Can you solve the computer virus riddle? - James Tanton

譯者: Camila Lin 審譯者: Amanda Zhu

Your antivirus squad is up against a particularly sadistic bit

你的防毒小隊正在和 某個特別殘虐的位元對抗，

of malicious code that’s hijacked your mainframe.

它屬於一段正在駭入 你大型電腦主機的惡意程式。

What you’ve learned from other infected systems— right before they went dark—

你在其他被感染的系統 螢幕黑掉前發現，

is that it likes to toy with antivirus agents in a very peculiar way.

它喜歡以一種特定的方式 玩弄防毒特工。

It corrupts one of the 4 disks that run your mainframe,

它會毀損控制主機運作的 四個硬碟之一，

represented by lights showing which are on and which off.

並以燈的亮暗來表示硬碟的啟動狀態。

Then it selects one member of the antivirus squad— this’ll be you—

然後，它會選出一位防毒特工 ──就是你──

and brings them into the mainframe.

並將你帶進主機中。

It tells them which disk it corrupted,

它會告訴你它破壞了哪個硬碟，

allows the agent to switch a single disk on or off,

並允許你開啟或關閉一個硬碟，

then immediately de-rezzes the agent.

然後立刻將你從虛擬世界中消去。

Your squad can make an all-out attack to break into the mainframe

你的小隊能強硬地開出 通往主機的一條路，

and destroy one disk before they’re wiped out.

並在被消滅之前毀損一個硬碟。

If they destroy the corrupted one, the malware will be defeated.

如果他們毀去中毒的那個硬碟， 就能戰勝這個惡意軟體。

Any others, and the virus will erase the entire system.

如果毀去完好的硬碟， 整個系統就會被病毒化為烏有。

The lights are only visible within the mainframe,

你只能在主機內部看見燈，

so you won’t know until you get there which, if any, are on.

所以在還沒進入前， 你不知道哪些燈亮著。

How can you communicate, with your single action,

你該怎麼只靠一個動作， 就能告訴你的隊員，

which of the 4 disks has been corrupted?

這四個硬碟中的哪個已經中毒？

Pause here to figure it out for yourself. Answer in 3

［如欲自行解題，請在此按暫停， 答案揭曉倒數：3］

Answer in 2

［2］

Answer in 1

［1］

The setting is a big clue for one solution.

設定方式是一份重要的線索， 提示了一個解決辦法。

Using binary code— the base two numbering system that only uses 1s and 0s—

使用二元碼，也就是只用 1 和 0 的二進位數字系統，

we can represent each of the 4 disks with a 2-bit binary number

我們可以用兩位元的二元數字 來分別代表這四個硬碟：

ranging from 00 for zero to 11 for three.

從代表 0 的 00 到代表 3 的 11 。

What we’re looking for now is some sort of mathematical operation

我們現在想找出某種數學公式，

that can take the lit disks as input, and give the corrupted disk as an output.

其中亮燈的硬碟編號是輸入端資料， 而中毒的硬碟編號是輸出端資料。

Let’s consider one possibility.

讓我們想想這個可能性。

Say that the corrupted disk was this one,

假設這是中毒的硬碟，

and when you come in, no lights are on.

但你進來的時候，沒有一個燈是亮的，

You could turn 11 on to indicate that disk.

你就能打開 11 號的燈來標示它。

Okay, what if you came in and 11 was already on?

好，那如果你進來的時候， 發現 11 號的燈已經亮了呢？

You have to switch one light.

你就必須轉一個燈。

Which seems like the most innocuous to change?

哪個燈就算改變狀態也不要緊？

Probably 00, in that if you were to add 00 and 11,

應該是 00 ，

因為把 00 和 11 加起來，

you’d still get 11.

總和還是 11 。

So maybe the key is to think of addition of binary numbers,

所以，也許關鍵就是二元碼的加總，

with the sum of the lit disks communicating the corrupted disk number.

讓亮燈硬碟編號的加總 代表損壞硬碟的編號。

This works great, until we start with a different hypothetical.

這個理論運作得很成功， 直到我們用另一個不同的假設。

What if 00 was the corrupted disk, and 01 and 10 were on?

如果 00 是中毒硬碟的編號， 但 01 和 10 的燈卻亮著呢？

Here, the sum of the lit disks is 11.

這樣一來，亮燈的 硬碟編號總和是 11。

But we need to change this to a sum of 00 with the flip of one switch.

但我們必須靠著一個動作， 把總和改成 00。

We have four options: turning switch 00 on gives us 11.

我們有四個選擇： 打開 00，得到 11。

Turning 01 off takes us back to 10,

關掉 01，總和是 10。

and turning 10 off gives 01.

關掉 10 則會得到 01。

None of those work.

沒一個行的通。

Turning switch 11 on gives us 110 by standard binary addition.

根據標準二元進位法， 如果打開 11，我們會得到 110。

But we don’t really want three digit numbers.

但我們不想要三位數。

So what if— to keep the result a two digit number—

所以，為了讓結果保持在二位數，

we break the rules a bit and let this sum equal 22.

我們稍微更動一下規則， 讓它們的總數成為 22。

That’s not a binary number, but if we regard 2s as the same as 0s,

這不是個二元制數字， 但如果我們把 2 當成 0，

that does indicate the correct disk.

還是能指出中毒的硬碟。

So this suggests a strategy:

這讓我們想出一個策略：

look at the sum of all the lighted disks we see,

把亮燈的硬碟編號加起來，

regarding 2s as 0s.

並把 2 視為 0 。

If it’s already the correct result, flip 00,

如果這是正確答案， 就改變 00 的燈號，

and if not, find the switch that will make the sum correct.

如果不是，就選擇讓總數正確的硬碟。

You can see for yourself that any starting configuration

你會發現，不管初始配置為何，

can sum to any number from 00 to 11 with a flip of a switch.

只要開亮或關閉一盞燈，就能讓總和 變成 00 到 11 中的任何數字。

The reason this works is related to a concept called parity.

這個策略之所以有用， 和「奇偶性」這個觀念有關。

Parity tells you whether a given value is even or odd.

「奇偶性」能告訴你， 一個被給定的值是奇數還是偶數。

In this case, the values whose parity we’re considering

在這個案例中，我們考慮奇偶性的值

are the number of 1s in each digit place of our binary sums.

就是加總中兩個位數中 1 的個數。

And that’s why we can say that 2 and 0, both even numbers,

這也是我們能將 2 與 0 一視同仁的原因：

can be treated as equivalents.

它們都是偶數。

By adding or subtracting 00, 01, 10, or 11,

藉著加減 00、01、10、11，

we can change the parity of either, both, or neither digit,

我們就能改變其中一個或兩個位數的 奇偶性，也能讓它保持原樣，

and create the disk number we want.

來得到我們想要的硬碟編號。

What’s incredible about this solution is that it works for any mainframe

這個解法的驚人之處，

就是只要主機的硬碟數量 是 2 的次方就能適用。

whose disks are a power of two.

With 64 you could turn each activated disk into a 6-bit binary number

如果是有 64 個硬碟，

你可以將每個運作中的硬碟 以一個 6 位元的二元碼代表，

and sum the 1s in each column,

並加總每個區塊的 1，

regarding any even sum as the same as 0 and any odd sum as 1.

將總和中的所有偶數視為 0， 奇數則視為 1。

1,048,576 disks would be daunting, but entirely doable.

1,048,576 個硬碟是個難題， 但這方法仍然可行。

Luckily, your mainframe is much smaller.

幸運的是，你的主機沒那麼大。

You make the valiant sacrifice and your team rushes in,

你英勇犧牲，而你的小隊衝了進來，

destroying the corruption and freeing the system.

毀掉了中毒的硬碟， 讓系統重獲自由。