Stanley的量性思考

Happy Pi Day.

Celebration of Pi Day — MMNA Nova Happy Pi Day and remember how mathematics interacts with our daily lives

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The Illusion of Area: What We Forget to Teach About Mathematics — MMNA Nova In many real life problems involving the sizes, it usually involves more factors that sizes alone, and all factors affect our decision making.

Introducing MMNA Nova 數養思行: Empower mathematical mind my new project.

Scheduling is one of the huge problems I am facing recently. It is also an everlasting problem since the birth of human civilisation thousands of years ago.

Interestingly, though it seems immediately mathematical which involves procedures and calculations, it is a relatively novel branch of mathematics to model scheduling universally.

Leonhard Euler looked into the nature of Konigsberg Seven Bridges Problem as a scheduling problem about the inter-relationships of stages, and simplified it with a mathematical representation of graphs with nodes and edges. By analysing these graphs mathematically, we therefore can understand how one schedule is feasible or not, or working better than another.

This is a step that we move from relying on experience to relying on analytics. Experience grants us the understanding of the relationship between procedures and stages to construct the graphs, and analytics helps us find the optimal solutions following certain rules and constraints.

This is a combination of mathematical mind and creativity to abstractise and bring out a problem’s true nature.

(Image created by AI)

It’s very much pleasurable to share my thoughts with pre-service teachers of students in Mathematics and Mathematics Education in the Chinese University of Hong Kong on how mathematical modelling education would interwoven with future mathematics education.

It is more edging than ever to rethink what mathematical knowledge, skills and values are the essential aspects for future generations.

As a tennis lover, the new 1-point slam in Australian Open has drawn some attention in the world. With the best pro tennis players, amateur champions, and stars around the world joining together without any boundaries, 48 of them fought a battle with just one point in tennis for a million-dollar championship.

Tennis lovers must have heard about Roger Federer’s speech saying that even though he won nearly 80% matches in his career, he only won 54% of the points he played. This made the results of the 1-point slam highly unpredictable, for even Alcaraz or Sinner would have a probability of around 0.6^5 = 0.08 to win, assuming that they will have 5 matches from Round 32 en route and already had a better winning percentage in points than Federer. The result was, as expected, surprising.

(Photo credit: Australian Open)

I always hear people say “math is absolute and is always right.” It all depends on which part of mathematics it is referred to.

Mathematical minds try to define and limit the scope of questions asked, so that the answers could be absolute and always right only within the scope.

At the same time, they ask for exceptions that are just out of the scope which will falsify the answers. And there are always plenty of exceptions that make “wrong” the “mathematics” people think of.

The process is just like when someone says with authority “the world is like this”, and a mathematical mind will humbly ask “what do you mean by ‘the world’ and ‘this’?”, and “what if ‘the world’ is something else, is it still like ‘this’?”

Equality relation is the most fundamental concept in mathematics. From Day 1 children start learning mathematics, they would come across the concept that one thing equals another, until the very last day.

But equality is not a trivial concept. We start from the physical object of an apple, a boy, a cat, etc are all equal to a numeral “1”, so that we only need one standard arithmetics to handle all counting problems of these objects instead of developing methods that count only apples but not boys, and vice versa.

Gradually, we follow the same arithmetics to continue to “count” quantities with unknowns, even with numbers that are uncountable.

Then we enrich the equality to an extend that one system, subject to certain criteria, equals (or, more technically, is equivalent) to another, so that we can even bring forward the understanding of one known system to the other being studied. With this, we are actually categorising objects and building a complex network that if we update our knowledge on one aspect, it immediately update our knowledge on many equivalent others.

That’s a wonder in mathematics.

One core of mathematics is the study the structure in patterns. And music, as organised sound with patterns, is always an interest in mathematical studies, from Pythagoras to Euler.

Tonnetz is an illustration proposed by Leonhard Euler to display musical intervals. The edges connecting nodes demonstrate either a relation of major third, minor third or perfect fifth intervals. Therefore, chords become polygons, such as a major7 chord would be the smallest parallelogram, and musical analysis can be considered as a study of geometry.

Same mathematical tool is also used by the same Euler in the famous Konigsberg Seven Bridges Problem. This also demonstrates the generalisation and abstraction nature of mathematics.

Daily language is usually ambiguous. This is where mathematical language kicks in: When one defines the meaning of “larger”, a mathematical measurement can be established, and one can compare things through its measurement, be it the numerical values, dimensions, or anything.

One can argue about what measurement is the appropriate one for discussion, but once the measurement is agreed the comparison follows logically. (I.e., ‘If we compare the numbers by its size, “6” is larger than “7”.’)

Learning to state one’s thoughts in such a way is the learning of mathematics.

Mathematical modelling is part of the development of mathematics: From an intuitive description of patterns and phenomenon to a formal and falsifiable language.

呢輪留意緊英國前PM Rishi Sunak 推緊一個名為Richmond Project嘅項目，期待能夠提升到國民嘅數學能力。暫時見到嘅內容主要都係講緊一啲好基本嘅運算、數感等等。令我諗到一個細個成日玩、又好符合英國國情嘅遊戲，就係望住下面個世界杯外圍賽分組排行，睇下可以推算到乜嘢資訊同個數學原因。

例如︰

1. 英格蘭出左線，因為剩兩場分差7分，多過兩場最多可以generate到嘅6分。
2. 英格蘭剩低兩場比場入面，一定有一場對Albania，因為英全勝而ALB只輸一場，所以第二輪未對。
3. 亦因此，ALB未輸過俾其他球隊。所以Serbia嬴果三場入面只會有Latvia同Andorra。基於賽程係打完一個循環先打第二個循環嘅假設，佢哋輸果兩場一定有一場係對英格蘭。剩低果場可能係英、ALB或者LVA。
3a. 如果係LVA嬴SRB，即係佢只可以和過AND一場而第二場未打。所以佢對AND就係最後一場波。亦即係頭三隊要互對兩場。同時，ALB一定和左SRB一場。而且，SRB嬴果三場一定係兩場AND一場LVA。所以AND輸果6場，會係ENG、ALB同SRB各兩場。而LVA輸果4場，就係2場ENG、一場ALB同一場SRB，同和左一場ALB。暫時無矛盾。
3b. 如果係ENG嬴SRB，即係SRB只對左ALB一次兼且和埋，而ALB和嘅第二場一定係LVA、LVA一定和左一場AND。所以ALB嬴果三場會係兩場AND一場LVA，而佢哋最後兩場會係對ENG同SRB。咁LVA嬴果場一定係對AND，即係佢哋對晒兩場AND，即係LVA同AND最後一場只可能對ENG或者SRB。LVA而家知嘅情況係輸嘅場次入面有3場係分別輸ENG、ALB同SRB。AND則係輸嘅場次入面2場輸ALB、1場分別輪LVA、ENG同SRB。兩個情況都暫時無矛盾。
3c. 如果ALB嬴SRB，即係ENG最後兩場嘅對手分別係ALB同SRB。所以佢哋一定兩勝LVA同AND，而LVA一定嬴左一場AND。如果ALB未對SRB第二場，就會變返LVA對AND係佢哋最後一場個情況，但LVA嬴過AND一場，咁佢哋和果兩場一係兩和ALB跟住AND和SRB、一係同ALB、SRB各和一場跟住AND和ALB。對應埋啲輸波情況，呢個都無矛盾。

所以，喺現有資訊入面，幾個情況都有可能，無得做落去啦。不過如果有多少少資訊，例如得失球數目、或者其中一兩場賽果，就有機會繼續推落去。

This is a logic game.