A googol particles

A googol particles - solution

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To solve this I used the fact that

the integers 1-9 add 1 digit each
the integers 10-99 add 2 digits each
the integers 100-999 add 3 digits each
the integers 1000-9999 add 4 digits each

and so on.

Using this one can easily arrive at the formula

n=d × 10^d - (10^d-1)/9

where n is the total number of digits in the sequence after you have added all integers up to and including those that contain d digits.

Using the above I find that setting d to 98 gives an n that is less than a googol, but it becomes more than a googol if I would set d to 99.

For that I used a calculator that can work with many digits. I found a java calculator online which was suitable for this:

http://www.alpertron.com.ar/BIGCALC.HTM

Setting d=98 gives

n=9 788 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 889

The remaining digits up to 1 googol = r = 1 googol - n

r = 211 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111

Those remaining digits will come from the integers that have 99 digits.

So by dividing r with 99 we get the number of such integers that will be used

r / 99 = 2 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435.464 646...

We notice here that the rest is 99 × 0.464 646... = 46

The first number with 99 digits in the sequence is 10⁹⁹ so the last number that partly fits into the sequence is the integer part of the above division plus 10⁹⁹ which is:

102 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435 465 768 799 102 132 435

And the 46th digit of that number is the last digit in the whole sequence.

So the answer is: 4

If you have some idea how to do this without the need for a calculator that can handle many digits, please let me know, send me a feedback via the link above, and then maybe we can do this with 1 googolplex digits instead.