home to the famous temple that's the destination of your pilgrimage.

那裡的著名寺廟是你 這次朝聖之旅的目的地。

Entering from the northwest,

你從西北邊進入，

you pass through the city gates and the welcome center,

你通過了城門以及歡迎中心，

where you're given a map and a brochure.

在歡迎中心，你拿到了 一張地圖和一本小冊子。

The map reveals that the town consists of 16 blocks,

地圖顯示，這個鎮 包含了 16 個街區，

formed by five streets that run west to east,

一共有 5 條東西向的街道，

intersecting five more that run north to south.

和 5 條南北向的街道交錯。

You're standing on the northernmost street facing east,

你站在最北邊的 那條街道上，面向著東邊，

with the two blocks containing the gate and the welcome center behind you.

你背後的兩個街區是城門 以及歡迎中心的街區。

The temple's only entrance lies at the very southeast corner.

寺廟唯一的入口 在最東南方的角落。

It's not a long walk, but there's a problem.

步行不用很久，但有一個問題。

As you learn from the brochure,

你從小冊子中得知，

Duonia imposes a unique tax on all visitors,

多尼亞對所有的訪客課一種稅，

which must be paid when they arrive at their destination within the city.

在訪客抵達城市內的 目的地時就要付稅。

The tax begins at zero,

稅額一開始是 0，

increases by two silver for every block you walk east,

每向東走一個街區， 就要多收 2 銀元，

and doubles for every block you walk south.

向南走一個街區， 多收的稅額加倍。

However, a recent reform to make the tax fairer

然而，近期有一項改革， 讓課稅更公平，

halves your total bill for every block you walk north

每向北走一個街區， 總稅額就可以減半，

and subtracts two silver for every block you walk west.

向西走一個街區， 則可以減 2 銀元。

Just passing through the gate and the welcome center

單單通過了城門以及歡迎中心，

means you already owe four silver.

就表示你已經欠了 4 銀元。

As a pilgrim you carry no money and have no way of earning any.

身為朝聖者， 你身無分文，也無法賺錢。

What's more, the rules of your pilgrimage forbid you

不只如此，你的朝聖之旅還規範

from walking over any stretch of ground more than once during your journey—

你在旅行過程中不可以走過 同樣的一段路超過一次，

though you can cross your own path.

不過你可以交叉經過已走過的路。

Can you figure out a way to reach the temple without owing any tax

你能不能想出一種方法， 能抵達寺廟且不用支付任何稅金，

or walking the same block twice in any direction?

也不朝任何方向 走過同樣的街區兩次？

Pause here if you want to figure it out for yourself.

如果你想要自己解解看， 請在這裡暫停。

Answer in: 3

答案即將公佈：3。

Answer in: 2

答案即將公佈：2。

Answer in: 1

答案即將公佈：1。

You look at the map to consider your options.

你看著地圖，想著有什麼選擇。

Walking towards the temple always increases the tax,

向寺廟的方向走，一定會增加稅額，

and walking away decreases it,

遠離它則會減少稅額，

so it seems like you can never reach it without owing silver.

似乎你永遠不可能 在不用付稅金的情況下到達寺廟。

But what happens when you walk around a single block?

但如果你繞著單一街區走會如何？

If you start out owing four silver and go clockwise starting east,

如果你一開始就欠了 4 銀元， 從向東方向開始順時鐘繞著走，

your tax bill becomes six,

你的總稅額是 6 銀元，

then 12,

接著是 12，

then 10,

接著是 10，

then five.

接著是 5。

If you looped again, you'd owe seven,

如果你再繞一圈， 你就會欠 7 銀元，

14,

14，

12,

12，

and six.

然後是 6。

It seems that each clockwise loop leaves you owing one extra silver.

似乎順時鐘的每一圈 就會再增加 1 銀元的稅。

What about a counterclockwise loop then?

那麼逆時鐘呢？

Starting owing four again and going south first,

回頭從欠 4 銀元開始，先向南走，

your bill changes to eight,

你的總稅額變為 8，

10,

10，

five,

5，

and three.

然後是 3。

Looping again you'd owe six,

再繞一圈，你會欠 6 銀元，

eight,

8，

four,

4，

and two.

然後是 2。

Each counterclockwise loop actually earns you one silver.

逆時鐘繞一圈其實 會讓你賺到 1 銀元。

That's because any tax doubled,

那是因為任何稅額會被加倍，

plus two,

加 2 銀元，

halved,

減半，

and minus two,

再減 2 銀元，

always ends up one smaller than it started.

最後一定會比一開始時少 1 銀元。

The key here is that while the different taxes for opposite directions

這裡的關鍵在於， 雖然相反方向有不同的稅額，

may seem to balance each other out,

似乎會彼此抵消掉，

the order in which they're applied makes a huge difference.

但套用的順序卻會 造成很大的不同。

You start off owing four silver,

你一開始先欠了 4 銀元，

so four counterclockwise loops would get you down to zero.

所以做四次逆時鐘的繞圈， 就能讓你的總稅額變成 0。

Unfortunately, it's not that simple, since you can't walk the same block twice.

不幸的是，沒那麼簡單， 因為你不能走同樣的街區兩次。

But there's another way to reduce your bill:

不過還有另一種方式 來減少你的總稅額：

walking one large counterclockwise loop through the city.

繞整個城市走一大圈。

From your starting position, walk three blocks south.

從你的起始點，向南走三個街區。

You need to leave the southernmost street clear for the final stretch,

你先不能走最南邊的街道， 那是最後抵達終點要走的，

so continuing counterclockwise means going east.

接著持續逆時鐘走， 也就是向東走。

Walk two blocks to the eastern wall and you owe a whopping 36 silver.

走兩個街區後會抵達東牆， 你就欠了 36 銀元之多。

But now you can start reducing your bill.

但現在你可以開始減少總稅額了。

Three blocks north and one block west cuts it to 2.5.

向北走三個街區，向西一個街區， 總稅額就只剩 2.5 銀元。

You can't go west from here —that would leave you with no way out.

從這裡，你不能向西走—— 向西之後你就無路可走了。

So you go one block south, and the remaining three blocks west,

所以你要向南走一個街區， 再向西走三個街區，

leaving you with a debt of -1 silver.

現在你負的債是 -1 銀元。

And since doubling a negative number still gives you a negative number,

因為將負數加倍仍然會得到負數，

walking the three blocks to the south wall means the city owes you eight.

向南走三個街區到南牆， 就表示城市會欠你 8 銀元。

Fortunately, that's exactly enough

幸運的是，那金額就足夠

to get you through the final blocks to the temple.

讓你通過最後 幾個街區，抵達寺廟。

As you enter, you realize what you've learned from your pilgrimage:

當你進入時，你了解到 這趟朝聖之旅讓你學到了什麼：

sometimes an indirect route is the best way to reach your destination.

有時，間接的路徑反而是 抵達目的地的最佳路線。