**Qióng:** Shīfu Shíjú! Shīfu Shíjú!

**Shīfu Shíjú:** Qióng, what do you want?

**Qióng:** Please, tell me why size matters?

**Shīfu Shíjú:** Idiot! Go finish your chores.

**Qióng:** I have done them, Shīfu! I am ready to know!

**Shīfu Shíjú:** Very well. Sit down. Now, first I will show you the way of integers. What is the next digit in this series? 12345…

**Qióng:** The next digit is '1', Shīfu!

**Shīfu Shíjú:** How can you say the next digit is '1'? Have you never brought Shīfu a six-pack?

**Qióng:** The next digit is '1' if the series is 1 through 5 repeating: 1234512345123–

**Shīfu Shíjú:** Idiot! If you introduce complexities such as grouping and blocks you will never understand! To follow the way of integers, you must not think in cliques and tribes; you must ask yourself what one, all on his own, can contribute.

**Qióng:** Thank you, Shīfu. Now I will go and rake the yard.

**Shīfu Shíjú:** Sit down, Qióng! You have asked to know. Now you will know.

**Qióng:** Yes, Shīfu Shíjú.

**Shīfu Shíjú:** On the shed, I have combination padlock. You have seen this padlock?

**Qióng:** Yes.

**Shīfu Shíjú:** Describe it

**Qióng:** It is red and has three wheels side by side. Each wheel has some numbers and if you turn each wheel to its magic number, then the padlock will open. But if any one of the three wheels is on the wrong number, it will not open. Here, I will paint a picture of it for you:

**Shīfu Shíjú:** Excellent, Qióng. And how many combinations are possible for that lock?

**Qióng:** Well, the highest combinations is 999, and 000 is also possible, so there must be 1000 combinations.

**Shīfu Shíjú:** That is true, but how can you know this on the abacus?

**Qióng:** Well. Let's see. Each wheel has 10 different digits. And there are three wheels. So the number of combinations is 10 x 10 x 10. That's 10^3.

**Shīfu Shíjú:** Very good, Qióng. Let us call the number of digits on a wheel the "girth" of the lock. And let us call the number of wheels the "length" of the lock. In that case, when we ask how many combinations, we also ask for the girth to the power of the length: girth^length.

**Qióng:** Thank you, Shīfu. I will go and rake the yard now.

**Shīfu Shíjú:** Idiot! Do you think the way of integers can be mastered in mere seconds? There is *much* mystery.

**Qióng:** I will sit and listen, Shīfu.

**Shīfu Shíjú:** Let us call the quantity of digits on a wheel the "girth" but let us call the digits themselves the "alphabet". What is the girth of a wheel, and what is its alphabet, Qióng?

**Qióng:** The girth of a wheel on this lock is 10, and the alphabet is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

**Shīfu Shíjú:** Very good. And how long will it take to open this lock if you do not know the magic numbers?

**Qióng:** That depends on how quickly the thief can try each combination and change to the next combination when he has guessed poorly.

**Shīfu Shíjú:** I have trained you well, Qióng.

**Qióng:** You are my Shīfu, Shīfu.

**Shīfu Shíjú:** Let us say that the thief can try one combination per second, including the time it takes to move to the next attempt. And let us say that the thief is half lucky and half unlucky, so that he will find the combination halfway through his trial of all combinations.

**Qióng:** If he were the unluckiest man in the province, Shīfu, it would take him almost 17 minutes. But since he is half lucky, it will take him only a little more than 8 minutes.

**Shīfu Shíjú:** That is right, Qióng.

**Qióng:** Shīfu, why do you use such a stupid, pointless lock on the shed? It takes on average only a little more than 8 minutes for a stranger to open it!

**Shīfu Shíjú:** The shed is empty, Qióng.

**Qióng:** I am now enlightened, Shīfu Shíjú. Please continue.

**Shīfu Shíjú:** What happens if we increase the length of the padlock from 3 to 4?

**Qióng:** The number of possible combinations also increases from 10^3 to 10^4. Instead of 1000 combinations, there would be 10,000.

**Shīfu Shíjú:** How long does it now take the half-lucky thief to open the lock?

**Qióng:** Now it takes him almost an hour and a half– over 80 minutes instead of 8.

**Shīfu Shíjú:** Good. But let us not increase the length of the padlock. Let us leave that at 3 and instead increase the alphabet of each wheel. We will just add the letter 'a': 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a. What happens if we increase the girth in this way, from 10 to 11?

**Qióng:** The number of possible combinations increases from 10^3 to 11^3. Instead of 1000 combinations, there would be 1331.

**Shīfu Shíjú:** And how long does it now take the thief?

**Qióng:** 22 minutes. Shīfu Shíjú?

**Shīfu Shíjú:** Yes, Qióng.

**Qióng:** Girth sucks! It does hardly anything for us. We should only care about length!

**Shīfu Shíjú:** No, Qióng. Girth is intelligence, and Length is wisdom. Much wisdom is very useful even for the unintelligent, but the very intelligent are only a little useful without wisdom. To follow the way, we need intelligence **and** wisdom.

**Qióng:** Thank you for correcting me, Shīfu.

**Shīfu Shíjú:** We need the size of the ship *and* the motion of the ocean.

**Qióng:** Yes, Shīfu.

**Shīfu Shíjú:** Nothin' from nothin' leaves nothin', Qióng.

**Qióng:** You're creeping me out, Shīfu.

**Shīfu Shíjú:** I am old, Qióng, and detached. But PBS keeps running those nostalgia concerts.

**Qióng:** I will go rake the yard now, Shīfu.

**Shīfu Shíjú:** You have not yet attained true knowledge. Sit down.

**Qióng:** Yes, Shīfu.

**Shīfu Shíjú:** By how many letters must we increase the alphabet in order for the change in girth to equal a change in length from 3 to 4?

**Qióng:** Adding 1 length gave us a tenfold increase in combinations. To get a tenfold increase in combinations by adding girth, we would have to add… carry the bead… we would have to add 12 more letters to the 10 numbers on the wheel.

**Shīfu Shíjú:** You have spoken well. 22^3 is 10,648, and 10^4 is 10,000. Close enough for imperial work. It is obvious that if we add two wheels to the padlock for a total of five, but leave the girth of each wheel at 10, that will be 10^5 combinations, or 100,000. How many letters of girth do we need, Qióng, in order to equal that on a 3-wheel padlock?

**Qióng:** We need an alphabet of at least 47, because 47^3 is 103,823.

**Shīfu Shíjú:** So, to the original 10 numbers we need to add at least 12 letters to reach 10,000, and we need to add at least 37 characters to reach 100,000. And to reach a million, we would need to add 90, because 100^3 is…

**Qióng:** An even million. Not too much different from the number of shells my family has paid for me to attend this academy.

**Shīfu Shíjú:** I, too, know something of raking the yard, little Qióng.

**Qióng:** I am now enlightened, Shīfu.

**Shīfu Shíjú:** Now, take up your abacus. How long will it take the half-lucky thief to open a padlock with a girth of 47 and 3 wheels or a girth of 10 and 5 wheels?

**Qióng:** Around 14 hours.

**Shīfu Shíjú:** And for a padlock with a girth of 100 and 3 wheels, or a girth of 10 and 6 wheels? How long to try half a million combinations?

**Qióng:** Almost 6 days, Shīfu.

**Shīfu Shíjú:** You have understanding. Now reach for wisdom. Imagine you have an alphabet of 7776 characters on a wheel.

**Qióng:** That is a mighty wheel, Shīfu!

**Shīfu Shíjú:** If the half-lucky thief must try half the combinations before opening the padlock, then how long will he need with a girth of 7776 and a length of 1 wheel? 2 wheels? 3 wheels?

**Qióng:** I'm gonna need a bigger abacus.

**Shīfu Shíjú:** Do your best.

**Qióng:** He will need a little more than 1 hour for 1 wheel, and 350 days for 2 wheels, and but for 3 wheels he will need 7,450 years.

**Shīfu Shíjú:** 4 wheels? 5 wheels?

**Qióng:** For a girth of 7776 and 4 wheels, he will need over 544 millennia. And for 5 wheels, he will need 450,460,353 millennia.

**Shīfu Shíjú:** Dayam, boy. You rockin' dat abacus!

**Qióng:** Thank you, Shīfu.

**Shīfu Shíjú:** And how long is a millennium?

**Qióng:** 1000 years.

**Shīfu Shíjú:** How many of those does the half lucky thief require for 7776^5 combinations?

**Qióng:** 450 million millennia. And 900 million millennia if he's very unlucky.

**Shīfu Shíjú:** What is the population of the Empire, Qióng.

**Qióng:** I believe it is 2 billion plus another 200 million.

**Shīfu Shíjú:** Suppose every person in the empire is a half-lucky thief trying to open this lock. That should reduce the required time, yes?

**Qióng:** I shall divide by 2,200,000,000, Shīfu.

**Shīfu Shíjú:** And with the whole Empire trying, how long will it take?

**Qióng:** It will still take them almost 205 years.

**Shīfu Shíjú:** Girth of 7776. Length of 6 wheels. Thieves numbering 2,200,000,000. How long?

**Qióng:** Some big-ass number, Shīfu.

**Shīfu Shíjú:** 1,592,172 years, Qióng. And even if there are a million such Empires with a population like our own, it will take them a year and 7 months.

**Qióng:** So we should secure our empty shed with a padlock having 6 wheels, each with an alphabet of 7776 letters.

**Shīfu Shíjú:** Letters are hard to remember, unless they form words. But words are easy to remember, even when they do not form sentences. Tell me how long it would take a million Empires of population 2-and-a-fifth billion to open a lock of girth 7776 and length 7.

**Qióng:** 7, Shīfu? Even with a million such Empires, that would take 12,390 years.

**Shīfu Shíjú:** Can you memorize 7 random words, Qióng?

**Qióng:** Plus or minus 2, Shīfu. 7 words are harder to remember than 6 when they have no meaning, but 12,390 years is so much more than 1 year and 7 months!

**Shīfu Shíjú:** 7776 girth and 8 wheels and 1,000,000 Empires of 2,200,000,000 half-lucky thieves! Quick, quick!

**Qióng:** 96,340,000 years. And even for a billion such Empires, that would be 96,340 years. By that time, the fields will no longer be fallow. I am now enlightened, Shīfu Shíjú.

**Shīfu Shíjú:** This is the first way of the integer: the book of eight wheels.

**Qióng:** Now I will go and rake the yard.

**Shīfu Shíjú:** Idiot! There is a second way!

**Qióng:** Teach me, Shīfu!

**Shīfu Shíjú:** How many numbers do the scribes use in Dàqín?

**Qióng:** 10, just like us.

**Shīfu Shíjú:** And how many letters do the scribes use in Dàqín?

**Qióng:** 26.

**Shīfu Shíjú:** And they use not only the small letters but the big letters. So 52, correct?

**Qióng:** Correct, Shīfu.

**Shīfu Shíjú:** So that is an alphabet of 62: the numbers and the small and big letters. But the scribes of Dàqín also use many special characters. How many?

**Qióng:** Well, let's see. They use ,./<>?;':"[]\{}|`~!@#$%^&*()-=_+ and that makes 32 plus the void.

**Shīfu Shíjú:** They also use some symbols that require special ink, but your everyday Jiǔ will not use those. In fact, he may not even use all that you have named. But you have done well.

**Qióng:** Thank you.

**Shīfu Shíjú:** The 33 you have named plus the 62 letters and numbers make an alphabet of 95. With a girth of 95, how many wheels do we need to match the book of 8 wheels?

**Qióng:** 15 wheels would be too few, but 16 wheels would be more than enough.

**Shīfu Shíjú:** Good. So which of these combinations will pose the greatest challenge to a billion Empires of billions of half-lucky thieves? The first option is **border dress 84 wp curve 4444 tumble cult** and the second option is **)YRmWd&}6lKKTRpk**.

**Qióng:** The second is slightly better than the first by an amount that does not matter.

**Shīfu Shíjú:** I love this job because of students like you.

**Qióng:** 96,340 years for a billion Empires of 2,200,000,000 half-lucky thieves! Our empty shed will truly be safe, for 7776^8 or 95^15.8 are truly huge numbers.

**Shīfu Shíjú:** No, they are not. They are small numbers. In the Book of Factorization, it is written that they are only 103 bits.

**Qióng:** What is a bit, Shīfu?

**Shīfu Shíjú:** Try to see the larger point.

**Qióng:** I am sorry.

**Shīfu Shíjú:** If the padlock has a girth of only 2, how many wheels must it have in order to be as strong as the strongest padlocks we have discussed?

**Qióng:** With an alphabet of only two characters, we would need 103 wheels.

**Shīfu Shíjú:** …

**Qióng:** I am now enlightened, Shīfu.

**Shīfu Shíjú:** After you have raked the yard, I will teach you about a padlock that has a girth of 2 and 4096 wheels. How long do you think it would take many empires of many thieves to find the magic number for such a lock?

**Qióng:** My abacus has no such number.

**Shīfu Shíjú:** It would take all the thieves in all the empires many lifetimes of the universe to solve such a puzzle. And if they had the magic to do it much faster, there would always be more wheels, each more powerful than the previous. Go now, and think on it while you rake.

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