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Whole Numbers Problem Solving and Puzzles

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Prod. Code: ISBN 0-9683967-5-5


Whole Numbers Problem Solving and Puzzles


For Math Enrichment


Frank Ho


A good question to ask me when you see this workbook is why there is chess stuff in it? It is something about how I got my son started in chess when he was about six years old.

I did now know how to play chess at all when he showed interest in a set chess, which I got as a gift. He was excited in finding out that he could be rewarded by taking the opponent’s piece because he made the right move, later we studied chess together and he become a Canadian junior chess champion and also a chess master.


My interest shifted from playing chess itself to studying the relationship between chess and mathematics to eventually create the Vancouver Math and Chess Learning Centre in 1995, perhaps the only learning centre where youngster could learn chess and mathematics after school at that time. My curiosity of researching in the relationship of math and chess led me and Andrew, my son, created the workbook Mathematical Chess Puzzles for Juniors in 1997. This workbook is the continuation of creating mathematical chess puzzles after my first workbook was published. So what is the relation between mathematics and chess?


My belief is most games are related to mathematics and a chess game which had been refined in the past several thousands years is no exception.


The chessboard and chess pieces themselves are geometry. The chessboard is symmetric in terms of its main diagonals.


The chessboard is made of 4 identical small boards if it is divided by one horizontal line and one vertical line going through the centre. The set-up positions of chess pieces are symmetric between black and white. The chess pieces set-up positions on either side is palindrome except the king and queen.


Rook’s move is a slide motion (left/right, up/down) in geometry. The between moves of rook before reaching the destination is using the concept of communitative property, for example, before Ra1 to Rh1, Rook could move from a1 to c1 (4 squares) then from c1 to h1 (3 squares) or from a1 to d1 (3 squares) then from d1 to h1 (4 squares). The complication is the player has to watch what would happen if the different choices are made and this is much complicated than adding 3 + 4 = 4 + 3 = 7. These complications involve the calculations of different paths and also the opponent’s possible responses. The deeper the player could calculate the paths, the higher possibility of playing better is. The calculation of path requires logic thinking which is very similar to the idea of using factor tree to find out what are the prime factors of 64, but chess is more complicated in a way, the opponent’s moves also have to be thought in advance.



l m

2     16

 l m

2                  8

   l m

2                4

      l m

2                2


If rook is at a1 and is free to make moves along file a and rank1, what has to be considered before moving? The most important is to see if there are any opponent’s pieces, which could intersect with the rook. Thinking in math way would be to see what is y when x = 1 and what would be x when y = 1, we will be looking for intersections. The idea of coordinates would be easier for chess players to learn if they already have acquired the practical experience of “intersections” coming from different chess pieces.


The idea of one-to-one cancellation of chess pieces left on the board is similar to the subtraction property of equation.


One would think that chess perhaps has nothing to do with fractional numbers since all moves are all in whole numbers. Why queen is the most powerful piece in chess and we usually move chess pieces toward the middle? The all have something to do with the ratio a/b, where b is the 64 squares and a is the squares under control.