Understanding and Counting Symmetric Integers: An Algorithmic Guide

Exploring numerical patterns is a common theme in programming challenges and practical applications. One interesting pattern involves “symmetric integers.” This guide delves into what symmetric integers are and presents a clear algorithm to count them within a specified range.

What is a Symmetric Integer?

An integer is defined as symmetric if it meets two specific conditions:

It must have an even number of digits (e.g., 2, 4, 6 digits). Numbers with an odd number of digits (like 123 or 9) are never symmetric.
The sum of the first half of its digits must be equal to the sum of the second half of its digits.

Consider these examples:

11: Two digits (even). First half sum (1) = Second half sum (1). Symmetric.
22, 33, …, 99: All symmetric for the same reason.
1221: Four digits (even). First half sum (1 + 2 = 3) = Second half sum (2 + 1 = 3). Symmetric.
123: Three digits (odd). Not symmetric.
1234: Four digits (even). First half sum (1 + 2 = 3) ≠ Second half sum (3 + 4 = 7). Not symmetric.

The Problem: Counting Symmetric Integers in a Range

The core task is straightforward: given two positive integers, low and high, determine how many symmetric integers exist within the inclusive range [low, high].

Example 1:

Input: low = 1, high = 100
Output: 9
Explanation: The symmetric integers are 11, 22, 33, 44, 55, 66, 77, 88, and 99.

Example 2:

Input: low = 1200, high = 1230
Output: 4
Explanation: The symmetric integers are 1203 (1+2=3, 0+3=3), 1212 (1+2=3, 1+2=3), 1221 (1+2=3, 2+1=3), and 1230 (1+2=3, 3+0=3).

Algorithmic Approach to Counting Symmetric Integers

A reliable way to solve this problem is to iterate through each number in the given range and check if it meets the definition of a symmetric integer.

Here’s a step-by-step breakdown of the algorithm:

Initialize a Counter: Start with a count variable set to zero. This will store the number of symmetric integers found.
Iterate Through the Range: Loop through every integer x from low to high, inclusive.
Check Number of Digits: For each number x, determine its total number of digits. The easiest way is often to convert the number to a string and find its length.
Filter by Digit Count: If the number of digits is odd, the number cannot be symmetric. Skip to the next number in the range.
Split the Digits: If the number of digits is even (let’s say 2n), divide the digits into two halves: the first n digits and the last n digits.
Calculate Sums: Calculate the sum of the digits in the first half. Calculate the sum of the digits in the second half.
Compare Sums: Compare the two sums. If they are equal, the number x is symmetric.
Increment Counter: If the number is symmetric, increment the counter initialized in step 1.
Return Final Count: After checking all numbers from low to high, the final value of the counter is the answer.

Illustrative Walkthrough

Let’s trace the process for the number 1221:

Number: x = 1221.
Convert to String: “1221”.
Digit Count: Length is 4 (even). Proceed.
Split: The number of digits is 2n = 4, so n = 2.
- First half digits: ‘1’, ‘2’.
- Second half digits: ‘2’, ‘1’.
Calculate Sums:
- First half sum: 1 + 2 = 3.
- Second half sum: 2 + 1 = 3.
Compare: 3 == 3. The sums are equal.
Conclusion: 1221 is a symmetric integer. Increment the counter.

Key Considerations

The constraints often given for this type of problem (e.g., high up to 10,000) mean that this direct iteration and checking approach is perfectly efficient. The number of operations per integer is proportional to the number of its digits (which is small, at most 4-5 in this range), making the overall time complexity very manageable.

Conclusion

Counting symmetric integers within a range is a problem that tests understanding of basic number properties and algorithmic implementation. The core strategy involves iterating through the numbers, checking for an even digit count, splitting the digits, summing the halves, and comparing the sums. This methodical approach ensures all symmetric numbers are correctly identified and counted.

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