[HIMCM暑期班]第2課:建模

第二節課從最簡單的模型開始入手：七橋問題。

首先，先去wikipedia上了解一些有關七橋問題的背景知識。http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

而這節課要作的事情，其實在wiki上已經有所介紹，建模分兩步：

1. 將地圖分隔開的部分染成四種顏色，而且標記橋：

2. 再將其抽象成node和edge...

而後是證實：

一筆畫的證實很容易，難點在於用英語。在說明清楚的同時，若是能配一些示意圖則會更佳。

比較好的說明有：

1.　一種說明的思路(from gx)：

假設，在至少具備2個節點，且具備n(n∈N*)個奇數度節點的連通圖中，存在一條路徑，通過且僅通過每一條邊一次.

根據條件可得：

A.每個節點出度一定對應另外一個節點的入度，所以全部節點的出度與入度之和爲邊數的兩倍，一定爲偶數。

B.當節點不爲起始或終點時，節點的入度應當等於出度，所以奇數度的節點只能是起始點或者終止點。

 

若假設成立，則

① n=0      圖中僅具備兩個偶數度節點，結論顯然成立。

② n=1      僅有一個奇數節點，度的總和爲奇數，與A矛盾，不成立。

③ n=2      a)一個或兩個奇數節點不是起點或終點，與B矛盾，不成立

                 b)兩個奇數節點爲起點與終點

                        ↑待證實

④ n>2      至少有一個奇數度節點不爲起點或者終點， 與B矛盾，不成立。

 

2. 比較好的Problem Restatement:

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges.

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. It is proved that the problem has no solution. There could be no non-retracing the bridges. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

 

3. 有人找到了柯尼斯堡如今的衛星雲圖：

惋惜橋已經不在了，被換成了高速公路了，另人唏噓不已。

P.S.　有人用flash高仿了一個wiki上的插圖：

很是高端有木有？！