[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 212
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Вероятно, все же аналитически доказывается существование максимального числа. А вот как оно конструируется - темный лес.
There's a "statistical" consideration, in fact, the obvious one:
For a 1-digit number = 10 solutions.
for 2-digit = 50 (10*5)
for 3 ~ 10*5*3.33 ~166.6
for 4 ~ 10*5*3.33*2.5 =~500
for 5 ~ 10*5*3.33*2.5*2 ~1000
...
for n =~ 10/1 * 10/2 * 10/3 * 10/4 * .... * 10/(n-1) * 10/n
Thus, as n increases, the number of "correct" numbers first increases (up to 10th place) and then starts decreasing, and finally inevitably becomes less than 1.
Seems to be a correct reasoning // Pretty cool, eh? :)
Of course it does not show the maximal solution, but at least proves its existence.
And you can even calculate where (in which digit) to wait for it approximately.
Do you reckon? // You've done a great job with mutsik!..!
таким образом при возрастании n количество "правильных" чисел сначала возрастает (до 10го разряда ) потом начинает убывать и в итоге неминуемо станет меньшим 1.
Притом можно даже посчитать где (в каком разряде) приблизительно его ждать.Пощитаешь? // С муциком вона как лихо разделался!..
In short, you need to find the minimum n such that (10^n)/n! < 1
I'll try it myself. :)
found:
1.612 at n=43
0.645 at n=44
Thus it is "proven" that the maximum "correct" number has no more than 43 digits.
// but can have less.
Total correct numbers are at most ~ 22025 // Excel rules
нашёл:
1,612 при n=43
0,645 при n=44
Таким образом "доказано", что максимальное "правильное" число имеет не более 43 разрядов.
Crap. Inattention again. Pay attention to the correct answer:
1.612 at n=24
0.645 at n=25
Thus, it is "proven" that the maximum "correct" number has no more than 25 digits.
Well, well, I see you're digging a little. Yeah, well, the statistical "proof" is on my mind too. Its drawback is that it calculates "probability", but it doesn't make reliable conclusions. Even with k=99 the probability of getting the number right is non-zero.
It seems to me myself that the maximum number is unlikely to go much further than 11 digits.
By the way, has the second problem (about n numbers) given anyone yet? It's definitely easier.
Я вот не удержался, на RSDN сходил. Там получили машинное решение 25, но аналитического таки нет
Shit, I should have seen the answer, but then I wouldn't have been interested. You could run through the "good girl Tanya" series of problems - there are rarely pure programming problems.
In general, the question "how many such numbers?" is really more like a question for a programmer in this case.
Там получили машинное решение 25, но аналитического таки нет
Alsu, does this mean that there is still no proof even of the boundedness of the set of such numbers?
Вообще вопрос "сколько таких чисел?" действительно в данном случае скорее похож на вопрос для программиста.
alsu, означает ли это, что даже доказательства ограниченности множества таких чисел все еще нет?
It's hard to understand programmers' comments :))), but a quick glance seemed to me to be a proof of the absence of such numbers for n>25
Вообще вопрос "сколько таких чисел?" действительно в данном случае скорее похож на вопрос для программиста.
alsu, означает ли это, что даже доказательства ограниченности множества таких чисел все еще нет?
Total correct numbers at most ~ 22025 // Excel rules // copy-paste from previous page also rules ;)
Alexey, my reasoning on page 212 quite proves (correctly) the boundedness of this set.
Maybe it's a bit sluggish, but it's quite rigorous.