## 2009年4月27日 星期一

### 一道組合題

最近我從朋友口中得知一道有趣的組合題，其題解十分有趣。各位可以試試。

在一個 10 × 10 的方格表裏，已由上至下，由左至右順序填好 1 至 100 這 100 個整數（即第一橫行順序填好 1 至 10，第二橫行填好 11 至 20，如此類推）。現在我們用 50 個 1 × 2 的小方塊沒有重覆地蓋著這 100 個方格（縱放或橫放皆可）。蓋好後，我們將每一個小方塊蓋著的兩個數字分別乘起來，然後求這 50 個乘積的和。

我找到了一個不消十五行的解答。這題解以不變量 (invariant) 為主要技巧。我會在 5 月 1 日公佈我的答案。

## 2009年4月26日 星期日

### 數學講座

A discussion of this issue would touch upon various aspects of science, involving deep and philosophical questions. With a far more modest aim and scope this talk looks at the issue through some episodes in the work of one mathematician of the last century, Godfrey Harold Hardy (1877-1947).

## 2009年4月24日 星期五

### Context Free Grammar Exercise

1) S -> SS
2) S -> aSbb
3) S -> bbSa
4) S -> bSaSb
5) S -> e（即empty的意思）

## 2009年4月18日 星期六

A few days ago I went to an exhibition called "Mathema". Below are some random pics I took at the exhibition. (Highlight the space to see the answer.)

What is the purpose of this machine?

Can you relate this picture to a famous parable?

Answer: "The parable of the rice grains".

What is the purpose of this machine?

Answer: The Germans used this machine for sending encrypted messages (known as the Caesar's code) during WWII. Note that the the three vertical wheels (just above the keys) will turn the keys, and they allow three degrees of freedom for doing encryption. The guide told us that the British actually broke the code during the war, as the Germans did not use the machine in a careful way. (They always begin their messages with Dear ... ==" )

Why are the above two maps different?
tance, an area r
Answer: Each of the two maps preserve a certain quantity: distance and area respectively.

These two books list out the first 4 million digits of pi!

You can "see" how the Pythagoras' Theorem works by turning the wheel!

This is what NASA sent to the space for aliens to read~

Why is this number so special?

Answer: This number is called "googol", and the name of the famous search engine "google" was inspired by it.

Along which path will a ball from the upper point reach the lower point in the least time?

Answer: Along the curve. This is the famous "Brachistochrone curve". This curve has another remarkable property: the time it takes for the ball to reach the lowest point is always the same, regardless of the starting point!

Mathematicians regard this as the most beautiful theorem in the world. This formula appears so simple but yet it involves some of the most important constants: 1, 0, e, i, pi.

「老師，佢過界！」