Nice Example on Schutte Problem

Some background first. A tournament of N players mean a competition that every pair of players play against each other exactly once. In our case, no draw is allowed; either one wins or the other wins.

A tournament is with Schutte property of order k if every set of k players are all defeated by one of the other players.

Using probabilistic method, it is easy to show that for any k, there exists sufficiently large N such that a tournament with Schutte property of order k is possible.

My focus here is a cute example of tournament with Schutte property of order 2: when N=7, name the players by 0,1,2,...,6, then a tournament with Schutte property of order 2 is given by:

i defeats j if and only if (i-j) is a quadratic residue of 7.

培正數學邀請賽初賽 -- 試題及答案

第九屆培正數學邀請賽初賽題目已經上載。

中一組：http://www.mathdb.org/resource_sharing/others/s_puiching09_H1.pdf
中二組：http://www.mathdb.org/resource_sharing/others/s_puiching09_H2.pdf
中三組：http://www.mathdb.org/resource_sharing/others/s_puiching09_H3.pdf
中四組：http://www.mathdb.org/resource_sharing/others/s_puiching09_H4.pdf
高中組：http://www.mathdb.org/resource_sharing/others/s_puiching09_H5.pdf
答案：　http://www.mathdb.org/resource_sharing/others/s_puiching09_HA.pdf

數學網頁資料設計比賽 2010

數學資料庫將於香港資優教育學院合辦「數學網頁資料設計比賽 2010」，詳情可參閱 http://www.mathdb.org/mac/2010/。

我們將於 1 月 29 日與資優教育學院共同舉行簡介會，讓老師和同學瞭解比賽詳情。有興趣的老師和同學快填妥簡介會回條並按指示遞交吧！

神奇教練

前排睇新聞，話國際米蘭係意甲聯賽賽事尾段連入兩球，以4-3險勝錫耶納，保住了國際米蘭教練摩連奴近八年執教球隊主場不敗的紀錄。

感覺這個神奇教練真的很神奇。但怎樣將這樣的「神奇」量化呢？有！假設他執教球隊主場每場不敗的機會是95%，而球隊每年主場賽事最少25場，那麼八年就有200場主場賽事。這可能有些誤差，那就當它是180場吧。那麼180場不敗的概率的機會率只有0.00009778。

（聲明：以上計算的假設毫不嚴謹，亦無任何統計數據backup支持，純為筆者吃飽飯沒事幹（現為紐約時間晚上九點左右）發表的文章。）

培正數學邀請賽：最新消息

數學資料庫協辦的培正數學邀請賽初賽將於下星期六（1 月 23 日）舉行。有關詳情已上載至比賽網頁。參賽同學及領隊老師請特別注意以下事項：

1. 准考證已寄給各參賽學校及個人報名的參賽者。
2. 參賽者必須帶備准考證及身分證應考。
3. 參賽者應使用大會提供的 HB 鉛筆作答（答題紙樣本），惟需自備橡皮擦及其他文具。
4. 如答案小於 1000，須「補 0」以湊足四位，例如：如答案為 39，應填「0039」。
5. 本年將不會派發答案予領隊老師。試題及答案將於初賽後盡快上載至數學資料庫（屆時會在本網誌公佈），亦會於一星期內上載至比賽網頁。

Polynomials and topology

Recently my friend is working on a problem in topology, and out of his work, it appears that the there is a special pattern in the coefficients of the following polynomial, when m,n are relatively prime:



It appears that if one expands this polynomial out, collects terms and arranges them in decreasing powers of x, then the non-zero coefficients are all either 1 and -1, and they appear to alternate as the power decreases. (e.g. when m=4, n=3, the polynomial is



It is not known whether this pattern really exists. But I thought this is cute and may be of interest to some of you. Does any of you have any idea about how to prove/disprove it?

(The case of interest in topology is when m > 3n, but it looks like this pattern persists as long as m,n are relatively prime.)

Elementary number theory

Someone say that 167588402882520529579353108764873470755823697 is the smallest positive integer k such that all digit of 1989k are the same.

Do you agree?