[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 275
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Well, no, you have to break your brain there as well. The other thing is that you have to suck it in with thoughtfulness, while at the same time protecting your soft spot and other places of origin.
Very often you do not know where the trouble will come from, especially if you are using the government as a source of additional income. But this option is in any case not the best one for applying one's talents. A person's ultimate purpose in this life is clearly not to gain benefit for oneself alone.
Mathemat писал(а) >>
Man's ultimate purpose in this life is clearly not to benefit himself alone.
What is it?
Well, you are familiar with Sciento, you must have heard about the eight speakers. Actually, I don't dare answer such a provocative question, especially from you.
Ну ты с саенто знаком, о восьми динамиках слышал небось.
Yes and there have been rumours of twelve... :) Okay, forget it, that's not what this song's about.
Some authors, on the other hand, think there are more dynamics. They add, say, ethics (in the Scientology sense).
Here's a problem I don't see any catch so far:
a) Well 10. If it had been less, we would have covered no more than 27 cards.
b) This one's harder. It looks like 12, but I haven't decided yet.
Странная у Вас рихметика, однако.
Но с задачкой я перебрал: цифр там 302, но доказать это будет нелегко. То, что их не меньше 301, доказывается легко. Но вот чтобы доказать, что их ровно 302, придется потрудиться.
there is no need to work hard here. The number of digits of a given number is the minimum integer greater than its decimal logarithm. In our case
lg(2^1000)=1000*lg(2)=1000*0.30102999...=301.02999...
So the digit is exactly 302.
No, no, we don't have any logarithms as there is no calculator. All with a piece of paper and a pen.
2^1000 = (2^10)^100 = 10^300*1.024^100
To prove 302, we have to prove that
10 < 1.024^100 < 100 - with a piece of paper and pen!
The right inequality can be easily proved. The left one, on the other hand, is a bit tricky because 1.024^100 ~ 10.715 is in fact, and needs to be evaluated carefully. With Binom we'll have to take at least the first 4-5 members.
Well, that's okay. But has anyone solved the 5^1000 problem?
About the card task, point b), 31 cards.
Looks like the first 27 cards can be discarded. That leaves 4. You have to find out the product of them in the minimum number of questions. I've been thinking and thinking - I can't get less than 4 questions...