When all the integers from zero (0) to n are written, the digit zero appears exactly n times?

Question

There are no leading zeroes. The sequence is:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, .etc., all the way to n. When you have finished, you will have written exactly n zeroes.

Find n. Here's a hint:

http://answers.yahoo.com/question/index;_ylt=AuKFq10_URe157KTNjQkaAjsy6IX?qid=20070621172550AACMXFC&show=7#profile-info-1c25f0773926f8cde3f71d5867bbbc1eaa

OK, alert answer by the folks who point out the trivial solution n = 1. I am looking for the *other* answer, which Scythian points out is somewhere betwee 100 billion and 1 trillion.

Answer 1

Oh, nice problem, even though nobody else is getting it yet. Let me get back to you on that one. There's more than 1 trillion zeroes from 1 to 1 trillion, and 98,888,888,889 zeroes from 1 to 100 billion, so the answer is a number between 100 billion and 1 trillion.

Addendum: This has been an extremely tedious iteration process, much too long to enter here, but when you get to

1) 100,559,404,365, there are 100,559,404,364 zeroes

2) 100,559,404,366, there are 100,559,404,367 zeroes

So, there does not seem to exist a number where it equals to the number of zeroes in all the numbers up to and including it.

I'll look at it again when I got time, and I hope I'm wrong.

Addendum: Well, if we say all the numbers from 0 to and including n, then the answer would be n = 100,559,404,365.

Answer 2

It's a little misleading to say that Romans didn't have a concept of 0. 0 is pretty intuitive in it's basic form, ie. "none" or "nothing". The idea was back then in the time of the Ancient Greeks and Romans, and very far after until the end of the dark ages and perhaps after, there were no mathematicians in the strictness sense Then, there was no separation between science, mathematics, and natural philosophy . An analogous of this that we can picture is that maybe 50 years ago there was very little speciality in the medical fields. Now, there are very different specialities and an internal medicine doctor is very different than other specialities. So, although everyone was clearly aware of the concept of "nothingness" (eg. if someone has 3 olives and someone takes 3 olives from him, how many olives does he have) they weren't willing to make a number out of it, because the philosophers/mathematicians didn't believe that nothing could be a number. It wasn't until much later when Arabs began to use the number 0 which they adopted from the debate in Greeks about whether there should be a 0. After retaking of the Iberian Peninsula from the Moors all this knowledge, as well as the ancient Greek knowledge in general, was incorporated in Europe. (which in general is a funny story because the Arabs had invented the word and number 0, which Latin did not have. Which probably made translating the books to Latin entertaining)

Answer 3

Find the n such that the sequence of integers from 0 to n contain exactly n zeroes among the digits? There's the trivial solution of n=1. This refers to the number 0 by itself. Other than that, people seem quick to rule out other possibilities, but I don't think that's too obvious. When you get to numbers like 1001, 1002, 1003 etc. the zeroes can add up fast. So there might be. I have to check this out.

Answer 4

n-1

Answer 5

n=1 works

Answer 6

Its n/10 + 1. Because, every 10th number will have a 0 in it, and plus there is the one at the begining.

Ashley

Answer 7

If n = 90,999,099,099

the the number of zeros between 0 and n is

91,786,836,110 which is *more* than n!

so there might be another answer.

I'll keep searching.

Answer 8

Interesting extention of that problem...

Hmm... scythian...
I have 8 and 2 as the first digits...

Answer 9

there's an infinite number of zeroes here :O
omg i can't write the way it's full of ^ signz and powers,, but i study in mathematics department so i knew the answer and how to find it out :->
u made me feel smart thank u