Two Interesting Questions

It is a long while since I wrote here last time. I have been to Europe and visited the Deutsche Museum in Munich. There is a small corner about Mathematics there and I took some photos. Maybe I post it next time.

Recently I read two interesting questions. Let me share them here.

1)

Find a smallest finite set of integers, such that every integer in the set is the sum of two other distinct elements in the set. There is one another rule: if x is an element of the set, then -x must NOT be an element of the set.

2)

A square matrix is said to be doubly stochastic if the entries in each row and in each column sum to the same value.

A permutation matrix is a square matrix where each entry in the matrix is either 0 or 1, and there is one and only one entry in each row and in each column which is 1.

Prove that any doubly stochastic matrix is a convex linear combination of permutation matrices.

Comment: The first question is interesting by itself. The second question is considered interesting because an easy solution by induction is possible, after knowing a combinatorial theorem called Hall's Theorem. The question itself may mislead you to some linear algebra arguments.

多買了，反而便宜了？

　　換季了。時裝店裏滿是新一季的秋裝，而未賣完的夏裝則大減價促銷。七折，五折，甚至三折，都不是新鮮事。為求盡快賣掉存貨，不少商店都提供「遞進式折扣」，那就是買得愈多，折扣愈大。我見過以下的折扣表：

「一件五折，兩件四折，三件或以上三折」

　　看過這樣的優惠後，我的第一個反應是：很少人會只買兩件吧？

　　在大多數情況下，很多人應該寧買三件也不買兩件。那是因為多買了，可能反而便宜了。試想想：假設我買了兩件衣服，原價共值　$300。四折後則為　$300 × 40% = $120。如果我多買一件原價　$50 的衣服，三件三折，折扣後為　$350 × 30% = $105。即使這件多買的衣服完全不管用，多買它也立即省了　$15。事實上，簡單的代數運算告訴我們，只要第三件衣服不超過　$100（也就是原來的總價格的三分之一），我們都可以免費拿走它，更可能省下了金錢。

下次遇上這樣的「遞進式折扣」時，不妨留意能否多買而省錢！

新網頁測試

數學資料庫的新網頁已在測試中

http://silver.mathdb.org:8080/silverstripe-test/
網頁的內容會不斷更新.