The Sum of All Four-Digit Palindromes: A Mathematical Odyssey

Introduction

Four-digit palindromes are a fascinating subset of numbers that read the same forwards and backwards. These numbers, such as 1221 or 4554, have captivated mathematicians and enthusiasts alike for their unique properties. In this article, we will explore the sum of all four-digit palindromes, drawing upon principles similar to those used by Carl Friedrich Gauss to compute the sum of the first 100 numbers. By leveraging elegant mathematical techniques, we will uncover the total sum and shed light on the underlying patterns.

Understanding Four-Digit Palindromes

A four-digit palindrome is a number that can be represented in the form ABCA, where A, B, and C are digits and A is not zero. For instance, 1221, 3443, and 9889 are four-digit palindromes. The first digit (A) ranges from 1 to 9 (since palindromes cannot start with 0), while the middle digits (B and C) can range from 0 to 9.

Counting Four-Digit Palindromes

Let's calculate the total number of four-digit palindromes. Given that A has 9 possible values (1 through 9), and B and C each have 10 possible values (0 through 9), the total number of four-digit palindromes is:

9 (for A) * 10 (for B) * 10 (for C) 900

The Sum of All Four-Digit Palindromes

Now, we will find the sum of all four-digit palindromes using a method similar to Gauss's technique for summing consecutive numbers. The key insight is to pair the palindromes in a way that simplifies the summation process.

Paired Calculation

Consider that if n is a four-digit palindrome, then 11000 - n is also a four-digit palindrome. This pairing allows us to use the following formula:

Sum (n (11000 - n)) * (Total Palindromes / 2)

Since each pair (n, 11000 - n) sums to 11000, and there are 900 palindromes, the total sum is:

11000 * (900 / 2) 11000 * 450 4950000

Therefore, the sum of all four-digit palindromes is 4950000.

Further Explorations

The method used to find the sum of four-digit palindromes can be extended to other number types and lengths. This technique, known in mathematics for its elegance and efficiency, is a testament to the power of paired mathematical thinking.

Additional Examples

For example, if we consider three-digit palindromes (ABA), the total number is 90 (9x11). The sum would be:

1100 * (90 / 2) 1100 * 45 49500

The same principle applies to five-digit palindromes (AXYXYA), which have 9000 possible combinations, and their sum would be:

110000 * (9000 / 2) 110000 * 4500 495000000

Conclusion

The sum of all four-digit palindromes is a beautiful example of the power of mathematical pairing and symmetry. By leveraging the elegance of Gauss's method, we have uncovered that the sum is 4950000. This discovery not only adds to our understanding of number theory but also highlights the simple yet profound ways in which mathematical techniques can solve complex problems.

For further reading on similar mathematical concepts, palindromes, and number theory, we recommend exploring resources on discrete mathematics and number pattern analysis.