The Three Words

The easiest words - A B C
When we feel the thing is easy, we say "It's as easy as A B C".

The most beautiful sounds - Do Re Me
You are probably already aware if you have seen the movie 'The Sound of Music'.

The basic foundation of everything - One Two Three
Numbers are the building blocks of Mathematics. Mathematics is the mother of all sciences. Science is the mother of all inventions. Invention produces everything we have today.

The first words when meeting a friend - How Are You
This is the most common, simplest and yet warm and polite greeting when meeting friends whom you know or not.

The most friendly words of appreciation - Thank You Mate
This is a bit Aussie, but it is the most friendly appreciation expression here down under.

The easiest way to get away doing something - I Don't Know
Don't use this too much. Try to be more positive and helpful next time.

The most difficult words to say - I Love You
Usually things are easier said than done. For this, it's easier done than said. We may be doing this already to people around us. But to most, it is difficult to say the words out from the mouth.

The mostly heard words everyday around us - O My God
We hear these words everywhere, at any time and almost in every occasions. We hear them when people are overjoyed, happy, surprised, shocked, frightened, sad, distressed, depressed, hurt, angry, furious, disappointed, annoyed, regretful, ......... , you name it. True, God is everywhere!

The Two Swallows

This is a translation of the Chinese poem - 'Swallows' (燕詩) from Po Chu-i (白居易) , (772 - 846) :

Under the beam live two swallows,
a happy, joyous and blissful couple.
Together a love nest that they build,
and give birth to four sweet little fellows.
Days and nights the four babies grow,
chirping to beg for food to swallow.
Worms are hard to catch and find,
but young mouths seem never satisfied.
Though torn and worn the bodies are,
willing and capable still in parent's minds.
Despite dozen times flying to and fro,
yet fear kids in nest still with hunger and in trouble.
So tirelessly they work, and work, and work,
and unknowingly a month's already come and go.
The children slowly get taller and stronger,
while thinner and weaker becomes the couple.
Muttering to teach the kids to speak,
the two brush their young feathers each by each.
Once the wings get into full strengths,
the two lead them flying in court bit by bit.
Now freely they fly higher and farther,
without looking back to their father and mother.
With the wind they're finally gone,
scattering everywhere and everywhere.
The couple shouts loud and hard to the skies,
calling each by their names but no one replies.
Nothing can they do but back to their empty nest,
sobbing into the whole night in sorrow and in tears.
Birdie. Birdie. Don't you cry.
You should reflect and come to realize.
When once you were kids, you did fly,
leaving parents behind with their broken minds.
What your parents saw the other day,
was exactly and truly what you are seeing in front of your eyes.

To all the children of the world!

Qianlong Emperor and the Widow - Sequel

In my blog - Qianlong Emperor and the Widow, 8 December, 2009, Qianlong Emperor was not able to complete a couplet as set out by the widow when he tried to ask for an overnight stay in the widow's house. He had to reluctantly leave and seek accommodation elsewhere.
That night, Qianlong Emperor felt extremely unsatisfied with his inability to complete the couplet and could not get into sleep. He was still thinking about the upper half of the couplet:

寂寞守寒窗 寡室寧容賓客寓

No matter how hard he tried, he was nowhere near success to complete it. Instead, he thought he managed to produce a perfect couplet based on this upper half.
So early next morning, Qianlong Emperor went back to the widow's house again, and presented to the widow the following couplet:

寂寞寒窗空守寡

俊俏佳人伴伶仃

The couplet is considered as perfect as, apart from the sentence matching, it has a same but different radical in each word in the upper and lower halves. The couplet can be translated as follows:

'In this lonely and tiny unit as a widow,
a beautiful fair has companies little.'

My story stops here as I don't know how the widow would react, nor whether Qianlong Emperor would take the widow back to his palace as one of his queens. Readers, you may use your own imagination to provide an ending to this story as you desire.

Paper Scissors Rock

We went to Westfield in Chatswood for shopping on a Saturday. It was near lunch time. Everybody circled round and round inside the car park looking for parking spaces. It was indeed very hard to find one.
Out of nothing, a car backed out from a parking slot in the middle of the aisle. So I hurriedly accelerated and signalled the left indicator light. Opposite from the other side, another car also hurriedly accelerated and signalled the right indicator light. The outgoing car then drove away, leaving our two cars stuck in front of the empty parking slot. I poked my head out of the window, and a lady in the other car also poked her head out of the window. We both argued that we were the one who spotted the parking space first. Temperature began to rise, and cars began to queue up behind each of our two cars. Suddenly, my son Johnson sitting next to me said, "Paper Scissors Rock?" So we proposed. The lady agreed. And the matter was quickly settled.

The result of who finally got the parking space is immaterial. The important thing is, even in resolving conflicts, we can always turn a hostile situation into a friendly one if we care to.

Kung Hei Fat Choy

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Wish you all a happy, prosperous, healthy and lively Year of the Tiger!

The Bright Side of Financial Crisis

People are usually distressed by negative news of financial crisis. But there is always a bright side of everything. Looking back at the financial tsunami two years ago:

. Countries were more united than ever; working in collaboration in combating world-wide recessions.
. Most Governments gave out financial stimulus packages such as cash bonus, subsidies etc; you had extra cash to spend at your own will.
. Housing became much more affordable to the general public, for residence and for investment.
. Financial products (e.g. stock, commodity etc) were inexpensive everywhere; you could easily pick a few and make a fortune as time went along.
. International oil prices declined as economy weakened, but it was good for your pump as local fuel prices fell accordingly.
. Aussie $ fell to historical low; it provided a good opportunity for holidaying down under.

There are always bright spots among dark clouds. Given time, patience and prudence, you can always get through the gloomy storms. You can even be better off in a financial crisis, if you have the money (still)!

Trump Card

I heard this story from one of my ex-colleagues in Cathay Pacific, and I took the liberty of doing some editing and giving it a title.

There were four Catholic men and one Catholic woman having coffee.
The first Catholic man stood up and said proudly, "My son is a priest. When he walks into a room, everybody calls him

'Father'."

The second Catholic man stood up and said proudly in a loud voice, "My son is a Bishop. When he walks into a room, everybody says

'Your Grace'."

The third Catholic man stood up and said proudly in an even louder voice, "My son is a Cardinal. When he walks into a room, everybody says

'Your Eminence'."

The fourth Catholic man stood up and said proudly in the loudest voice, "My son is the Pope. When he walks into a room, everybody says

'Your Holiness'."

The Catholic woman was sipping her coffee in silence. The four men looked down at her and gave her a subtle "Well ...... ?". The woman then raised her head and said shyly, "I have a daughter. She's

slim,

tall,
38D breast,
24" waist,
34" hips.

When she walks into a room, everybody screams

'My God'."

Techniques of Teaching Mathematics

I taught private tuition in Mathematics to students junior than me while I was studying in secondary school. Over time, I came to become aware of some skills in handling some special situations during the lessons. Here are things you may say in class under certain circumstances:

. You don't want to go through all the possible cases. You just show one, and let the student figure out the rest,

"Without loss of generality, .......... "

. You know it's true, but you've lost your notes and you don't know how to prove it,

"It is obvious that .......... "

. You are struggling on how to explain, and your student suddenly reminds you the critical point,

"It now becomes clear that .......... "

. You feel tired after a long day's study yourself, and you don't want to explain in details,

"This is trivial."

. You are desperate to go on Friday afternoon,

"The rest are left as exercises to students."

Quite Easily Done - Is It?

When mathematician (or anyone working with mathematics) finishes a problem, he or she tends to tell the whole world that the work was quite easily done (Q.E.D.).

More often than not, they have already worked on it for days, if not weeks!

Gauss in His Childhood - A Glimpse of Talent

Talent is something you can do easily while others find it difficult to do.
Genius is something you can do easily while others find it impossible to do.

When Gauss (1777-1855) was a child, he attended the primary school in his local town. One day, the teacher found the class was too unsettled. In order to keep the class quiet for a while, he asked the children to sum the numbers from 1 to 100 before they could be dismissed. The class did stay silent while everyone was busy sketching their calculations on papers. Gauss didn't move. He stared at the blackboard for a few seconds, raised his hand and said, " The answer is 5050." The teacher looked at Gauss in disbelief, thinking he was just keen to leave the classroom. Gauss then explained step by step as follows:

"We call the sum of the numbers 1 to 100 S. Then,

S = 1 + 2 + .................. + 99 + 100

If we reverse the order of the numbers and add, the sum is still the same S.

S = 100 + 99 + .................. + 2 + 1

If we take the two together (i.e. add them vertically), we have two times the sum 2S, but the terms become 101 all the way for the 100 terms.

2S = 101 + 101 + .......... + 101 + 101

Since the numbers are the same, we don't need to add 100 times but we can multiply by 100. Simpler still, multiplying by 100 is just to add two zeros at the end of the number.

2S = 101 x 100 = 10100

To get back to the sum S, we just divide by 2.

S = 10100/2 = 5050."

Q.E.D.

Following on Gauss idea, we can develop further to generalize to sum the numbers from 1 to N (no matter what N is):

1 + 2 + ............... + N = N(N+1)/2

or, to sum from the number M to number N:

M + (M+1) + .......... + (N-1) + N = (N-M+1)(N+M)/2

If we still want to go further, we can research more to sum their squares:

1 + 2² + 3² + ………. + N²

or their cubes,

1 + 2³ + 3³ + ………. + N³

or, to push to the extreme,

1 + 2^m + 3^m + ………. + N^m

(no matter what m is).

But these require further mathematical techniques like recursive formula and Bernouilli numbers. I can touch on more as time permits.