Implementation of a new variant of chess

BACKGROUND OF THE INVENTION

The present invention relates to chess games and, more specifically, to a new chess variant and method of play thereof affecting the beginning of the game, along with its implementation.

DESCRIPTION OF PRIOR ART

Classical chess (or simply "chess") is a game of skill for two players, played upon an eight-by-eight square, checkered board. Chess rules and movements of the pieces have been essentially unchanged for several hundred years, except some minor unifications more than a century ago which have resulted in a game which is played across the globe in the same manner.

Chess from its origins has been an "almost infinite" game, where the combinatorial explosions of possibilities gives raise to so many different positions that playing skill and talent have been more important, and overwhelmed attempts at memorising most possible courses of the game.

There have been proposals to modify classical chess, or alter its rules. The reason behind this is that widespread and intense study of the game has made many strategies and aspects of it deeply analyzed, compromising creativity and, crucially, "over-the-board" discovery of the moves on the part of the players. Most importantly, in the last 15-20 years, the growth of opening theory together with generalised access to computer-aided preparation have enabled rote memorization of opening lines (i.e. initial sequences of moves by both players) to have a strong impact on elite competitions, as well as on competitive professional and amateur events (see the work by Kasparov in the References).

Chess variants proposed in the past have often entailed the creation of new pieces for use along with classical chess pieces, as well as new boards or rules. However while on an abstract level these variations may occasionally be as interesting (or more interesting for some) as the classical game, and have achieved some popularity, they have not made a major impact on the chess-playing world. The reason, we believe, is that the history and tradition of the game is so strong and has created such a paramount shared heritage, that most chess lovers find any substantial change to the way the game is played (change of any of: board, pieces or rules) as "alien" and devoid of any attraction. We will thus not follow this first line of invention in the present work. We have concluded that changes to the game of chess that feature such profound alterations result simply in wholly different games, and do not offer any hope of creating a variant of the game satisfactory for the current classical chess community.

There has been another, very different, second line of invention, Fischer-Random ("FR") chess,

(advocated by former American chess World Champion Robert J. Fischer) now-called chess960, which in the pursuit of a variant of the game has avoided changing the board or adding new pieces, or any rule of the game except the initial position of the pieces. This initial position is obtained from a random reshuffling of pieces before the start of the game, within each of two rows where pieces lie in the initial classical chess position (i.e. the first and last ones, or the two that are closest to each player, where all the pieces but the pawns lie). From then onwards, the game proceeds as in classical chess. The resulting game is much closer to the original game of chess than the variants discussed above, and has gained acceptance and some popularity. FR chess has also been accepted as a relevant, if still a minority part of the chess world (see References herein about the Fide Handbook) while it has not challenged in practice the dominance of the original form of the game.

We believe it is logical and deserving that FR has achieved a higher standing in the chess world than variants from the aforementioned, first line of invention. FR chess is not an "alien" game to chess players. However, still the weight of chess tradition and the profound influence of the initial position of the pieces in the development of the opening and middle-game phase of the game, have made widespread popularity of FR chess elusive. It is noteworthy to what extent the importance of tradition for the chess community has been yet again confirmed. Note that as the initial classical chess position is one in 960 positions, the history of the classical game is roughly 0.1 % of the possible development of FR going forward, hence history is not fully obliterated but almost so - it has an extremely minor role in FR chess, as has been indeed the intent of its advocates. Another chess variant along the same line of invention than FR is Random Opening Chess, also interesting but which has not gained any of the popularity of FR chess, and has the same issues as FR chess but to a greater extent, so we will only mention it here.

We consider the present invention to be in the same line of invention as Fischer-Random chess, as only the starting position of the game is modified, but not any other rule or its board and pieces, while such starting position is determined at random just before the start of play.

We conclude the description of prior art with the observation that no chess variant that has ever been invented up to now seems to be close enough to classical chess for the tastes of the current chess global community, i.e. providing the empirical lesson that no modification of chess that does away with its legacy and tradition is good enough, in this sense, as an alternative for such community.

PURPOSE OF THE INVENTION

To provide a new variant of chess and its implementation, that is as similar to classical chess as possible and preserves its legacy and tradition, while it substantially diminishes the role of rote memorization of the opening phase, which has recently been steadily increasing due to the impact of computer-aided preparation. Specifically

1 . There is no change to the board, the pieces, or the rules. 2. Only the initial position may differ from classical chess, and is uncertain, i.e. not known until just before the beginning of play, and taken at random from a known collection of positions which is representative of actual contemporary master (i.e. skilled) competitive play.

3. It fully integrates and preserves the legacy of chess within the realms of the new game.

See 2.4 below for a more accurate description.

We rephrase the purpose of the invention in the following aspirational way:

To provide an " evolution" of classical chess, i.e. not a mere variant but a method of play that introduces uncertainty in the initial position while the legacy of chess is preserved, hence along with it the revealed preferences of the current classical chess community. We intend such an evolution to have, for the first time in the history of chess variants, the foundation to aspire to become a widespread alternative or complement to classical chess.

The author has denominated his invention "Neoclassical chess". SPECIFIC PROBLEM SOLVED BY THE INVENTION

The objetive of the present invention is to develop a new variant of the game of chess that solves the issue of the excessive memorization of the opening phase while preserving the rules, tradition and legacy of the classical game of chess, along with its implementation. We have defined in what follows the requirements that such a new game (which we call "neoclassical chess") should satisfy:

(i) It is a game equal to chess, except in its initial position, and the playing set-up process is simple as in chess.

(ii) It starts from a balanced initial position i.e. as balanced as that of classical chess and includes and preserves its legacy. All master chess games are possible "neoclassical chess" games (except for rare exceptions: if they feature an incorrect opening or are not relevant from a competitive viewpoint), a property which we denominate "backward compatibility" with classical chess. Reciprocally, all future "neoclassical chess" games are possible as games of classical chess ¹ , which we call "forward compatibility" with classical chess.

(iii) It must reflect the opening preferences of contemporary skilled human play, with the capacity to evolve and incorporate future preferences through a systematic, non-arbitrary process.

¹ i.e. considering its development as appended to the concrete, initial sequence of moves leading from the initial position of classical chess to the initial position of neoclassical chess, as it will be clarified later. Additionally, we have aimed to modify classical chess in the least possible degree, the strict minimum needed to solve the problem, and to accomplish this in a reproducible, non-arbitrary manner.

Mathematical framework

In order to develop the new game, we have built a logical framework that delineates our objective from a mathematical perspective. The problem has been reformulated as a "constrained optimization" problem within the set of all variants of chess which fulfill the three requisites mentioned above. The objective is to find the one among these which is closest to chess and verifies a certain constraint: there must be sufficient uncertainty about the initial position of the chess variant so that the value of opening preparation based on rote memorisation is in practice greatly diminished, as it is very difficult to accomplish and unlikely to be useful (because of the large number of possible lines and the high likelihood of forgetting such preparation in time). The constraints are expressed as restrictions on the cumulative distribution function of chess positions (taking into account transpositions) at several key probability levels of the validity of the preparation. Please see Appendix I for an illustration of the mathematical formulation of such constraints.

BRIEF SUMMARY OF THE INVENTION

NOVELTY AND UNIQUENESS OF THE INVENTION

The invention we propose is the only known chess variant that achieves randomness by choosing its initial position in the following manner: a game is chosen at random from a database of relevant contemporary master chess games, and the position after an initially specified number of moves is selected, to be used as initial position in the new invented game.

Hence it uses a single reference database of games that represents the state of the art in relevant games between players of sufficiently high skill, so that a randomly chosen, correct initial sequence of moves gives rise to a position which is in turn used for the start of the game. We claim that this chess variant uniquely both integrates and preserves the legacy of chess (i.e. is forward compatible as well as backward compatible with the classical game, as defined in 2.4(ii) above).

SHORTCOMINGS OF CURRENT SOLUTIONS WHICH THE NEW INVENTION ADDRESSES

These have been considered in 2.2. The current chess variant which most closely resembles classical chess is Fischer-Random chess (a.k.a chess960), to the exclusion of the invention herein presented. Referring to the three objectives specified in 2.4 above Fischer-Random chess fulfils (i) and regarding (ii), is "backward compatible", while it is not "forward compatible": many, in fact the vast majority (statistically, more than 998 in 1000) of Fischer-Random chess games are not possible games of classical chess. Additionally, it does not fulfil (iii) either, as it does not reflect opening preferences of contemporary classical play (in fact, it does away with chess opening theory). Note that other variants that introduce new board, new pieces or new rules after the start of the game occurs do not even fulfil condition (i) (and of course, nor do they fulfil (ii) completely, nor (iii) ).

UTILITY IMPROVEMENTS OF THE CURRENT SOLUTION WITH RESPECT TO PRIOR ART

The author claims that switching from classical chess to the new game, or complementing its practice with the new game, particularly from moderate up to the highest skill levels, likely achieves substantial utility improvements.

The major ones, from which others partly stem, is a significant increase in OTB ("over-the-board") ratios of play as high-depth, player-specific opening preparation becomes of little use. This is extremely valuable for the game as a sport (i.e. based on live actual performance of the players) and for the interest and delight of the game spectators and results in more creative play also in the opening.

The players themselves at all levels from strong amateurs upwards, and particularly at the top competitive level, experience substantial utility improvement in:

Not having to exert themselves for an inordinate amount of preparation time in rote opening memorisation, which is generally seen as most unpleasant (while in the new game emphasis shifts to understanding of the whole possibilities of the opening and to pattern recognition, which is the hallmark human skill used throughout the game)

Longer value of the study of the opening made in the past, or less time needed for it after the opening repertoire in the new game is built

A larger value of making the personal "human capital" investment of becoming a chess professional or devoted amateur, with plausibly a longer-duration career, as rote memorisation ability decreases markedly at senior ages and much less short-term memory refreshing is needed.

All this happens as the new game preserves the legacy and essence of chess, which has been seen empirically very important to the chess community and the major obstacle to the growth of chess960. Our investigations also indicate that the possible initial positions are not excessively forcing or conditioning to a player's style: as strong randomness is reached for departures from the initial position of a very low number of moves.

Another important feature of the new game is that contemporaneous classical chess collective opening preferences and their future evolution are naturally integrated into the modified new game, as the distribution of opening sequences mimicks that of contemporary master play.

The final, all-encompassing utility improvement is achieved by network effects as a new, improved game is made available for an already massive following of chess players. See paragraph 5 for other benefits, including those affecting casual amateurs and learning players.

All this utility improvements may be the base for generating value, economic and otherwise, through the promotion of new chess events and methods of implementation described below.

DETAILED DESCRIPTION OF THE INVENTION

CONCEPTUAL PROCEDURE AND METHOD OF PLAY

The invention herein consists of series of steps, together with its implementation by software within several types of apparatus and/or its transmission by electronic or digital means. The procedure includes a randomization step that will result in the selection of a legal classical chess position from where the game would be started. The new game would be exactly equal to the game of classical chess from thereon (i.e. following all the classical chess rules), thus differing solely in the initial position from where the game starts, which will be one among many possible ones, as different from classical chess, which starts from a unique position of the pieces, crucially known in advance by the players.

Definition of "Neoclassical chess"

In order to better describe the procedure we define the collection of different "Neoclassical chess" games, considering all "depths" (number of full moves), where D is an integer number D = 0, 1 , 2, 3... For the purposes of the current patent application, depth D may go from 0 to 10 in increments of 1 .

Example of meaning: Depth 2 refers to a position in a game two full moves after the initial classical chess position, i.e. after the second move of Black (note that a full move comprises a move by White and the move by Black which immediately follows after it). In this position White plays next.

Neoclassical chess at depth D (where D is an integer) as the following way of playing: A game of chess that proceeds in the following way. Just before play, a classical chess game is selected at random from a reference database of all relevant, correcf contemporary human master games, and the position in it after D full moves is designated as initial position of the Neoclassical chess game. The players start the game from such position thus obtained. The position in the selected classical game is obtained after D moves by White and D by Black, so that the White player chooses the next move in the Neoclassical chess game.

Note that Neoclassical chess at depth D = 0 is precisely classical chess, which we retain for internal consistence of the denomination but which is not a new game.

We can similarly define slightly different games when D is an integer plus 0.5. Note that this is equivalent to depth being an odd number of "half-moves", (a "half-move", as different from a "full move", is a move

² Correct at least up to including full move D. from only one of the players, either White or Black). We denominate this variant "Neoclassical chess with Black pieces" or "Neoclassical Black chess".

Where appropriate, "Neoclassical chess" will refer in all what follows indistinctly to "neoclassical chess" or "neoclassical black chess" in the context of two forms of the same, new game or of the same invention.

We define the collection of different "Neoclassical black chess" games, considering all "depths" (number of full moves) D = 0.5, 1 .5, 2.5, 3.5,.... For the purposes of the current patent application, depth D may go from 0.5 to 9.5 in increments of 1 .

Example of meaning: Depth 2.5 refers to a position in a game two-and-a-half full moves after the initial classical chess position, i.e. after the third move of White (hence it would be the position reached at Depth 2 plus an additional 0.5 move, which is a half-move by White, so its third one). In this position Black plays next.

Neoclassical Black chess at depth D (where D is an integer plus 0.5) as the following way of playing: Equally as "Neoclassical chess at depth D" above, except that in the "Black" variant, D is not an integer but equals N+0.5 (with N being an integer), so that the position in the selected classical game is obtained after N+ 1 moves by White and N by Black, hence the Black player chooses (moves) first in the

Neoclassical Black chess game.

This will result in a variant of the new game in which positions are obtained such that the last move is a move by White. It is more different from chess than the variant resulting from the whole integer depths described above. However this additional variation of the new game has independent interest as White would retain the advantage of having moved first, but Black as partial compensation has the slight advantage of making the first choice, within the neoclassical game, after the initial position is obtained.

For an illustration of the new proposed game in these two forms, see Figure 1 at the end of the application.

Further explanations on the terms of the definition

How is depth D chosen: depth is chosen according to the type of new game that is to be played. It can be chosen optimally to be the minimum possible that obtains a sufficiently large number of positions, in the sense described at the end of 2.4 above, so that preparation by rote memorisation becomes impossible, either in the normal or "black" variants described above. See Appendix I in the current patent application for an illustration on how to explore what are the theoretical optimal depths. Notwithstanding this, even if the depth(s) that best work in practice are to be confirmed by sufficient experimentation in practical events, the theoretical investigation undertaken for the current invention strongly suggests the best options. Our preliminary analysis, to be confirmed by practical experimentation, suggests that best uses appear to be: a. Depths 0.5, 1 , 1 .5, 2.0 for school chess applications and amateur training and b. For "neoclassical chess" as is to be customarily played: depth 3 for a single standard of play at all levels, including top-level chess.

c. For "neoclassical black chess" as is to be customarily played:

i. Depth 2.5 for amateur and non-professional master competition

ii. Depth 3.5 for a single standard of play at all levels, including top-level chess. d. Depths 4, and above through 10 in 0.5 increments, (including fractional depths as 4.5, 5.5 etc.) possibly mainly used for exhibition and opening and training, while depths above 10 mainly for training purposes.

In order to clarify the procedure, we further explain several terms used in the definition of "neoclassical chess".

How a game is selected at random: as many games of the reference database will repeat positions at a certain depth, those positions are not equal-weighted, but appear in the same proportions as those in the collection, reflecting the distribution of collective opening preferences of "masters", which was the condition specified in 2.4 (iii). Note that as we draw from the "empirical distribution" of games (the database), for a sufficiently large number of games we obtain the same (a priori) probability distribution of openings within the new, neoclassical chess game.

What are contemporary games: A sliding "time window" is used. The last N years up to the current moment are considered as defining contemporary play, where N could be any reasonable number; for example, 5, 10 or 20 years. Every year or every few years, the reference database of games is updated, taking again the previous N years. This also provides for the evolution of chess opening preferences as requested in 2.4 (iii). Example: In January 2016 we would construct the database with all relevant games from the previous 10 years, 2006 to 2015, both included. In January 2018 the database would be updated and would comprise relevant games from 2008 to 2017. In an alternative process, games from all recorded history could be used.

What are correct games up to a certain move: We would consider them correct if they appear frequently enough in top-level games (where both players are above a certain official rating number, higher than for just qualifying for the database) or the computer evaluation (using a specified highly-rated compute "engine" or software program) after each move changes negligibly, so that the attained position does not deviate significantly from that of the initial classical position, or a combination of both criteria What is relevant human master play: master in a broad sense, i.e. players of sufficiently high skill level, and taking (relevant) rated play competitions. The proposal is to take only "rated" ³ games where both players are above a certain rating (as a possibility 2400 or 2500 which correspond to conventional "International Master (IM)" or "Grandmaster (GM)" ratings), while other ratings could be considered. Also reasonable criteria are whether to use only standard ("slow") rate of play (i.e. time limits used for the chess clock), i.e. eliminating blitz, rapid and blindfold games which can bias the distribution due to the specificity of these forms of play, or include at least rapid play. This could be further sophisticated using different variants, a percentile of top-rated players and the rating limit could evolve slightly over the years; this complication adds nothing significant because of the slow evolution of preferences. Also, relevant play could exclude short draws (games that end in a draw after at most a certain number of moves) where not much competition may be intended by the players, so that they do not bias the database.

SPECIFIC NEW PROCEDURE

This section describes the above procedure in more detail regarding the order of the steps, where we assume the maximum depth of interest Dmax is given (see above for the meaning of depth in

"neoclassical chess" and "neoclassical black chess").

Construction of the "reference database"

1 . A public or commercial database of all relevant games (for example "rated" human games, and/or from official or so deemed "serious" chess competitions) is collated and used, in a standardised public format ⁴ .

2. An initial filtering process is done on the database. This consists of the following collection of steps, where steps "a" through "c" can be realised in any different order.

a. Eliminating non-contemporary games, where contemporary games encompass a

previous time period before a relevant reference date.

b. Eliminating all games where one of the players is at time of play below a certain level of rating representing a sufficiently high skill level. The minimum rating defining the skill level can be constant over time or can be slightly adjusted to account for rating "inflation", taking different levels in different time frames. Alternatively, accredited titles can be used as a criteria for qualifying players over the minimum.

³ This "rating" can be obtained from any public sources (international or national federations) or can be assigned by the inventor with a systematic criteria considering several of these.

⁴ The PGN format is one such public format for annotating chess games. c. Eliminating non-relevant games: for instance, use standard ("slow") rate of play, eliminating blitz and blindfold games which can bias the distribution due to the specificity of these forms of play; in this example rapid-play may or may not be eliminated.

Computer games are also eliminated.

d. Early draws can also be considered for elimination.

e. Thus an initially filtered game database is obtained.

Optional rare opening sequence determination step. Listing and classifying all distinct positions arising from "opening sequences" with their corresponding absolute and relative frequencies (different move orders can give rise to the same position). Establishing which are "rare occurrences" of such positions, (i.e. where they appear in less than a pre-specified, small percentage of cases as e.g. 0.01 % or 0.001 % and/or in less of an absolute number of games) ⁵ .

Optional filtering by truncation step. A second filtering process is done on the database

("truncation process"), where all games which result in an opening sequence which is deemed a "rare occurrence" are eliminated.

For each game in the database, each position obtained at the maximum depth of interest Dmax and the sequence of moves leading to it is examined, and is approved if:

a. The position appears in a sufficiently large fraction of relevant games where both players have sufficiently high rating (top-level human chess approval criteria), or b. alternatively, a highly rated software chess engine (e.g. over 3000+ equivalent human rating, more than any human player in history), or several such engines via an average, is used to assess if the resulting position has an evaluation which differs from the evaluation of the initial classical chess position less than a certain threshold (which may or may not be calibrated by the above criteria in 9a).

c. In the unlikely case that it is not approved, the game is discarded, along with all others that lead to the same position at the maximum depth of interest.

Thus the final, single "reference database" is obtained, which is not dependent on the choice of depth. Although the resulting reference database of games seem to depend on several design choices, the author claims that these are not too important for the resulting distribution of openings in the database, which is highly robust, i.e. differs little, within broad reasonable ranges of the parameters used above. Hence the result is non-arbitrary except perhaps by a very small

⁵ This is performed for Dmax, so that the process does not depend on which is the depth as long as it is below the maximum. margin of approximation. Note that the construction of the database is performed once every year or every few years.

Obtaining the initial position in neoclassical chess

The depth D is given.

1 . A game is chosen at random from the reference database thus obtained, with all such games having equal probability of being chosen (uniform probability distribution), using a high-quality random generator for the problem at hand. Thus any such game will have a 1/N probability of being chosen, N being the number of games in the reference database.

2. The first D full moves of the game are selected, determining the "opening sequence". If D is an integer, the sequence will end with a move by Black. If D is non-integer, the sequence will end with a move by White. Note that opening sequences have different probabilities of being selected, as some may be more frequent than others at such depth within the database.

3. When such D full moves are played as in normal classical chess from its fixed initial position, a legal classical chess position will result.

4. The position thus accepted is the outcome of the procedure and will be used for the players of the new game ("Neoclassical chess") to start play. In the case that depth D is an integer, the White player will start play as in the classical game; otherwise the Black player will. From then on, play will be conducted as in classical chess.

For other variations of the procedure, please see Appendix II. IMPLEMENTATION OF THE INVENTION

It will use software that will store the "reference database" and a high-quality random generator, apt for the large number of games or sequences of moves within the database. Actual implementation of the invention will proceed along the following lines, including without limitation as a representative sample.

1 . A spreadsheet or software program is implemented in a computer, personal or otherwise,

including the reference database (which can be simplified to include only opening sequences or positions) as described above. A random choice is made on these (uniform in the former, nonuniform in the latter and according to opening sequence frequencies) and the selected opening sequence and initial position for the new game is obtained for the players to start play from. The sequence may appear in standard chess notation, or the position may appear in a chess diagram or other equivalent means. This could or could not be used in conjunction with usual dedicated tournament chess software for organizing tournaments (be it closed, e.g. Round-Robin or open tournaments of the new chess form). The purpose of this implementation is the use for collections of players: chess clubs, tournament organisers and arbiters. 2. Application for websites that manage game playing by large collections of players online, possibly including computer programs as players. Similar software programs implementing the procedure and putting in contact any computer connected to the internet, in order to play online the new game (neoclassical chess or neoclassical black chess, each at any depth). The purpose of this implementation is that commercial and non-commercial chess-playing websites enable on their sites the new game.

3. An application for smartphones, portable or wearable computers or physical devices (for instance game consoles, new or existing). Alternatively, additions or modifications to already existing physical devices. Such procedures may install the "reduced form" implementation described in the Appendix II whenever small memory resources in the computer or device make it advisable, to avoid a full database of games. The purpose of this personal implementation is that an individual or small group of individuals be able to obtain in a fast manner its random initial position for play.

4. A chess program (software) for installation in any computer o device that would include as an additional feature the possibility for the buyer of playing the new game with its chess engine.

5. Other additional features to the core capabilities (e.g. select highest-rated games of the database which feature the three possible results of win by White, win by Black, and draw).

For an illustration of the implementation, see Figure 2 at the end of the application.

OTHER USES AND BENEFITS OF THE INVENTION

The invention could potentially be used similarly in other board games of skill where in order to decrease the importance of memorisation of the initial position, this is obtained from a random choice from a database of actual human games up to a certain move. Some other additional or incidental social benefits of the invention are arguable as they include plausible but yet-to-be-proved empirical claims, in which the author strongly believes:

The adoption of the new version of the game of chess invented plausibly enhances the personal well-being of the aggregate of professional and elite players once it becomes the standard, as a result of less exerting and more pleasurable preparation. We also predict that a transition "investing" period in opening preparation, whether initially more time-consuming or not, will lead to shorter total lifetime preparation time (even including time discounting) at the depths associated with competitive professional play, and probably at other levels too.

The smaller relevance of memorisation for the general, amateur chess population, results in seniors' relative ability not being artificially downgraded by the boost that computer-aided preparation has given to an arguably already-too-large importance of rote memorisation. In the current state of affairs, this artificial degradation of the chess performance of very experienced players (inevitably seniors at some point) results in discouragement, plausibly reducing their interest in chess. This is socially negative not only from the interest as a competitive pastime, it becomes more socially relevant in the context of recent investigations of chess practice as a tool in the prevention of Alzheimer.

The benefits described within 3.3 may likely result in increased levels of interest from the chess and wider community. Also, if the new game becomes popular, although it may partly displace ("cannibalize") some interest from classical chess, the aggregate result (sum total of interest in the classical and neoclassical game) is also highly likely to increase with the total number of chess events. It does not seem far-fetched to speculate that the total economic value of the chess as an economic activity (and hence salaries and profits over and above wide amateur satisfaction) can also increase in a more competitive and less predictable game.

Neoclassical chess might well boost the well-established benefits of chess in schools once an initial chess learning phase is completed and the opening becomes a subject of study, as it strongly emphasizes the skill of pattern-recognition (with its unambiguosly positive impact on intellectual youth development) over rote memorisation. The benefit is compounded by the fact that the usefulness of the latter is being increasingly challenged in the digital era.

NON-OBVIOUSNESS

We claim that although once known to the public, the current invention may in the future be deemed by some to be not excessively complex in retrospect, it is in fact highly non-obvious in its purpose, i.e. as a solution to the current predicament of the game of classical chess.

The concerns that the current invention addresses are very well-known and publicized and are found in declarations not only of chess observers, but in those of many of the top current and recent chess players, and have been well-known for at least two decades. For the invention to be a success, it has to find acceptance from a substantial part of the current chess-playing community; otherwise its value (economic and other) will be small, as value can only be derived from a quantitatively substantial number of chess players wishing to engage in the new form of the game and to use the new invention. If such success were in fact obtained by the new invention in the near future, it would demonstrate that it is highly valuable. It would be hard to believe that a highly valuable invention would be simultaneously "obvious" and simple enough to be at reach by many of the massive number of followers of the game. We argue that the core of the current invention is not only the procedure itself but the fact that it serves the purpose of solving the multifaceted, highly constrained problem herein described, thus involving a strong counterfactual (leap of imagination) as to the future collective reaction to the invention of a given large group of individuals (the current chess community) and their evolving reaction as they test the invention. Thus the invention will be subject itself to the verdict of an already existing, large group of users, and the very fact of eventual success would confirm its non-obviousness.

REFERENCES

R. DESJARLAIS, "Counterplay: an anthropologist at the chessboard". University of California Press. (201 1 )

R. EALES, "Chess: the History of a Game". Hardinge Simpole Limited, revised edition. (2002)

S. GIDDINS, "How to build your chess opening repertoire". Gambit Publications (2003).

[Kasparov] G. KASPAROV, "Garry Kasparov on Modern Chess. Part I: Revolution in the 70s "(Chapter 24). Everyman Chess (2007).

The rules of chess960 (Fischer-Random chess) are recognised in the FIDE Laws of Chess of the international chess federation (page 22).

STATISTICAL ANALYSIS OF OPTIMAL DEPTH

For each depth, we obtain the probability density function of resulting positions, taking into account transpositions (different move orders arriving to the same position), both in the cases of integer depth (neoclassical chess) and non-integer (neoclassical black chess).

For each depth, we order all positions in decreasing frequency. With this order, we construct the cumulative distribution function F, statistical term that allows us to state that, for example, after Black's seconde move the 7 most frequent positions cover 50% or more of the games in the reference database. In our formal framework, the "restrictions" of the optimization are those that such function F must verify. From the results in the database, we construct the table seen as figure 2 (using a concrete,

representative database for illustration):

The table shows for different depths the probability of the most frequent initial position, and how many positions (taking them in order of decreasing frequency, hence first those more important) a player must prepare to be assured, with a certain probability, that the initial position obtained is among those he/she has prepared. For example, at depth 2 a player must study 7 positions to be have a chance of at least 50% that his/her preparation has been useful, and 13 for a 66.6% probability. In statistical terms, the restriction on the uncertainty is expressed as: F(7) > 0.50 where 7 is the smaller integer that verifies the equation. Also, the most frequent position at depth 2 appears with a 1 1 .9% probability, and is that obtained after the sequence of moves in algebraic chess notation 1 . d4 Nf6 2. c4 e6 (where N stands for "knight"). This number 1 1 .9% includes less frequent move-orders leading by transposition to the same position. An analysis of the results indicates that the jump from depth 2 to 3 is significant at all relevant probability levels of validity of preparation, in particular the very relevant 66.6% and 75%, reaching for the first time at depth 3 sufficiently high levels (more than 50 positions), and strongly indicates that 3 is the minimal depth that achieves sufficient uncertainty for all levels of play, including top-level chess. Similar analyses are carried out in the case of Neoclassical black chess, and are confirmed by the examination of other databases obtained with different parameters.

VARIATIONS OF THE PROCEDURE

Modifications of the procedure: "Variable-depth" Neoclassical chess

It may be possible to use different depths, while they result in a complex game specification, where the depth of the initial position of the new game could vary. This is another possible variant, that cannot be described as a process independent of the actual statistical distribution of games, whereas the main form of the new game that we have described can be determined by the simple choice of the first D moves of a random relevant master game, i.e. independent of the actual distribution of games. However, it is yet a further simple modification of the original invention.

A modification may be to select initially at random the depth to be used, instead of using a different, fixed depth for different purposes. For example, a random draw would be done from an initial distribution that assigns a 10% probability to using depth 1 , 15% to depth 2 and 75% to depth 3. Then once the depth is obtained, the rest of the steps from 1 would follow similarly. This would combine to a "variable depth" version where maximum depth of any initial position is 3.

Another interesting modification of the procedure, at the expense of conceptual and practical complexity is the following "variable depth" version: construct a database of positions for a certain depth, let us say 3. Then, keep all the positions unchanged up to a significant cumulative probability (of the most frequent ones) above a significant high threshold (75% or 90%). For the remaining marginal low-probability positions, reduce all sequences of moves that can be grouped at a lower depth6 (also an integer as 3 is one, so White plays next). There is an interest in simplifying and grouping in a natural way this infrequent openings to lower depths, as they do not affect the restrictions on the uncertainty at the key probability levels, which are below the threshold.

Other "variable-depth" variations are similarly possible. For example, most of the opening sequences could be of depth 3 except one where the sequence has been replaced with 2 sub-sequences of depth 4 with 2 respective frequencies adding up to the frequency of the sequence they replace. This possible modification may make possible a variant with opening possibilities conforming to some conventional opening classifications, preexisting or otherwise.

Modifications of the procedure: different order of steps and use of a "reduced form"

The description in 4.2 condenses the steps of the procedure, but different variations of the procedure may be used. Many of the steps can be performed in a different order. Without being exhaustive, many permutations are possible: as a matter of statistical sampling, the steps in 4.2.1 : 2a, 2b, 2c, 2d could be performed later, i.e. after step 1 in 4.2.2, by discarding a game that does not pass the criteria and choosing at random another game, i.e. re-enacting step 1 . Similarly, step 3 can be established before steps 1 or 2. Some of the steps are optional as described.

However, the most significant alternative way in which the procedure described could be enacted, leading to very similar results, is by postulating a distribution of either opening sequences (with their relative frequencies) or initial positions, as representative of reference play at a certain depth D. This we may call a "reduced-form" distribution, in place of the game database or distribution itself. Hence this non-uniform probability distribution or frequency distribution of opening sequences (or initial positions) could be then used to obtain directly the opening sequence by means of generating a random selection in a nonuniform manner, i.e. taking into account its frequency (or "weight" of each distinct opening sequence) ⁷ . Hence the alternative way of the procedure is:

Given depth D and database obtained as after 2e (optionally after 5 above), and using the classification of opening sequences similarly as above, obtain an opening sequence database at depth D with its corresponding (absolute, relative) frequencies.

Generate a selection from the non-uniform statistical distribution of opening sequences.

Follow from step 4 of 4.2.2 above.

For example, all marginal 3 ^r move positions originating in the opening 1 . b3, e5 may accumulate to it, so that they would be replaced by it and their combined probability.

⁷ Note that it is not necessary to unify opening sequences that lead to the same position after a certain number of moves, although it is also possible to do so. We claim that the first and the alternative procedure, using a "reduced form" of the database, is essentially the same.

An additional alternative or different order of the procedure would be the following: "truncating" all games included in the database, so that they only contain the full moves of the game up to the selected depth, or that they exclude moves over a certain, single, sufficiently high depth in order to save memory space. Similarly, in the game database additional auxiliary information (chess event in which the game takes place, result of the game, etc.) which is not relevant for the procedure above, is eliminated in order to save on unnecessary information. These seem to us to be trivial modifications of the procedure.

Another modification would involve using the reduced form above to implement the "varuiable-depth" procedure described in 4.2.3. Also, using computer play or a mixed human / computer game collection are possible, although they do not fulfil all the dimensions of the problem herein described.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 shows a scheme of the implementation showing all elements:

A central processing unit for storing a database of reference (1 )

A database of reference created from all recorded chess games in history divided by categories (2)

A selection and filtering process based on a specific algorithm that selects one single game in order to stablish the initial position of the new game (3)

A chess software for installation in any device (4)

Electronic devices to play the new game online (5)

Unlimited number of users to play the game live (6)

Figure 2 shows a chart showing the statistical analysis of the optimal depth.