-

# Mathematical domino combinations.Numbers in sets and digital systems of dominoes.

The domino costs in a row of other playing constructions and has attributes inherent in any gaming systems, but it is a much greater game design rather than others such as a chess or playing cards or draught checkers, because numbers of dominoes can be organized very diverse. Namely ratio of chessmen figures with squares of chessboard in the classical chess is optimum, if to not consider Japanese or Chinese variants of this ancient game, and also colors (suits) and numerical values of classical playing cards are optimum, but ratio of playing elements (tokens or tiles) and digital systems of dominoes with squares of checker-boards can have variations, and research of various mathematical domino combinations is an outstanding conundrum.
The traditional dominoes represents complete sets of pieces which look like rectangular tokens or tiles, or differently to tell playing cards or dies with numbers which form pair numerical combinations designated by points.
Quantity of dominoes in complete sets settles all possible pair or otherwise to tell twofold combinations of numbers or digits from 0 to N. Complete sets of dominoes with combinations of numbers from 0 to 6 or from 0 to 9 are most known, but hypothetically digital intervals can be from 0 ad infinitum. Moreover, some combinations of numbers in complete sets can repeat or be absent, and also decks of dominoes can be distributed on digital proportional groups.
According to mathematical systems of domino combinations in different complete sets it is possible to differentiate checker-boards on various rows of squares, and thus sizes of gaming space can be multifarious.
Namely the package of this board game has optimal checker-board which has 8 vertical and 8 horizontal rows, that similarly to a classical chessboard, and complete sets of dominoes which are organized accordingly to play on 64 squares. But besides, the mathematical system of domino allows to calculate different ratios of digital tokens with configurations of various playing boards.
Variants of mathematical domino combinations and digital systems are shown in the table.

 Lines of the table show intervals of domino combinations from 0 to N. Right columns show quantities of dominoes in complete sets corresponding to lines, and also quantities of doubles and fractions. And also quantities of dominoes in complete sets with repeated doubles or fractions, namely amounts of numerical domino combinations in the event that doubles or fractions will be presented twice. For example, the complete set with combinations of numbers from 0 to 9 includes 55 dominoes of 10 doubles and 45 fractions. If the complete set includes twice doubles then contains 65 tokens, and if twice fractions then contains 100 pieces of dominoes. The third complete set of dominoes in the package of the present board game on pages of this website corresponds to the third line and fourth column in the shown table, namely two decks of 16 elements with twice fractions of digital combinations from 0 to 3. And main complete sets do not correspond with numerical values of the shown table, as their design has only some repeated doubles. Actually shown table is the elementary scheme of mathematical domino combinations, but numbers in sets and digital systems of dominoes can be organized in more complex designs.

The table shows only general numerical combinations and digital systems according to which it is possible to change quantity of numbers in complete sets and design constructions of dominoes, but actually changes can be very different. Detailed research of various mathematical domino systems I shall try to publish in the future on pages of this website.

-

Traditional or classical complete set of dominoes with digital interval from 0 to 6 includes 28 tokens. Such mathematical system of domino numbers demands a checker-board having 8 horizontal and 7 vertical rows of squares, as 7x8=56/2 =.28.
Actually sizes of gaming space depends on rules of concrete games or conundrums, but configurations of playing boards should correspond to systems of numbers in complete sets, namely should be proportional for arrangement of dominoes on squares in view of mathematics.
The design of traditional or otherwise to tell classical complete set of dominoes within the limits of proportional checker-board is shown in picture:

 Tokens of dominoes are distributed on proportional playing groups which are located on opposite halves of schematic board as chessmen. 14 tokens of each proportional group completely arrange squares of two extreme horizontals, that similarly to arrangement of figures in a chess, and consequently 7 verticals and 8 horizontals are proper for this digital system of dominoes. Or it is possible to apply a checker-board having 7 verticals and 7 horizontals. Arrows at the right show how focusing designations should be put on tokens of two proportional groups, that is actually shown picture is the technical domino design according to which this variant of board game can be made. Numbers behind schematic board show sums of digits on vertical and horizontal rows, that corresponds to the magic square which form digital system in the classical complete set of dominoes in the shown playing positions, because sums on each vertical is 24, and on each horizontal is 21, or 21+21=42. Look information on magic squares and numerology of numbers on pages of other website: www.numeralgame.64g.ru.

It is necessary to tell that the shown magic square of dominoes does not grow out calculations of anonymous mathematicians or unknown scientists as telecasts told on the TV in one of television channels. But this or other possible magic squares are constructive parts of invention which is confirmed by patent RU2259223 according to which design of the present board game is organized.
And also it is necessary to tell that the shown traditional complete set of 28 domino tokens and playing board of 7 verticals and 8 horizontals can be taken as model for games in program of world championships in the International Federation of Dominoes. As such mathematical system of digital domino combinations is classical.

-

Complete set of dominoes with combinations of numbers from 0 to 7 and twice quantity of fractions.
Digital system of such deck corresponds to the checker-board of 8 verticals and 8 horizontals as contains 64 domino tokens which correspond with 64 squares, that precisely corresponds to the classical chessboard.
Playing construction or otherwise to tell numerical design of this complete set looks as shown in picture.

 Dark and light triangles designate tokens of two proportional groups. And also triangles are focusing designations, according to which dominoes should be focused on a chess board. Doubles are designated by differently directed triangles as their orientation has no value, but triangles of different directions are dark and light that designates belonging of doubles to proportional groups. Tokens of dominoes of two proportional groups too are designated by contrast triangles on backs. Mathematical domino combinations of this complete set can have interval from 1 to 8 instead from 0 to 7 and signs of digital system will correspond to numerical notations of verticals and horizontals of the checker-board. Then it is possible to play games and puzzles in which rules are based on parities of dominoes with numerical values of squares. Rules and instructions of such games with dominoes are not described in the manual in PDF files, and also images of this complete set for printing are not presented. But I can give necessary pictures by e-mail if someone wishes to print this game variant and to try playing process. This variant of game design too can be considered as digital model for world championships and tournaments under aegis of the International Federation of Dominoes. As mathematical system of 64 domino tokens corresponds to a classical chessboard that allows to play with completely open information and equal chances of players to win as in a chess, that is necessary criterion of sporting games. As a result games with dominoes can be recognized as a kind of sports, and can be included by the International Olympic Committee in programs of Olympiads as official sporting discipline.

It is possible to remove tokens with number 7 from this complete set. Then mathematical system of digital combinations includes 49 dominoes and approaches for a checker-board having 7 vertical and 7 horizontal rows. But distribution to proportional groups at this case cannot be considered, as the reduced deck contains not even quantity of dominoes.
Or it is possible to remove tokens with numbers 7 and 6. Mathematical system of digital combinations in this case contains 36 dominoes and approaches for a checker-board having 6 verticals and 6 horizontals. Distribution to proportional groups is correct at that, namely it is possible to consider dark and light triangles during games or conundrums.
In a similar way mathematical systems of digits in this complete set of dominoes can be reduced further, and as a result consecutive removal of tokens with numbers 7, 6, 5, 4, 3 will have quantitative changes according to sizes of checker-boards having 7x7, 6x6, 5x5, 4x4, 3x3 rows. In essence this mathematical system of domino combinations possesses unique digital feature according to which different intervals of numbers correspond with checker-boards of equal verticals and horizontals, or otherwise to tell with equipotent numerical squares. It can be considered in rules of various puzzles and conundrums with dominoes.

Digital matrix of dominoes.

 Numbers are put on domino tokens or playing cards of the shown complete set in the unusual way, namely hexadecimal digital matrix instead of decimal is used. This matrix allows to designate numbers from 0 to 15 by means of points. Namely points of dominoes can have 16 numerical values, and very great complete sets or designs of playing constructions can be organized owing to it. Picture shows some variants according to which positions of points in the matrix designate numbers, but designations can differ from offered, namely design of dominoes can be miscellaneous.

Design of this game package can consist of different complete sets and decks of dominoes in total.
Complete sets with big intervals of numbers can comprise small digital intervals, so smaller decks can be received at reduction of greater, and each system of digitss can be distributed on proportional game groups at that. Namely some complete sets can be incorporated in one, and everyone can be distributed on proportional groups, or otherwise to tell digital system of dominoes of one common complete set can include mathematical combinations of domino numbers which are grouped according to numerical ratios of different orders.
Thus design of domino tokens can have corresponding designations that players could choose different complete sets and proportional groups of different orders, and also could distinguish own and another's dominoes.
In addition design of tokens or playing cards can have focusing designations which correspond to distribution of proportional groups and allow to have dominoes on squares of a checker-board in playing positions as figures in a chess game, that provides equal chances of players to win a prize.
Many possible variants of dominoes can be in detail calculated and prepared for industrial production if manufacturers of board games or book-publishing firms, or private publishers will be interested in commercial application of any gaming designs which are described on pages of this website.
Hypothetically this intellectual logic game can be considered as a package of constructive digital pieces which allow to form diverse complete sets or decks of dominoes and different configurations of playing boards. Or most simple complete sets of mathematical domino combinations can be prepared for industrial production of the present board game.

All shown constructive decisions and digital systems of dominoes are confirmed with patents for inventions, and also with certificate for effective model RU 30332.

Pieces of the shown complete sets can be made not only of paper as printed playing cards, but also in the form of tiles and tokens of various materials, and including of natural stones. In particular by means of marble and granite, or various semiprecious stones as souvenir and gift copies.
Playing boards too can be made of various natural and semiprecious stones in the form of game tables that can be interesting as unusual and extraordinary gifts to presidents of publishing firms and manufacturers of toys, and also managers of industrial companies which can carry out mass production and trading distribution of the shown variants of dominoes.

upwards