A New Chess Set for Teaching Mathematical Chess

A New Chess Set for Teaching Mathematical Chess

Frank Ho

Teacher at Ho Math and Chess Learning Centre, www.mathandchess.com

If one thinks that chess game is made of warriors or commanders and kings and queens battling in the field then this notion does not really address the reason why chess pieces move in a pattern-like direction for example, rook moves up and down or left and right and bishop moves diagonally. Is chess game a reflection of ancient war or is it an invention based on a mathematical principle? The author believes that chess game is invented by using the geometry symmetry concept and this conjecture is based on  analyzing the moves of each chess piece and thus concludes that this symmetry property is how chess was created from a mathematical point of view.

To be a fair game, the positions of chess pieces must be placed in symmetry and so is the layout of a chessboard. Perhaps it is not coincidental that the play field of chess game is all about squares and the Chinese character of “rice field” is also a 2 by 2 square. Given fixed parameter, the square gives the largest area. The chessboard is a tessellation of 8 by 8 squares. There are 4 symmetry lines in each square and these 4 symmetry lines constitute the moves of rook, queen, king, pawn and bishop. It makes sense that each chess piece moves along the symmetry lines to divide each small square evenly and fairly. How about knight, why knight is the only piece in all chess pieces, which jumps?

How chess moves are originated

To play a symmetric game the smallest board required is 5 by 5. I believe that the possible moves of each chess piece are originally intended to be a 360 degrees of circular movement. For example, take a look at a chess diagram (See Figure 1.), If a chess piece is placed at c3, how many ways can this chess piece reach out to the side of a square to form a shape of inner circle (inscribed circle c2, b2, e4, d4) or outer circle (circumscribed circle c1, a3, e5, e3)? Depending how points are connected, a shape of square could also be formed in addition to circle.

The first “easy” way would be to move top-down or bottom-up and left-right or right-left, from c3 so as to reach the limit of a square and an inner circle is born and thus the moves of rook is born. The motion of its move is called translation or slide in geometry. Connect the 4 out reached points with 4 straight lines, the shape is actually a square; but with contour curve then it forms a circle.

The second way of moving to the outer limit of a square and form a circle is to move in the directions of two main diagonals, thus an outer circle is produced and the moves of a bishop is born. Arguably, the four points also make a shape of a square. This motion from c3 to each of the 4 diagonal points is also a double-slide. The bishop can view in 360 degrees. 

Combine the above two ways of rook and bishop moving, we have the most powerful move in all chess pieces that is a queen and this is the birth of a queen’s move. King can only move in one square in each move and follows the moves of a queen. 

In a 5 by 5 chessboard, we notice that all chess squares on each of 4 sides are covered by the moves of rook and bishop except a2, a4, b1, c5, d1, d5, e2, e4, so from a game point of attacking or defending view, this is a problem – there are 8 squares which are not covered. This is the reason of the birth of another chess piece called knight, which covers the 8 squares by jumping to those 8 squares because it does not move by following the same moves of rook or bishop to reach the 8 squares. This perhaps is the reason why knight jumps since knight does not trace any squares in one straight line to reach any one of those eight unreachable squares.