Endless Love

Joe lived in a small town. He has been looking for job for some time. One afternoon, he went out for the job hunt as usual. On his way driving back home, Joe saw an old lady standing helplessly beside her car which had just broken down. Joe stopped his car and said to the old lady, "Madam, it's cold out here. Take your seat in the car and let me fix it for you." After Joe fixed the car, the old lady wanted to give some money to Joe as appreciation, but Joe declined. He said, "If you really want to show appreciation to me, can I ask you that next time you saw others needing help, you lend your hand too." The old lady nodded and happily drove her car away.
Later, the old lady went to a coffee shop. A waitress came up to serve her, and saw her hair was all wet with sweat. The waitress gave her a towel. The old lady noticed that the waitress was very tired from her look, and further noticed that the waitress was pregnant. Despite having to work the whole day for making a living, the waitress still served the old lady eagerly with warm smiles. After the old lady finished with her meal, she paid the bill and, in addition, she took out a $100 from her pocket, put on the tray and gave to the waitress before leaving the coffee shop. On the tray, there was written a small note, "Today I have been helped by a kind gentleman. I wish I can offer in turn some help to you. Please extend this piece of love by lending your hand to others needing help too." The waitress read the note and was deeply moved.
That evening, the waitress went home after work. Her husband was there. She took out the $100, told him what happened and gave it to him. She then sat beside him and gave him a kiss, "Don't worry. Things will soon be alright, Joe."

Joe was the kind man who helped the old lady to fix her car in the afternoon!

The Seven Bridges of Konigsberg

Konigsberg was a small town in East Prussia (now Russia), divided by a river into several parts which were connected by seven bridges as shown in the diagram. The citizens of Konigsberg crossed these bridges for leisure walks on Sundays. One day, they wondered, "Can we take a walk in Konigsberg in such a way that we cross each of the seven bridges once and only once?"

At first sight, we seem to be faced with a tedious and daunting task of tracing out all the possible routes with the seven bridges, and showing whether there is a particular route that works. To address such problem more systematically may require techniques of topology and the like. But when Leonhard Euler (1707-1783) looked at the problem, he immediately claimed that it was NOT possible to have such a walk. Despite Euler (pronounced as "oi-ler") was a great mathematician, he was able to prove his claim by a simple and clever way which can be understood by almost anyone. Euler's strategy to tackle the problem was by method of proof by contradiction. Now let's see how the genius was at work:

First note that Konigsberg is divided into FOUR regions, A, B, C and D interconnected by the seven bridges as shown in the diagram. Next assume that it IS possible to have a walk in the town by crossing each of the seven bridges once and only once. The walk may start in any one of the 4 regions, A, B, C or D, and end in any one of them (which may or may not be the starting region). In any case, we must have at least TWO regions which are neither the starting region nor the ending region.
Now consider any one of these regions. Since it is not the starting region nor the ending region, if we go into this region to visit, we must go out from it accordingly. To visit this region once, we have to go into the region through one bridge and out through another since we cannot cross the same bridge more than once. So we have to have 2 bridges connected to this region in order to visit it once. If we visit this region a couple of times, we have to have an even number of bridges so that we don't cross the same bridge more than once. But looking at the diagram, NO region in this town has such a property (i.e. connected with an even number of bridges: the island C has 5 bridges while the other regions A, B and D all have 3 bridges each), let alone there are at least 2 such regions. Hence, our original assumption that it IS possible to have such a walk leads to some contradiction to the given facts, and thus it cannot possibly be true. In other words, we cannot have a walk in Konigsberg by crossing each of the seven bridges once and only once.

Q. E. D.

空籠

最近記起一個關於亞嘛的故事.

有一天，亞嘛, 大哥, 大嫂，威伯伯，姑媽和我們幾人飲早茶. 吃過一輪點心後, 亞嘛想將吃完了空的點心籠, 放到枱邊, 以便侍應容易收拾. 亞嘛看到坐在對面的大嫂, 正在津津有味地吃着蝦餃. 亞嘛指着她說, "亞嫂, 你嗰邊有冇籠空（隆胸)?" 大嫂嚇了一跳. 蝦餃差點兒哽在喉嚨裡. 她連忙向亞嘛灑手, 急促地說, "細聲, 細聲點 ! 我冇, 我冇呀 !" 亞嘛看了她一眼, 更加提高聲響地說, "乜話? 我明明見到你有籠空. 仲話冇?"
全場一時為之嘩然 ! 大嫂差一點要躲到枱底下去了.

甚麼？

有一天, 媽媽,志瀚和我到一間餐館吃飯. 點過菜後, 我發覺枱面只得兩對筷子, 於是便揮手向一位侍應小姐示意, 打算向她再取多一雙. 那小姐慢慢地走過來問我, "先生, 你想要些甚麼?" 我説, "我想要筷子." 那小姐愣了一下, 走到柜面找了一會, 然後拿着一張空白的 A4 紙回來給我.

老天！我明白了那位小姐為何會猶豫. 原來她以為我説, "我想要塊紙."!