A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.
1. **Verify that 7 is a lucky number:**
   We need to show that repeatedly summing the squares of the digits of 7 eventually leads to 1.
   \[
   7 \rightarrow 49 \rightarrow 97 \rightarrow 130 \rightarrow 10 \rightarrow 1
   \]
   - \( 7^2 = 49 \)
   - \( 4^2 + 9^2 = 16 + 81 = 97 \)
   - \( 9^2 + 7^2 = 81 + 49 = 130 \)
   - \( 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \)
   - \( 1^2 + 0^2 = 1 \)

   Since 7 eventually leads to 1, 7 is a lucky number.

2. **Construct a number \( L \) of the form \( 211\underbrace{0 \cdots 0}_{n}1 \):**
   - For any non-negative integer \( n \), consider the number \( L = 211\underbrace{0 \cdots 0}_{n}1 \).
   - We need to show that \( L \) is a lucky number.

3. **Show that \( L \rightarrow 7 \):**
   - The number \( L = 211\underbrace{0 \cdots 0}_{n}1 \) has the digits 2, 1, 1, and \( n \) zeros followed by 1.
   - Sum of the squares of the digits:
     \[
     2^2 + 1^2 + 1^2 + 0^2 + \cdots + 0^2 + 1^2 = 4 + 1 + 1 + 0 + \cdots + 0 + 1 = 7
     \]
   - Therefore, \( L \rightarrow 7 \).

4. **Show that \( L+1 \) is also a lucky number:**
   - The number \( L+1 = 211\underbrace{0 \cdots 0}_{n}2 \).
   - Sum of the squares of the digits:
     \[
     2^2 + 1^2 + 1^2 + 0^2 + \cdots + 0^2 + 2^2 = 4 + 1 + 1 + 0 + \cdots + 0 + 4 = 10
     \]
   - \( 10 \rightarrow 1 \):
     \[
     1^2 + 0^2 = 1
     \]
   - Therefore, \( L+1 \rightarrow 10 \rightarrow 1 \).

5. **Conclusion:**
   - For any non-negative integer \( n \), the pair \( (L, L+1) = (211\underbrace{0 \cdots 0}_{n}1, 211\underbrace{0 \cdots 0}_{n}2) \) are consecutive lucky numbers.
   - Since \( n \) can be any non-negative integer, there are infinitely many such pairs.

The final answer is \( \boxed{ (211\underbrace{0 \cdots 0}_{n}1, 211\underbrace{0 \cdots 0}_{n}2) } \) for any non-negative integer \( n \).