frankho ba gua mathematics (chinese eight diagrams mathematics)

何数棋谜数学智能开发与思维训练 The Frankho Ba Gua Math™ 何氏八卦数学™

Frankho Eight Diagrams Math™Trademark® TMA771400

Ho Math and Chess, www.mathandchess.com Made in Canada

© 2011 Frank Ho, Amanda Ho, all rights reserved. Canada copyright number 1069744

Student Name ___________________________________ Date ______________________________

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七巧板八卦九宫数不见题只见谜中谜数学国际象棋谜题全球第一独特发明

只见棋谜不见题劝君迷路不哭涕数学象棋加谜题健脑思维眞神奇

In 2011, I read a news piece (1) about a city (八卦城 bā guà chéng) in China, which

has streets designed in Chinese Ba Gua shape and there were no traffic lights installed

but traffic can flow with no major headaches. This piece of information got my interest

on the mystique of Ba Gua. Since I have been creating mathematics workbooks using

my invention of Geometry Chess Symbols (2) so I very quickly realize that the 8

directions the Ba Gua points to have a direct link to the queens’ 8 moves in chess.

Intuitively I know I could use Ba Gua to create a unique kind of mathematical puzzles to

entertain children for education purpose. Because of the close relationships between a

Ba Gua and chess, here I created the world first mathematics puzzle which links Ba Gua,

chess, and mathematics.

I like the philosophy that Ba Gua’s view about universe. This very much resembles to

the concept of math exponential number with base 2. I also like the idea of using Taiji to

start with the puzzle as a focal point. The Chinese Yi-Ching describes Taiji as follows.

無極生有極、有極是太極、

太極生兩儀、即陰陽;

兩儀生四象: 即少陰、太陰、少陽、太陽、

四象演八卦、八八六十四卦

The above translation in English is as follows.

The Limitless (Wuji無極) produces the delimited (Yuji 有極), and this is the Absolute

(Taiji太極).

The Taiji produces two forms, named yin (陰)and yang (陽).

The two forms produce four phenomena, named lesser yin, great yin (taiyin also means

the Moon), lesser yang, great yang (taiyang also means the Sun).

The four phenomena act on the eight trigrams (ba gua), eight eights are sixty-four

hexagrams.

http://en.wikipedia.org/wiki/Wuji_(philosophy)

http://en.wikipedia.org/wiki/Taiji

http://en.wikipedia.org/wiki/Yin_and_yang

http://en.wikipedia.org/wiki/Phenomena

http://en.wikipedia.org/wiki/Moon

http://en.wikipedia.org/wiki/Sun

http://en.wikipedia.org/wiki/Hexagram_(I_Ching)






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In creating the Ba Gua math puzzles, I try to convey a few ideas to children who work

on these puzzles. These ideas are:

1. There are no obvious written questions presented when first looking at the puzzle.

This is an excellent training for children to learn on how to mine data in today’s

information age. In other words, data may be presented to our future generations

in tables, patterns, diagrams, or through hand-held devices. Many children are

using cell phones to get information they want and the miniature 5 by 5

chessboard which children will see in Frankho Ba Gua puzzle is actually

simulating a hand-held device. By using the chessboard (simulating cell phone

screen), a student starts a math puzzle.

2. The chessboard and math puzzle have no written directions but children are able

to figure out what to do by using Geometry Chess Symbols and comprehending

their abstract meanings which these symbols represent after a teacher’s

explanation and demonstration. The instructions are directed by using Ba Gua and

chess moves and this trains a child to be comfortable with symbols and learns to

decipher symbols and transfer them in to concrete ideas at very young age.

3. In today’s information era, pieces of information come not only from books, but

also from other diversified sources such as internet. Children learn to analyze and

synthesize all kinds of information by information gathering from different

sources. Frankho Ba Gua puzzle is designed to train a child to take a divergent

(outflow for the centre of a chessboard) view and also a convergent view (by

working backward to the centre of Ba Gua on math problem to get further

answers). This thinking process is very similar to the view how a chess player

shall play chess that is not only just looking at how to make a move but also

looking at what the opponent is trying to do. This divergent and convergent view

is much similar to the math concept of function and Frankho Ba Gua puzzle is

actually teaching the concept of function through a math puzzle.

The mathematical function children are doing is as follows:

))), ) = value of each 8 math problems,






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Frankho Ba Gua puzzle offers an excellent opportunity for a child to learn how to

do math visually by moving data around and by working backward and

conducting table look-up to get the intermediate answers. It also teaches

directions and keeps children engaged while working on math at all times until the

final answer is reached.

4. While children work on Frankho Ba Gua puzzle, they will have many questions

throughout the time they work on, so it gives ample opportunities for children to

engage conversations with their teachers and as a result, teachers get to know

children’ math ability in depth by observing how each child works on math

problems.

5. Ba Gua shows binary trigrams so they can be used to teach number system.

6. Frankho Ba Gua puzzle is different from all other puzzles because it is a project-

oriented puzzle, not a single puzzle problem. They are not designed to be used as

daily work problems for children, rather each problem is a project for children to

work on with fun and challenging,

7. Chinese Ba Gua is very closely related to the international chess in terms of how

each piece moves and how they came together. I found out the relationship

between Chinese Ba Gua and international chess after I read the article written by

Mr. 郭彧 (3). Mr. 郭彧 did not mention any relationship between Ba Gua and

international chess in his article since his purpose of writing the article was to

prove the mathematical soundness of Ba Gua by using a cube. However, through

the diagram presented in his article I suddenly see the relationship between Ba

Gua and international chess and I was amazed! I will use郭彧’s diagram (after

some modifications) to illustrate how Ba Gua is related to international chess.

Since his article is difficult to get so I include it at the end of my article for

reference.

郭彧 is using a cube to represent Ba Gua and from its eight directions, he

concludes the relationship between math and Ba Gua.






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The relationship between international chess and Ba Gua can be explained as the

following.

1. The Ying and Yang of the centre of Taiji produces the Black side (Ying) and

White side (Yang).

2. The eight-Gua produces the side length of 8 of a chessboard and also the moves of

the mighty queen chess piece and Yang produces the piece of a king.

3. The top and down, left and right directions produce a rook, and the Ying and

Yang produce 2 rooks.

4. Where does the knight fit into the picture in Ba Gua? The knight actually was not

produced by Ba Gua but a global view produces the knight. My article (4)

explains how knight was produced.

5. The Ying and Yang produces two bishops and directions of each bishop is

produced in the following way:






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Two Chinese characters connecting

two opposite vertices

Connecting direction

and

and

and

and

From the above analysis, one can see clearly the reason why Ba Gua, math, and chess all

nicely fit together in harmony.

References

(1) http://www.yangtse.com/lypd/rwzr/200808/t20080812_483169.htm,

http://v.ku6.com/show/tgdG0DivRhwxBvy5.html

http://baike.baidu.com/view/164881.htm

(2) Geometry Chess Symbols trademark approved.

http://www.mathandchess.com/releases/release/1441781/21558.htm

(3) Why Chess Knight Jumps? Teaching Chess Using Geometry Concept A new idea of teaching chess http://searchwarp.com/swa146589.htm

http://www.yangtse.com/lypd/rwzr/200808/t20080812_483169.htm

http://v.ku6.com/show/tgdG0DivRhwxBvy5.html






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从数学角度论证八卦由来的科学性

郭彧

本文则尝试从数学角度对八卦的由来进行初步的科学论证。

我们说八卦具有时空属性，就必须把它放到时空里面进行验证。随着时间的流失，空间也在变换。

在某一特定的时间里，我们可以得到下面的空间图：

把几何上的八个象限同中国古代方明立方体结合起来，则有如下关系：

+X——天玄

-X——地黄






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+Y——东方

-Y——西方

+Z——南方

-Z——北方

乾——天玄（初爻），东方（中爻），南方（上爻），象限为：+X +Y +Z

兑——天玄（初爻），南方（中爻），西方（上爻），象限为：+X +Z-Y

震——天玄（初爻），西方（中爻），北方（上爻），象限为：+X-Y-Z

离——天玄（初爻），北方（中爻），东方（上爻），象限为：+X-Z+Y

坤——地黄（初爻），西方（中爻），北方（上爻），象限为：-X-Y-Z

艮——地黄（初爻），北方（中爻），东方（上爻），象限为：-X-Z +Y

巽——地黄（初爻），东方（中爻），南方（上爻），象限为：-X+Y+Z

坎——地黄（初爻），南方（中爻），西方（上爻），象限为：-X +Z-Y

从阳光照耀角度来说，方明正立方体有三个阳面和三个阴面；从数学象限角度来说，八个象限正好容纳八卦之象

，其卦爻的阳阴属性正与象限的正负属性相吻合。

《周易·系辞》说“八卦以象告”，可见当初翻译数学的人用“象限”一词，是多么地符合中国传统文化的内涵

。

《周易·说卦》：“天地定位，山泽通气，雷风相薄，水火不相射。”

为什么古人能有如此的说法？我们从上图看到：乾与坤相对，艮与兑相对，震与巽相对，坎与离相对，正是“天

地定位，山泽通气，雷风相薄，水火不相射”。难道这是偶然的巧合吗？






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如果不能说出于偶然的巧合，那就是说，古人之所以有这样的说法是有一定客观根据的。显然，其根据就在于“

方明”。所以《周易·系辞》方说：“古者包牺氏之王天下也，仰则观象于天，俯则观法于地，观鸟兽之文，与

地之宜，近取诸身，远取诸物，于是始作八卦，以通神明之德。”通神明之德，就是通方明神器之德。古圣人仰

观俯察天地四方之“六合”，就作出了八卦。《周易》里之所以不说圣人“画卦”，其理由也正在于此。

写于2008年4月18日






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七巧板八卦九宫数不见题只有谜中谜数学国际象棋谜题全球第一独特发明

见棋谜不见题劝君迷路不哭涕数学象棋加谜题健脑思维眞神奇

You are positioned in the middle of a 5 by 5 chessboard, which is at c3 and it is a miniature octagonal

figure of Chinese Ba Gua (八卦). In Chinese. "Ba" means eight (8). "Gua" means trigram. The Ba Gua

is divided into 8 Gua (trigrams/sections) consisting of the most original and basic binary form of "Yin"

(female - two short hyphens) and/or "Yang" (male - one long hyphen).

The following Ba Gua is an enlarged view c3 and it will guide you to solve the mysterious

mathematical chess puzzle. First you solve mathematical problems according the 8 symbols (made of

broken and unbroken lines) of Ba Gua or the directions each Gua points to.

After you solve the math problem then you find a further answer by associating it to the Yin Yang

(Taiji) in the middle of Ba Gua. At this step the answer is obtained by working backward.

Finally you will go back to the chessboard which has Ba Gua in the middle as a controller and it shows

how you shall move by following the chess symbols to get the final answer of each Gua.

Once you get the 8 final answers for eight Guas, then you will have to solve a finale Ba Gua puzzle.

This is Ba Gua (八卦).






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Step 1

Students have a condensed version of chessboard (5 by 5) and in the middle of the chessboard there is a

diagram of Ba Gua. Each trigram guides students which math problem shall be solved. The math

problem consists of 8 major areas: Chinese Tangram, Frankho Puzzle, Chinese magic square, smart and

speedy computation, word problems, Olympia math contest problems, Math IQ problems, and

recreational math problems.

Once the problems are solved, they must pass back to Ba Qua for further processing. Each answer is

passed on to the origin of universe which is YinYang (Taiji). An intermediate result is obtained by

using its answer from trigram to interact with the answer from the Yin Yang.

Step 2

At this point, you shall go back to the chessboard and stat to work on divergence by following chess

move, each answer obtained as the result of chess move will then be processed with the result of the

step 1 to get the final answer.

You are at c3.

5 15 12

4 10 19 11 13

3

2 15 27 23 14

1 16 11

a b c d e

S= 64

*1= + *2 = + *3 = + * 4 = +

*5= + *6 = + *7 = + * 8 = +






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Frankho Eight Diagrams Math™ 何氏八卦数学™ (The Frankho Ba Gua Math™)

七巧板八卦九宫数不知题只见谜中谜何氏国际象棋谜踪全球第一独特发明


You are at c3. Work from inner convergence to outer divergence. 从百川入海到一门独秀

Answer _____

Answer _____

Answer _____

Answer _____






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Frankho Eight Diagrams Math™ 何氏八卦数学™ (The Frankho Ba Gua Math™)

七巧板八卦九宫数不知题只见谜中谜何氏国际象棋谜踪全球第一独特发明


You are at c3. Work from inner convergence to outer divergence. 从百川入海到一门独秀

Answer _____

Answer _____

Answer ____

Answer ____






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☰111

Chinese Magic Square (幻方) is called 纵横图，九宮数 in China.

According to ancient Chinese legends, a giant tortoise surfaced from the River Lo in central China around 4,000

years ago. The ancient Chinese found a pattern on a tortoise shell (refer to the picture). There were circular dots of

numbers on the giant tortoise that were arranged in a three-by-three nine grid pattern on its shell.

The pattern of numbers on the giant tortoise in any given direction i.e. horizontal, vertical or diagonal all add up to a

total of 15.

Place 11, 12, 13, 14, 15, 16, 17, 18, and 19 in each of the 3 by 3 box such that each row, each

column, and each major diagonal all has the same total.

Return the answer of the middle square to Ba Gua.






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☴110 You are at b2 and the area of the entire square

is 16.

+ = _____+_____ = _____

+ =

– = _____–_____ = _____

+ =_____+_____ = _____

– =_____–_____ = _____

The Chinese Tangram. (七巧板; literally "seven boards

of skill") invented by the ancient Chinese hundreds of

years ago is a dissection puzzle consisting of seven flat

shapes (2 large right triangles, 1 medium right triangle,

2 small right triangles, 1 square, and 1 parallelogram)

which are put together to form shapes. The objective of

the puzzle is to form a specific shape (given only in

outline or silhouette) using all seven pieces, which may

not overlap.

3 9

13

15

2 18

12

1 17

16

14

a b c

Return the last answer to Ba Gua.

Return the answer also to the northeast circle of

the finale Ba Gua puzzle.






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☵010 Frankho Puzzle.

All the digits 1 to 3 must appear in every row and column. The number appears in the bottom right-

hand corner is the end result calculated according to arithmetic operator(s), inequality signs, and

chess move(s).

Return the answer of b3 to Ba Gua.

Return the answer to the southeast circle of the finale Ba Gua puzzle.






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☶100 Replace each letter by a number such that sum of the top three numbers in a row is equal the sum of

the three numbers in the middle column.






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☷000 ☳001 Tara has 6 red stickers and 7 blue stickers in

her pocket. What is the minimum number of

stickers that she has to take out of her pocket

such that at least 2 stickers will have different

colours?

Return the answer to the southwest of the

circle of the Ba Gua finale puzzle.

Suny’s soy sauce and its bottle together weigh 800

g. After using half of sauce, the sauce and its bottle

together weigh 450 g. How much does soy sauce

without bottle weigh?

☲101 ☱011 Emily had had a total of $180 of some $2

notes and $4 notes. She had 6 more $4 than $2

notes. How many $2 notes did she have? How

many $4 notes did she have?

Roxana has $48 and her sister has $20. How much

money does Roxana have to give to her sister so

both of them have equal amount of money.

Return the answer to the northwest circle of the Ba

Gua finale puzzle.






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Ba Gua Math Finale Puzzle

Fill in the remaining three empty circles with the symbols and and calculate the result in the

clockwise direction with the result equal to the number in Taiji.

S= 10

frankho ba gua mathematics (chinese eight diagrams mathematics)

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