pike perch which allows to solve problems of any level of complexity.

So, you in hands have now everything that will be required to you for this the fascinating and developing logical thinking game!

Resources to a pike perch

to Sudok online.

to Sudok offline. Club of fans to a pike perch

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