Palindromic number facts for kids

A palindromic number is a special kind of number that looks exactly the same when you read it forwards or backwards. An example of this is 16361. If you reverse the order of the digits, it remains 16361. This is similar to a word palindrome, like "racecar" or "level," which is spelled the same way in both directions.

In mathematics, these numbers are fun to study because they have a type of symmetry. The first few palindromic numbers in our standard counting system are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121...

How Palindromic Numbers Work

Most of the time, we look at palindromic numbers using the decimal system (Base 10), which is the standard way we count with digits 0 through 9. However, a number can be a palindrome in other systems too.

A number is palindromic if its digits match when flipped. For example:

• One digit: All numbers from 0 to 9 are palindromes because they only have one digit.
• Two digits: Numbers like 11, 22, and 99 are palindromes. In the decimal system, all two-digit palindromes are multiples of 11.
• Three digits: A number like 121 or 505 is a palindrome. The first and last digits must be the same.

Counting Palindromes

There are patterns to how many palindromic numbers exist:

• There are 10 one-digit palindromes.
• There are 9 two-digit palindromes.
• There are 90 three-digit palindromes (from 101 to 999).
• There are 90 four-digit palindromes (from 1001 to 9999).

As the numbers get bigger, there are more palindromes, but they become harder to find. For example, below the number 100,000, there are only 1,099 palindromic numbers.

Fun Math Properties

Palindromic numbers often appear in puzzles and recreational mathematics. They can have other special properties at the same time.

Palindromic Primes

A prime number is a number that can only be divided by 1 and itself. Some prime numbers are also palindromes.

• Examples: 2, 3, 5, 7, 11, 101, 131, 151.
• The number 11 is the only palindromic prime with an even number of digits.

Palindromic Squares

A square number is the result of multiplying a number by itself (like 2 × 2 = 4). Some squares are palindromes.

• Examples: 0, 1, 4, 9, 121 (which is 11 × 11), 484 (which is 22 × 22).
• There is a cool pattern with the number 1:
  • 1² = 1
  • 11² = 121
  • 111² = 12321
  • 1111² = 1234321

This pattern continues for a while, creating a "pyramid" of numbers.

Perfect Powers

Besides squares, other powers can be palindromes too.

• Cubes: A number multiplied by itself three times. Examples include 8 (2×2×2) and 343 (7×7×7).
• Fourth Powers: Examples include 14641 (11⁴).

Palindromes in Other Bases

We usually count in Base 10, but computers use Base 2 (binary). Numbers can be palindromes in binary even if they are not in Base 10.

• The number 3 in decimal is written as 11 in binary. This is a palindrome.
• The number 9 in decimal is written as 1001 in binary. This is also a palindrome.

Some numbers are not palindromes in any base (from Base 2 up to slightly less than the number itself). These are called strictly non-palindromic numbers. The first few are:

0, 1, 2, 3, 4, 6, 11, 19, 47, 53, 79...

All strictly non-palindromic numbers larger than 6 are prime numbers.

The Reverse-and-Add Game

You can try to create a palindrome from any number using a simple process called the Lychrel process.

1. Take a number (for example, 56).
2. Reverse its digits (65).
3. Add the two numbers together (56 + 65 = 121).
4. 121 is a palindrome!

Sometimes you have to repeat this step many times.

• Start with 87.
• 87 + 78 = 165.
• 165 + 561 = 726.
• 726 + 627 = 1353.
• 1353 + 3531 = 4884. (It took 4 steps).

Lychrel Numbers

Some numbers, like 196, never seem to become palindromes no matter how many times you reverse and add them. These are called Lychrel numbers. Mathematicians have tested 196 millions of times using computers, but it has never turned into a palindrome.

In 2017, a very large number (with 19 digits) was found that took a huge number of steps to finally become a palindrome. It is known as a "delayed palindrome."

Scheherazade Numbers

There is a group of numbers called Scheherazade numbers. They are named after the storyteller from One Thousand and One Nights.

• These numbers are related to the number 1001.
• 1001 is interesting because 7 × 11 × 13 = 1001.
• When you multiply 1001 by itself, you get palindromes that have patterns in groups of digits.
• For example: .

Sums of Palindromes

In 2018, mathematicians proved an interesting fact: every positive integer can be written as the sum of three palindromic numbers. This works for every number system with a base of 5 or higher.pl:Palindrom#Palindromy liczbowe