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1o) The empty chessboard : ranks and files A chessboard is constituted by a square board compound of 64 squares, organized like it's indicated on the following diagram, representing the empty chessboard, in
DIAG 1 :
Hence any square is defined with the help of a letter and a numeral. That way : the e2-square is the intersection between the e-file and the 2th rank ; in other words it's the only square situated simultaneously on the e-file and the 2th rank. In the same way the g5-square is the intersection between the g-file and the 5th rank. 2o) Diagonals At the previous straight lines, we had defined («files» and «ranks») it's convenient to add other straight lines named diagonals. This ones are constituted by squares of the same colo(u)r. There is at all 30 diagonals. The empty chessboard is divided in : a7,b8 a5, b6,c7,d8 a3, b4,c5,d6,e7,f8 a1,b2,c3,d4,e5,f6,g7,h8 (large black diagonal) c1,d2,e3, f4,g5,h6 e1,f2,g3,h4 g1,h2 a8 a6,b7,c8 a4,b5,c6,d7,e8 (Spanish diagonal) a2,b3,c4,d5,e6,f7,g8 (Italian diagonal) b1,c2,d3, e4,f5,g6,h7 d1,e2,f3,g4,h5 f1,g2,h3 h1 a1 a3,b2,c1 a5,b4,c3,d2,e1 a7,b6,c5,d4,e3,f2,g1 b8,c7,d6,e5,f4,g3,h2 d8,e7,f6,g5,h4 f8,g7,h6 h8 a2,b1 a4,b3,c2,d1 a6,b5,c4,d3,e2,f1 a8,b7,c6,d5,e4,f3,g2,h1 (large white diagonal) c8,d7,e6,f5,g4,h3 e8,f7,g6,h5 g8,h7 Hence, for example, on the DIAG 2, the 6th white-climbing diagonal is represented in  ,
the
5th black-climbing diagonal
is represented in
and the
6th black-downward diagonal
is represented in
DIAG 2 :
In the same way that every square is the intersection between a rank and a file, take notice each square is equally the intersection between a climbing diagonal and a downward one. As an example the g5-square is the encounter point between the 5th black-climbing diagonal and the 6th black-downward one. 3o) (Straight) lines and points The term line is, in this lecture, where we have to use a precise language, the common denominator of rank, file, and diagonal. This way we have 8 + 8 + 32 = 48 lines on the chessboard. It's convenient to identify each square with its central point. This way each line is clearly defined like the single straight line joining points corresponding at all squares constituting it.
1o) The centre The four squares e4, e5, d4, d5, like they are represented (in  )
in the following diagram
(this is the
DIAG 3
situated some lines under, in the next sub-column)
, constitute the centre.
Its primordial significance was recognized, a long time ago, by chess theorists, with clear-cut opinions : 2o) Caracteristics of the centre I's quite clear the caracteristic of the «centre», in consideration of the chess game, is that is exactly the place on the chessboard where each piece ruch its better radiance and influence (we say that in a theorical point of vue ; nevertheless further developments will lead us to qualify greatly this opinion of principle). Then it'is not difficult at all to understand the geometrical reasons of this fact. Clearly : The four squares of the centre are the only ones, on the chessboard, possessing the two following properties : Now we'll go on specifying this interesting ideas by the introduction of the new concept of «distance».
1o) Sides and corner squares There is 4 squares commonly named corner or angle squares a1, h1, a8, h8, wich are represented in .
The side of the chessboard is contituted by the a-file, the h-file, the 1th rank and the 8th rank. That is to say : 28 squares. This ones are indicated in
excepted the 4 corner squares.All square which is not on the side of the chessboard is said to be interior ; we'll say it's an interior square, by opposition to the concept of side square. Bring together this different kind of squares (we have to observe that «central squares» are «interior squares» and «corner squares» are «side squares») :
TAB 1 :
The following diagram represents, on the chessboard, the «side», the «centre» and the two «large diagonals». You see then «central squares» and «corner squares» belong to the larges diagonales. But this ones contain equally squares visualized in
b2, b7, g2, g7 (privilegied squares for the - fianchetto -
development of Bishops) and c3, c6, f3, f6 (privilegied squares
for the first development of Knights).
DIAG 3
:
2o) Adjacent squares All «square» is naturally a square in the mathematical meaning. Consequently this one has 4 sides and 4 vertex. We'll say that two squares are adjacent if they have at least a vertex in common. In fact, two situations may occur :
TAB 2
:
1o) Distance The distance between two squares is the minimal number of squares it's necessary to go across to make one's way, step by step (like a King), from one to the other, including the arrival square . Consider two squares A, B. The distance between both is noted : d(A,B) . For example :
d(e4,e4) = 0 ; d(e4,g8) = 4
The greatest distance between two squares is :
Although this number is a little one, it has rather a great significance in a chess game. Hence, Let us consider d = d(A,B) the distance between the two chessboard squares A and B. Then we'll say that this distance is : This terminology is connected with the movement of pieces ; we'll subsequently see the reasons for which the notion of middle distance would be without any interest in chess game study. 2o) Sectors of the chessboard By sector we mean all part of the chessboard constituted by the union of a set of squares. Sometimes, for a reason or another we are lead to consider such or such particular sector. For giving one or two arbitrary examples we may observe the "red sector" and the "blue sector":
Those sectors are constituted by adjacent squares, but it is not a necessity. Notice that the number of sectors is frankly very large, for don't say considerable:
In any event, we will give later on several typical examples of sectors, related on positions and/or the evolution of a chess game, showing all the interest of this concept. 3o) Zones of the chessboard By Zone we mean a sector intersection between an union of contiguous files and an union of contiguous ranks. We gladly admit that this definition can appear complicated, but really it is not the case. And anyway we will give below several very simple examples. The number of zones is a medium one: Number of Zones = 36x36 + 1 = 1297 A "trivial" example of zone is given by the Board: it is obviously the intersection between the union of all (the eight) files and the union of all (the eight) ranks. Another "trivial" one is the Center in the strict meaning: it is the intersection between the Center constituted by files d, e and the Middle constituted by ranks 4,5. This way it is equally clear that all union of contiguous files is a zone and similarly all union of contiguous ranks is a zone... And undoubtedly only mathematicians among you may understand very well that the empty set is also a zone... But, don't worry! This theoretical fact is practically without any importance.
PZ_(i,j) = set of all (k,l)-squares: k=i-1, i, i+1 & l= j+1, j+2, ..., 8
PZ_(i,j) = set of all (k,l)-squares: k=i-1, i, i+1 & l= j-1, j-2, ..., 1
INZ= set of all (i,j)-squares: i=b, c, d, e, f, g & j= 1, 2, ..., 8
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