Nth number in a infinite sequence of numbers

Question

This was interview question.

When the input is a infinite sequence of numbers starting from 1, what is the nth digit?

e.g.) 123456789101112131415161718192021.....

here 28th digit is 1.

Answer 1

Just to add a concrete implementation to what was already said:

(defun nth-digit (n)
  (loop :with previous := 0
     :for i :upfrom 0
     :sum (* 9 (expt 10 i) (1+ i)) :into digits
     :if (> digits n) :do
     (multiple-value-bind (whole remainder)
         (floor (- n previous) (1+ i))
       (return (aref (write-to-string (+ (expt 10 i) whole)) remainder)))
     :else :do (setf previous digits)))

The idea for implementation is to:

1. Sum the length of all 1-digit numbers (of which there are $9*10^0$), then sum all 2-digit numbers (of which there are $9*10^1$), 3-digit numbers, of which there are $9*10^2$ and so on. This gives:

$$ N = \sum_{i=0}^m 9\times 10^i \times (i+1) $$

2. Notice that $x = \lfloor\frac{n - N}{i+1}\rfloor$ will be a positive integer which counts the number of numbers having $m+1$ digits in them.
3. Finally, after you've already found what number contains your digit, you can find $e$ s.t. $r - e = 10^i + \frac{n - N}{i+1}$, and $e \leq i$ the $e'th$ digit of $x$ is going to be the one you are looking for.

Answer 2

The idea here is as follows:

• The first 9 digits are 123456789.
• The next 90 pairs of digits are (10)(11)...(99).
• The following 900 triplets of digits are (100)(101)...(999).
• And so on.

Given an index $n$, you calculate which group you are (singles, pairs, triplets, ...), then which tuple you belong to and which digit inside the tuple. For example, if $n=28$ then this is digit no. 19 in the second group, which is the first digit of the 10th tuple 19, that is, 1.