To solve this problem, let's break it down step by step.

Step 1: Understanding the definition of a lucky number

A lucky number is a positive integer whose digits can be divided into two groups such that the sum of the digits in each group is the same. For example, for a number with four digits, we need to find two groups of digits whose sums are equal.

Step 2: Identifying a 4-digit lucky number

We are tasked with finding the smallest 4-digit lucky number. A 4-digit number can be written as: $abcd = 1000a + 100b + 10c + d$ We need to find two groups of digits (from $a, b, c, d$) such that the sum of each group is the same.

Step 3: Searching for the smallest lucky number

Let's start by trying various combinations of digits for $a, b, c, d$ that satisfy this condition. We can experiment with splitting the digits into pairs, triples, or any valid groupings and check if their sums are equal.

Step 4: Identifying the neighboring lucky number

Once we find the smallest lucky number, we also need to identify if its neighboring number is also a lucky number. We must check numbers adjacent to it, either one less or one more, to see if they also satisfy the condition of being a lucky number.

Let me proceed to find the smallest lucky number and its neighbor.("smallest 4-digit lucky number its neighbor")click([0, 1, 2, 6])The smallest 4-digit lucky number whose neighboring number is also a lucky number is 1010.

Explanation:

• A lucky number is defined as a number where the digits can be split into two groups whose sums are equal.
• For 1010, we can split the digits as follows:
  • Group 1: $1 + 0$
  • Group 2: $1 + 0$
  • Both sums are equal to 1, so 1010 is a lucky number.
• The neighboring number is 1009, which is also a lucky number because:
  • Group 1: $1 + 0$
  • Group 2: $0 + 9$
  • Both sums are equal to 1, making 1009 also a lucky number.

Thus, 1010 is the smallest 4-digit lucky number whose neighboring number (1009) is also lucky【7†source】【8†source】.

Would you like more details on how to verify other numbers, or need help with any related concepts? Here are some related questions for you:

1. How do you check if a 4-digit number is a lucky number?
2. What other 4-digit numbers are considered lucky numbers?
3. How can you generalize the definition of a lucky number for larger numbers (e.g., 5-digit)?
4. How would you check if a 5-digit number has neighboring lucky numbers?
5. Can a lucky number have more than two groups of digits?

Tip: When searching for lucky numbers, start by examining numbers that have symmetry in their digits, such as numbers with repeated digits or alternating patterns.