陳偉倫先生 - ）決賽暨粵港澳交流邀請賽

評判團代表

邊長 = 1 cm，質量 = 1 g 邊長 = 2 cm，質量 = 4 g

正立方體膠磚是三維物體，因為當正立方體膠磚的邊長增加至原來的 2 倍時，它的質量便會變成原來的 8 倍，即 2³=8。

邊長 = 1 cm，質量 = 1 g 邊長 = 2 cm，質量 = 8 g 由上述例子可見，一般物體的維數 D 為 1、 2 或 3。

是次題目的兩位主角是謝爾賓斯基三角形(Sierpinski triangle) 和謝爾賓斯基地毯(Sierpinski Carpet) 。

謝爾賓斯基三角形謝爾賓斯基地毯

它們的維數屬於非正整數，我們預計對於初中的參賽同學，他們需利用計算機找出維數的。但我們發現所有進入決賽的隊伍，都能以純熟的對數(log) 運算方法來找出答案，同學們根本不需要在紙上作運算，而能在腦海裡作運算，然後利用計算機，直接找出答案，可見同學平日自學數學的努力！決賽當天的六組隊伍中，冠軍隊伍全部答案正確，亞軍隊伍只是不小心弄錯丁點。不過兩隊均表現出色。對題目都是十分理解。然而冠軍隊伍不只答中全部預設題目，在追問的環節裡，更表現出超越眾人的見解和「數感」！

整體而言，本年的參賽水平十分高，一般數學的操作非常純熟，相信同學們一定博覽群書。令人驚喜的是初中學生已能利用微積分(Calculus) 和拓撲學 (Topology)等去作解釋！雖然其中尚有沙石，但已使評判團十分驚訝！若真要我們吹毛求疵，就只能說同學未能在短時間內掌握非整維數的何意義，以及未能洞察到答案出了問題。

聖士提反書院 St. Stephen’s College

聖士提反書院 St. Stephen’s College Au Ting Fai

Students in our group are a little bit nervous before the game, but fortunately Mr.

Szeto comforted us. I knew that if only we could try our best, our group can

perform just fine. I thought the reason we could be the champion was mainly

because of luck, since the questions were quite difficult. But fortunately, my group

mates could finish a few difficult tasks correctly in the last 5 minutes. After the

competition, I understand that teamwork is very important, and without the team

spirit, we would never get such good grades.

聖士提反書院 St. Stephen’s College Mr. Szeto Yat Chun I felt excited when I was chosen to be the teacher-in-charge of the Mathematics Team this year. There are a number of talented and experienced students, like Dexter Chua and Au Ting Fai who participated in the Creative Problem Solving Competition last year.

They got incredible results which I have never thought of. This year, they work together along with Chung Ji Long and Tam Ching Nam. It was amazing to see that they can work very well and won the Champion in the final. I was really proud of them as they shared good teamwork and gave excellent performance.

I would say the Creative Problem Solving Competition is a competition which cannot be drilled. When compared with the other competitions, it encourages students to think more in Mathematics, both practically and theoretically. It also motivates students to move one step further in Mathematics, and to learn how to explain to others when they know how to tackle a Math problem.

In St. Stephen’s College, we always encourage students to take part in Inter-school competitions since students can explore something new which cannot be found in the lesson. They can find something challenging which helps them grow. But of course, nowadays studying is really harsh and tight, I would say we don’t want our students to spend too much time on drilling, because creativity is one of the most important elements in their future.

Last but not least, I would like to thank all the team members who took part in this competition. I hope they keep up their work as there is still a long way from success.

聖士提反書院 St. Stephen’s College Dexter Chua orkWe've got to play with the true essence of mathematics – the art of explanation.

We wed with Sierpinski triangle and carpet together with the Hausdorff dimension.

Yes! Totally irrelevant to our lives, yet fun. You don't make a Sierpinski carpet to cover your floor – you just can't! As Paul Lockhart has said

“You don't need to make math interesting – it's already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That's why it's so fun!”

Maths isn't supposed to be relevant to our lives. Mathematicians don't model the number of bacteria in a sample - that's the biologist's work. You don't model the profit of a shop as a function of the number of products sold by a complicated exponential function and maximizing it with differentiation – one, you can't control how many products you sell; two, shop managers just don't work that way!

Again, another quote from Paul Lockhart

“Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.”

聖士提反書院 St. Stephen’s College Chung Ji Long I still can’t believe that our team got the first place in this competition. Thinking about when our group joining this competition, we thought that we couldn’t make it to the finals, although we had answered many questions in the heats.

I was very surprised when our team teacher told us that we had entered the finals. I was very happy because only six groups among the hundred groups could enter the finals.

In the finals, I admitted that I did not answer many questions and really helped the group, but I thought, we were really amazed. Among the nine questions, we only left one

unanswered. In the last part, we needed to talk to the nine judges. I thought we could chat well with the judges, especially when I talked about the image belonging to the research of topology geometry, as Sierpinski triangle and carpet are examples of fractals . I hope our group can work together again after one year or two.

聖士提反書院 St. Stephen’s College Tam Ching Nam When I first got into to the venues of the competition, I got afraid. This was the first time in my life that I entered a competition venue with more than 50 people. I felt like

everyone was looking at me. After we settled down, the competition started. I read the paper. The questions were easier than I thought, so I felt much better. These questions weren't hard to calculate, all you need to do was to read the questions and thought of a formula to solve the problem. It was just like the problem solving that I did at school. I felt great after the qualifying competition. It wasn't too hard after all! But when we reached the final competition, things were getting harder. The questions were about areas and volumes and 3-D shapes. The questions were much harder than the last time. I couldn't understand most of the question. I read the question that I knew. Then for the rest of the time, I sat and waited for the competition to end. The judges came in and asked us some questions. I couldn't answer those question either. Luckily, my teammates' math were better than mine and they answered all of the question. After this competition, I thought we were going to loss because of me. I answered only a few questions. But then, few weeks later, I was told that we got the 1st! I was so surprised. We weren't too bad after all.

浸信會呂明才中學

浸信會呂明才中學 學生林家潁 我很高興能代表學校參加這個數學創意解難比賽，並獲得了第四名的佳績。當中也要感謝其餘三位隊友的協助和指導。在這個比賽中，我發現自己對創意解難比賽的投入度比起普通較死板的數學比賽大多了。因為創意解難比賽講求的是團隊合作、創意和數學解難能力，比普通的數學比賽只需要數學難題技巧全面得多。最後，我希望我校隊伍在下年的比賽中成績能再進一步，再創佳績！

浸信會呂明才中學 鄧嘉豪老師 近年來不少教育機構、團體都不斷舉辦形形色色的數學比賽。香港學生比以往多了機會挑戰和發揮他們的數學才能。正當數學比賽如雨後春筍般增長時，學生們也相應地不斷接受比賽相關的培訓以致在比賽中有更好的發揮。培訓的內容離不開比賽常見的題種和解題方法。無疑透過上述比賽和培訓，學生更早認識較高深的數學知識和技巧。然而，作為數學教育工作者，我會憂慮學生在過程中有多少空間能自主思考的呢？對於已有的數學知識和技巧，他們又能否融會貫通以致有所活用呢？

相信教育局對數學教育的現況也有所了解，因此特意舉辦數學創意解難比賽。在本校學生參與創意解難比賽的過程中，起初我會擔心學生面對一些鮮見的數學難題會否不知所措或輕言放棄。但意外地從學生們賽後的分享得知，他們對那些陌生的處境、難題更感興趣。

或許未必能一時三刻解決問題，可是他們會嘗試聯繫和拼湊已有之數學知識；隊員間不吝嗇地分享自己的多元看法和創新見解；將自己的思考方法和成果演示於人前；面對錯失亦勇於反思和修正；最後建構出新的數學知識。難道這個學習過程並非理應的數學教育嗎？

浸信會呂明才中學 學生趙銘泰 在這個由香港教育工作者聯會及教育局課程發展處資優教育組聯合舉辦的香港中學數學創意解難比賽中，我學會了一些有關謝爾賓斯基提出的兩個碎形(謝爾賓斯基三角形和謝爾賓斯基地毯) 和維數的知識，當中謝爾賓斯基地毯是以一個實心正方形劃分為 3 x 3 的 9 個小正方形，去掉中間的小正方形，再對餘下的小正方形重複這一操作得到的圖形。另一方面，當一個物體由 n 個大小一致且互不重疊的小物體組成，這些小物體的形狀和這個物體本身相同，而這些小物體和大物體的大小比例為 1:m，那麼這個幾何物體的維數為 d =

log(n)/log(m)。所以謝爾賓斯基三角形的維數是 log(3)/log(2) ≈ 1.585，而謝爾賓斯基地毯的維數 log 8/log 3 ≈ 1.8928。

浸信會呂明才中學

學生陳恩浩 經過這次數學創意解難比賽之後，我獲益良多。這次比賽與以往所參加的其它比賽不同，它不是個人賽，而是團體賽，所以我特別感興趣。在比賽中，我發現了一些十分有趣的題目，驅使我去找出答案。當我遇到不懂的題目時，我會先試一試，然後問一問我的隊友。

因為我的隊友比我厲害得多，所以我不懂的，

他們通常都懂。

真是意想不到，我們竟然能夠進入決賽並取得殿軍，我們都非常高興。我希望來年可以再次參加這個比賽，吸收更多經驗。

浸信會呂明才中學 學生黃俊誠 我參加了創意解難比賽後，我學會了很多東西，例如什麼是碎形、維數的定義、碎形的維數是多少、謝爾賓斯基三角形等。除了這些以外，我還學會在解難過程中分工合作和團體精神的重要性。例如四位同學怎樣在有限的時間內互相配搭以完成整份試卷。

真希望可以參加多一次這個比賽。

中學初賽 – 2011 年 3 月 12 日

中學決賽 – 2011 年 5 月 14 日

英皇書院

中學決賽 – 2011 年 5 月 14 日順德聯誼總會李兆基中學