Well, you're in a real pickle. You see, a few years ago, the king decided your life would be forfeit unless you tripled the gold coins in his treasury. Fortunately for you, a strange little man appeared and magically performed the feat. He placed handfuls of coins in and out of a magical bag, and sang a strange rhyme:
Xo’sh, sen juda mushkul muammodasan. Agar qirol xazinasini ko’paytirmasang qirol seni o’lishingni qaror qilgan. Baxtga ko’ra, gnom paydo bo’lgan, va buni sehr yordamida amalga oshirgan. U sehrli xalta ichiga tillalar soldi, va g’alati bir qo’shiq kuyladi: aminmanki
“The more gold goes in and more comes out, as sure as I am me. And in again, and out again, and now it’s it times three!”
“Qancha ko’p tilla, shuncha ko’p tilla. Kirib, chiqib tillalar, mana 3 barobar!”
Incredibly, that tripled the coins and saved your life. Were you grateful? Yes. Were you desperate? Yes. Did you promise him your first-born child in exchange for his help? Yes.
Ajablanarlisi, u hayotingni asrab qoldi. Xursandmisan? Ha. Muhtojmiding? Ha. Evaziga birinchi bolangni va’da berding.
Fast forward to today. No sooner have you given birth to a beautiful baby boy than the little man shows up to claim his prize. You cry and beg him not to take the baby. Softening, he begins, “If you can guess my name—” “Banach-Tarski?” you say. “It’s on the front of your shirt.”
Hozirga kelsak. Go’dakni dunyoga keltirishing bilan, Gnom sovg’ani olish uchun paydo bo’ldi. Yig’lab undan bolani olmasligini so’rading Ohista, “agar ismimni aytsang__” “Banak-Tarski”. “U kiyimingda turibdi.”
“What! That won’t do. Aha. My bag,” he explains, “increases the number of gold coins placed inside it in a very special way. If I take any number of coins and place them in, more will come out. And if I place those in the bag again, the total that comes out will be three times whatever I began with.”
“Nima! Bo’lishi mumkinmas. Qopim,” dedi u, ichidagi tillalarni noodatiy ko’paytiradi. Solgan tillalaring ko’proq bo’lib chiqadi. Va agar ularni yana solsam, umumiy soni boshida solganimdan 3 martaga ko’payadi.
He takes 13 coins and places them in the bag, then removes the contents. “I’ve used the magic once, not twice,” he says. “Tell me how many coins are in my hand and I’ll have mercy.”
U 13 ta tillani qopga soladi. Va chiqqanini oladi. Va u deydi, “Men 1 marta sehr qildim, 2 marta emas.” “Nechta tillam bor va men shafqat qilaman”
How many coins is he holding?
Unda nechta tilla bor?
Pause here to figure it out yourself. Answer in 3
Shu joyda to’xtab, nechtaligini ayting. 3
Answer in 2
2
Answer in 1
1
The bag’s magic works just like what in mathematics is called a “function,” and it’s convenient in both cases to use an arrow to denote the transformation. We can write what we know like this.
Qop sehri xuddi “funksiya” tushunchasidek, Ko’payishni ko’rsatgich orqali belgilaymiz Bilganimizni shunday yozsak bo’ladi.
We want to know what goes in this particular blank.
Bosh joyga nima kerakligini bilib olamiz.
Maybe the bag just multiplies the number of coins by some number. In that case, multiplying by that number twice would be the same as multiplying by 3, which means the multiplier would be the square root of 3. That’s not a whole number, though. And we don’t have bits of gold coins coming out of the bag. Something else is going on.
Qop tillalarni qaysidir songa ko’paytirar. Bunda, o’sha songa 2 marta ko’paytirish xuddi 3ga ko’paytirishdek bo’ladi, ya’ni ko’paytiruvchi 3ning ildizi bo’ladi. Ammo, bu butun son emas. Va tillalar bo’laklarga bo’linib qopdan chiqmayapti. Yana nimadir sodir bo’lyapti. Xo’sh, agar
Well, if filling in the blank between 13 and 39 is too hard, maybe we can start with something easier. Can we figure out what’ll happen to 1 coin?
bo’sh joyni 13 bilan 39 orasida to’ldirish qiyin bo’lsa, boshqa osonroq yo’l tutamiz. 1 ta tillaga nima bo’lishini bilasizmi?
If you use the bag on a single coin twice, you end up with triple; that’s three gold pieces.
Agar 1 ta tillani 2 martda solsez, 3 ta bo’ladi: bu 3 tillalar degani. Chunki
Because the bag always increases the number of gold coins, the blank must be between 1 and 3, so 2. It’s a start. What’s next?
qop tillarni sonini doim oshiradi, bo’sh joydagi son 1 va 3 orasida, shu uchun 2. Bu boshlanishi. Keyinchi?
Let’s think about a few other possible starting places.
Qani boshqa boshlanish joylarini o’ylaymiz
We already know 2 becomes 3 and that lets us fill in the next blank as well. Now we’re getting somewhere!
Allaqachon bilamizki, 2 soni 3ga aylanadi va bu keyingi bo’shliqni to’ldiradi. Mana endi, javoblarni topyapmiz.
We just need to extend this out to 13. Remember the other rule, though: when you put more coins in, you get more coins out. That means the numbers in every column must go in increasing order as well. In other words, because 6 coins become 9, it’s not possible for 4 coins to become 10. Nor could 4 become 5, since 3 becomes 6.
Buni shunchaki 13gacha cho’zishimiz kerak. Ammo, qoida yodda tursin: ko’proq tilla solsang, ko’proq olasan. Bu degani, har ustundagi son o’sishi kerak Ya’ni, chunki 6 ta tilla 9 ta bo’ladi, 4 ta tilla 10 ta bo’lishi mumkinmas. 4 5 bo’la olmaydi, chunki 3 6 bo’ladi.
So 7 and 8 fill those blanks on the right of 4 and 5, which in turn gives the answer for two more blanks.
Demak, 7 va 8, 4 va 5 ning yonida bo’ladi, bu boshqa 2 ta bosh joyga sonlar beradi.
Knowing that the numbers go in increasing order in every column, the only choices for the remaining blanks are 19, 20, 22, and 23.
Hamma son o’sishini hisobga olgan holda, 19, 20, 22, va 23 javob sifatida qolyapti.
And look! We have our answer! There must be 22 gold coins in his hand.
Va qara! Javoblar tayyor! Uning qo’lida 22 ta tilla bo’lishi kerak.
“I’ll give you three guesses,” the little man begins to say.
“3ta imkon beraman,” dedi gnom.
“22 coins,” you respond.
“22 ta” deding.
“What?! How did you know?”
“Nima? Qanday bilding?”
“I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”
“Topishmoq yoqdi. Javob kiyiming orqasida”