How the Königsberg bridge problem changed mathematics - Dan Van der Vieren

譯者: Songzhe Gao 審譯者: 瑞文Eleven 林Lim

你很難在任何現代地圖上 找到柯尼斯堡

You'd have a hard time finding Königsberg on any modern maps,

它怪異的地理位置

but one particular quirk in its geography

has made it one of the most famous cities in mathematics.

使它成為數學界最知名的城市之一

The medieval German city lay on both sides of the Pregel River.

這座中世紀的普魯士城 位於普列戈利亞河兩岸

At the center were two large islands.

河中間有兩座大島嶼

The two islands were connected to each other and to the river banks

以七座橋互相連接

by seven bridges.

Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town,

數學家卡爾·戈特利布·依拉 是附近小鎮的準鎮長

grew obsessed with these islands and bridges.

他從小痴迷於這些島和橋樑

He kept coming back to a single question:

他在同一個問題上反覆打轉

Which route would allow someone to cross all seven bridges

到底怎麼走才能跨越七座橋

without crossing any of them more than once?

卻不會重覆走過任何一座？

Think about it for a moment.

大家來想一想

7

7

6

6

5

5

4

4

3

3

2

2

1

1

Give up?

要放棄嗎？

你一定很想

You should.

It's not possible.

這怎麼可能？

But attempting to explain why led famous mathematician Leonhard Euler

但是數學名家萊昂哈德·歐拉 單純為了求證

to invent a new field of mathematics.

發明了全新的數學領域

Carl wrote to Euler for help with the problem.

卡爾寫信請歐拉幫忙解答

Euler first dismissed the question as having nothing to do with math.

歐拉起初認為 這個問題與數學無關

But the more he wrestled with it,

但當他愈投入，卻愈感其中的蹊蹺

the more it seemed there might be something there after all.

The answer he came up with had to do with a type of geometry

他的答案與當時還不存在的

that did not quite exist yet, what he called the Geometry of Position,

某種幾何學有關 歐拉命名為位置幾何學

now known as Graph Theory.

現在稱為圖論

Euler's first insight

歐拉第一個見解是：

was that the route taken between entering an island or a riverbank and leaving it

這跟出入島嶼之間的路線沒有關係

didn't actually matter.

Thus, the map could be simplified with each of the four landmasses

他把地圖簡化成四塊陸地

represented as a single point,

並標示成單點

what we now call a node,

也就是現在的「節點」

with lines, or edges, between them to represent the bridges.

連接它們的「線」或「邊」代表橋

And this simplified graph allows us to easily count the degrees of each node.

這種簡化的圖形 讓我們能輕易計算節點的分支

That's the number of bridges each land mass touches.

也就是是連接每塊陸地的橋樑數

為什麼分支很重要？

Why do the degrees matter?

Well, according to the rules of the challenge,

根據問題的規則

once travelers arrive onto a landmass by one bridge,

一旦行人由橋走上陸地

they would have to leave it via a different bridge.

就必須從另一座橋離開

In other words, the bridges leading to and from each node on any route

換句話說，在節點上來去的橋

must occur in distinct pairs,

都必須成對才行

meaning that the number of bridges touching each landmass visited

意味著連接陸地的橋數

must be even.

必須是偶數

The only possible exceptions would be the locations of the beginning

唯一的例外可能是起點和終點

and end of the walk.

圖表上，四個節點都是奇數

Looking at the graph, it becomes apparent that all four nodes have an odd degree.

所以不論選哪條路

So no matter which path is chosen,

at some point, a bridge will have to be crossed twice.

還是會經過某一座橋兩次

Euler used this proof to formulate a general theory

歐拉用這個證據制定了一個

that applies to all graphs with two or more nodes.

適用所有兩個以上節點的通論

A Eulerian path that visits each edge only once

只行經各邊一次的「一筆畫定理」

is only possible in one of two scenarios.

唯有兩種情況才有可能

The first is when there are exactly two nodes of odd degree,

第一種是有兩個奇數邊的節點

meaning all the rest are even.

意味著其餘節點都有偶數邊

There, the starting point is one of the odd nodes,

其中，起點是奇數節點

and the end point is the other.

終點也是奇數節點

The second is when all the nodes are of even degree.

第二種，所有節點均有偶數邊

Then, the Eulerian path will start and stop in the same location,

一筆畫路線的起點和終點 是同一個節點

稱為歐拉循環

which also makes it something called a Eulerian circuit.

So how might you create a Eulerian path in Königsberg?

所以要怎麼在柯尼斯堡 規劃一筆畫路線呢？

It's simple.

很簡單

Just remove any one bridge.

只要拆掉任何一座橋即可

And it turns out, history created a Eulerian path of its own.

結果，歷史竟然 真的創造出一筆畫路線

二戰期間，蘇聯空軍摧毀了兩座橋樑

During World War II, the Soviet Air Force destroyed two of the city's bridges,

making a Eulerian path easily possible.

形成一筆畫路線

Though, to be fair, that probably wasn't their intention.

不過，這應該不是他們的本意

These bombings pretty much wiped Königsberg off the map,

柯尼斯堡幾乎全毀，從地圖上消失

and it was later rebuilt as the Russian city of Kaliningrad.

它隨後重建成俄羅斯的加里寧格勒

So while Königsberg and her seven bridges may not be around anymore,

儘管柯尼斯堡與七橋已不復存在

they will be remembered throughout history by the seemingly trivial riddle

它們仍因這微小的謎題 催生出全新的數學理論

which led to the emergence of a whole new field of mathematics.

永存於歷史之中