# Z=123456789101112131415...

• Jul 27th 2006, 11:05 PM
Bartimaeus
Z=123456789101112131415...
I was given this question last week and have spent some time on it without any results. :(

The number Z=123456789101112131415... is made using an obvious pattern.
What is the 10 000th digit in this number?

Hope you can help. :)
• Jul 28th 2006, 07:36 AM
Soltras
Well there's probably a more rigid way than this, but I think this will work:

The first 9 digits each take up one space (1, 2, ... 8, 9)
The next 90 digits each take up two spaces (10, 11, ... 98, 99)
The next 900 digits each take up three spaces (100, 101, ... 998, 999)
The next 9000 digits each take up four spaces (1000, 1001, ... 9998, 9999)

So we want the 10,000th digit.

Well 10,000 = 9 + 2*90 + 3*900 + (7111).

So the 10,000th digit is somewhere in the section where the thousands are being enumerated.

We have reduced the problem to finding the 7111th digit of 1000100110021003. . .

Well 7111/4 = 1777 with a remainder of 3.

Well, the 1777th "number" (after the 0th being 1000) in the sequence is 2777. The third digit is 7.

(To illustrate, imagine we were looking for just, say, the seventh digit of this new sequence. Using this method, we go 7/4 = 1 with remainder 3. Then it's the third digit of 1001; a zero. If we were looking for the twelfth digit, take 12/4 = 3 remainder 0. Take the 0th digit of 1003, which is whatever digit comes before it - i.e. the 2 from 1002...)

In any case, for your problem, I believe the 10,000th digit of 123456789101112131415... is 7.