Can you solve the penniless pilgrim riddle

你能解决身无分文的朝圣者之谜吗

来源视频

“流浪的人未必都迷茫......” 

                               ——J.R.R 托尔金 《魔戒同盟》

After months of travel, you’ve arrived at Duonia, home to the famous temple that’s the destination of your pilgrimage. 

Entering from the northwest, you pass through the city gates and the welcome center, where you’re given a map and a brochure. 

The map reveals that the town consists of 16 blocks, formed by five streets that run west to east, intersecting five more that run north to south.

经历了数个月的旅程之后 你来到了多纳。这里有一座著名的神殿， 也是你此次朝圣之旅的目的地。

你从西北方向进入该区域，穿过城门和游客中心，在这里你得到了 一份地图和一个小册子。

地图显示这个城镇 由 16 个方块区域构成，五条自西向东的大街， 与五条自北向南的大街彼此相交。

You’re standing on the northernmost street facing east, with the two blocks containing the gate and the welcome center behind you.

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

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

As you learn from the brochure, Duonia imposes a unique tax on all visitors, which must be paid when they arrive at their destination within the city. 

The tax begins at zero, increases by two silver for every block you walk east, and doubles for every block you walk south.

你正位于最北边的街道，面向东边，在你身后是大门和 游客中心这两个街区。

神殿的唯一入口位于城镇的东南角。

去那里的路程并不长， 但有一个问题。

你从手册中看到，多纳对所有到访者 都实行一种独特的征税制度，这项税额在游客到达 城市内某处目的地时收取。

税额起初为零，每向东走一个街区 就增加两个银币，每向南走一个街区 就使当前税金翻倍。

However, a recent reform to make the tax fairer halves your total bill for every block you walk north and subtracts two silver for every block you walk west.

Just passing through the gate and the welcome center means you already owe four silver. As a pilgrim you carry no money and have no way of earning any. 

What’s more, the rules of your pilgrimage forbid you from walking over any stretch of ground more than once during your journey— though you can cross your own path. 

Can you figure out a way to reach the temple without owing any tax or walking the same block twice in any direction? 

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

Answer in: 3 

Answer in: 2

Answer in: 1

然而，最近的一项改革 使得缴税更加公平，每向北走一个街区会使税金减半，每向西走一个街区 就可以减掉两个银币。

因为你已经走过了大门和游客中心，你目前已经欠下了 4 个银币。作为朝圣者， 你身无分文，也没有办法赚钱。

此外，朝圣有规定，禁止重复已经走过的道路，但新的路线可以 与以前的路线相交。

你能够想出不用交税、 不重复走过的路线，就能到达神殿的方法么？

如果你想自己解题，请在此处暂停。

答案倒计时：

3

2

1

You look at the map to consider your options. 

Walking towards the temple always increases the tax, and walking away decreases it, so it seems like you can never reach it without owing silver. 

But what happens when you walk around a single block?

If you start out owing four silver and go clockwise starting east, your tax bill becomes six, then 12, then 10, then five. 

If you looped again, you’d owe seven, 14, 12, and six.

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

你看着地图思考如何前进。

朝着神殿走总会增加税金，背离神殿走则总会减少税金，似乎你永远也不能 零税金到达神殿。

但如果你绕着单个街区 走一圈会发生什么呢？

如果你开始欠下 4 个银币 并且以顺时针方向先向东走。

要交的税金变成了 6 个银币，然后是 12 个银币，然后是 10 个银币，最后是 5 个银币。

如果你再走一圈， 你所欠的银币就是 7，14，12，6。

似乎每次顺时针行走 会让你多欠一个银币的税。

What about a counterclockwise loop then? 

Starting owing four again and going south first, your bill changes to eight, 10, five, and three. 

Looping again you’d owe six, eight, four, and two.

Each counterclockwise loop actually earns you one silver. 

That’s because any tax doubled, plus two, halved, and minus two, always ends up one smaller than it started.

The key here is that while the different taxes for opposite directions may seem to balance each other out, the order in which they’re applied makes a huge difference.

那逆时针走会怎么样呢？

仍然以 4 个银币的税金开始， 先向南走，税金变成了 8，然后是 10，5，最后是 3。

再转一圈，你欠的税金变成 6，8，4，最后是 2。

每次逆时针行走 都会减少一个银币的税金，

这是因为任何数字先翻倍，再加 2，再减半，然后再减 2，都会比原来的数字小 1。

关键在于， 税金额度在相反方向上的差异似乎会相互抵消，但调整方向的先后顺序 会产生很大不同。

You start off owing four silver, so four counterclockwise loops would get you down to zero.

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

But there’s another way to reduce your bill: walking one large counterclockwise loop through the city. 

From your starting position, walk three blocks south. 

You need to leave the southernmost street clear for the final stretch, so continuing counterclockwise means going east. 

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

开始时你欠着 4 个银币，因此 4 次逆时针行走 就可以让你缴纳的税金为 0 。

但这并没有如此简单， 因为你不能重复绕同一个街区两次。

但还有一种方法 可以降低要缴纳的税金：在城市中走出一个大的逆时针圈。

从起始点开始向南走三个街区。你需要留出最南边的街道 作为最后的路线，走逆时针路线意味着你要继续向东。

向东走两个街区到达城墙， 此时税金达到了 36 个银币。

But now you can start reducing your bill. 

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

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

So you go one block south, and the remaining three blocks west, leaving you with a debt of -1 silver. 

And since doubling a negative number still gives you a negative number, walking the three blocks to the south wall means the city owes you eight.

但从现在开始 你的税金就要开始下降了。

向北走三个街区再向西走一个街区 使得税金降至 2.5 。

你不能再继续往西走， 不然就没有路可以走了。

因此你继续向南走一个街区， 再向西走三个街区，此时的税金为 -1 个银币。

由于负数翻倍仍然是负数，向南走三个街区 意味着城市欠你 8 个银币。

Fortunately, that’s exactly enough to get you through the final blocks to the temple. 

As you enter, you realize what you’ve learned from your pilgrimage: sometimes an indirect route is the best way to reach your destination.

幸运的是，这些金币的数量足够让你走到最后的神殿。

当你进入神殿时，你意识到 自己从这次朝圣中领会了一个道理：有时候，绕路才是 到达目的地的最好办法。