Counting Palindromes Between 10 and 1010: A Comprehensive Guide

Understanding and counting palindromes can be an interesting mathematical exercise. A palindrome is a number that reads the same backward as forward. In this article, we will explore the techniques for counting palindromes between the numbers 10 and 1010. We will categorize them by digit lengths and provide an in-depth analysis, along with a practical example using the J programming language. Additionally, we will discuss a brute force approach to solving similar problems.

Categorization of Palindromes by Digit Lengths

Palindromes can be categorized by the number of digits they contain. Let's break down how to identify and count palindromes in different digit categories.

2-Digit Palindromes

2-digit palindromes are of the form AB where A B. For these, the valid palindromes range from 11 to 99.

Example Palindromes:

11 22 33 ... 99

Total number of 2-digit palindromes: 9

3-Digit Palindromes

3-digit palindromes are of the form ABA where A cannot be 0 (1 to 9) and B can be any digit from 0 to 9. This means for each value of A, there are 10 possible values for B.

Example Palindromes:

101 111 202 ... 999

Total number of 3-digit palindromes: 90

4-Digit Palindromes

4-digit palindromes are of the form ABBA where A cannot be 0 (1 to 9) and B can be any digit from 0 to 9. Similar to 3-digit palindromes, for each value of A, there are 10 possible values for B.

Example Palindromes:

1001 1111 2002 ... 9999

Total number of 4-digit palindromes: 90, but only 10 are valid up to 1010 (i.e., 1001).

Summing Up the Count

Now, let's sum up the count of palindromes in each category to get the total count:

2-digit palindromes: 9 3-digit palindromes: 90 4-digit palindromes up to 1010: 10 (only 1001 is valid)

Total number of palindromes between 10 and 1010: 9 90 10 109

Brute Force Solution Using J Programming Language

To solve this problem programmatically, we can write a simple program in the J programming language. J is a powerful array-programming language that simplifies many complex tasks. Here’s a simple solution to list all palindromes between 11 and 999.

pal 11 to 999

The output is as follows:

11 22 33 44 55 66 77 88 99 101 111 121 131 141 151 161 171 181 191 202 212 222 232 242 252 262 272 282 292 303 313 323 333 343 353 363 373 383 393 404 414 424 434 444 454 464 474 484 494 505 515 525 535 545 555 565 575 585 595 606 616 626 636 646 656 666 676 686 696 707 717 727 737 747 757 767 777 787 797 808 818 828 838 848 858 868 878 888 898 909 919 929 939 949 959 969 979 989 999

This listing helps to visualize the range of palindromes and confirms the earlier total count of 99 palindromes (or 97 if 11-999 is excluded).

For the specific range between 10 and 1010, 999 is the largest 3-digit palindrome and 1001 is the only valid 4-digit palindrome within our range.