TA求其記 (2)

MATH232第一份功課。之前有學生問我：「我係咪求其寫D野落去，如果idea啱，咁你就俾分？」我答：「如果個idea係啱既，我無理由唔俾分你既。」他回應：「咁我真係求其寫D野落去啦喎。」

少說也有七年改卷經驗，雖然大部分都是answer only，但偶然也會改改有證明的功課，所以問題本來不大。不過combinatorics是一個很特別的課題，有時一題的solution可以一個數字或數式都沒有，有的就是一大堆英文字母，或者一堆圖。因此其邏輯推進更難被看出。學生表達能力較差的，可以將一個簡單的solution寫得亂七八糟。亦因此，曾有IMO HK committee指IMO最難改的題目就是combinatorics的題目。

結果改了三份功課，最少有五處的邏輯推進是錯的。功課第一題是這樣的：

A m times n chessboard, with m and n both odd, has the upper left corner coloured white. All boxes neighboring to a white box are black, and all boxes neighboring to a black box are white. Now, an arbitrary white box is removed from the chessboard. Prove that the remaining board can be perfectly covered by 1 times 2 dominoes.

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稍有經驗的都應該猜到第一個「證明」是這樣的：

In the original board there are (mn+1)/2 white boxes and (mn-1)/2 black boxes. After removing a white box, the number of white boxes is same the number of black boxes. Hence, the remaining board can be perfectly covered.

以上的「證明」有甚麼不妥，自己想一想吧。我只是想說，到了改功課後期，當我見到學生寫這樣的「證明」時，我打從心底裏發出變態的高興。總好過見到一些在這題寫了兩版（還要連圖都沒有一張……）的人，然後在中間有一處是錯誤的邏輯推進。

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看了第二個「證明」後，我自己傻笑了十幾秒：

Consider a 1 times 3 board. Obviously there is a perfect cover after removing any of the 2 available boxes. Now, we assume that we have a perfect cover for 1 times (2k+1) board. Then for 1 times (2k+3) board, we can divide the board into 2 parts, one part being 1 times (2k+1) and the other part is 1 times 2. Both parts can be perfectly covered by induction assumption, so 1 times (2k+3) board can be perfectly covered after removing any white box. By mathematical induction, 1 times n board, where n is odd, can be perfectly covered after removing any white box.

An induction on x coordinate again will lead to the result.

曾經聽過有人說，在香港和英國讀會考高考的人，要證明一個命題（編按：特別是涉及正整數m、n之類的命題），他們會很努力地嘗試用induction。

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以為上面的「證明」已經夠完美，怎知出了第三個「證明」：

Without loss of generality, we can assume that the removed white box is the left-upper corner one. Blah blah blah...

又曾經聽過有人說，IMO training出身的人很喜歡"WLOG"這四隻字母。看來我該查一查這位學生是否曾參與IMO training。