Palindromic numbers

What is a palindrome?

Look at 66, 848, 575, 797 and 1111. Read each one from left to right and then from right to left: you get the same number. Such numbers are called palindromes or palindromic numbers. The word is also used for words that read the same both ways, like ‘madam’ and ‘level’.

A quick test: write the number backwards. If nothing changes, it is a palindrome. 4554 backwards is 4554, so it is one; 4545 backwards is 5454, so it is not.

All 3-digit palindromes using 1, 2 and 3

A 3-digit palindrome has the same first and last digit. Using only 1, 2 and 3, the first digit (which also fixes the last) can be chosen in 3 ways, and the middle digit in 3 ways, so there are 3 × 3 = 9 of them:

First and last digit	Palindromes
1	111, 121, 131
2	212, 222, 232
3	313, 323, 333

Reverse and add

Start with a 2-digit number. Add it to its reverse. If the answer is a palindrome, stop; if not, reverse the answer and add again.

• 25 + 52 = 77, a palindrome in one step.
• 19 + 91 = 110; the reverse of 110 is 011, that is 11, and 110 + 11 = 121.
• 68 + 86 = 154, 154 + 451 = 605, 605 + 506 = 1111.

Does every 2-digit number end in a palindrome? Yes. Some take a long time: 79 needs 6 steps (ending at 44,044), and 89 needs 24 steps, ending at 8,813,200,023,188. For 3-digit numbers, the answer is unknown: starting with 196 is suspected never to give a palindrome, but nobody has proved it.

The palindrome puzzle

“I am a 5-digit palindrome. I am odd. My tens digit is double my units digit, and my hundreds digit is double my tens digit.”

Call the units digit u. The tens digit is 2u and the hundreds digit is 4u, and 4u must be a single digit, so u is 1 or 2. The number is odd, so u = 1, giving tens 2 and hundreds 4. In a palindrome, the ten-thousands digit equals the units digit (1) and the thousands digit equals the tens digit (2). The number is 12,421, twelve thousand four hundred twenty-one.

Smallest and largest 5-digit palindromes

The smallest 5-digit palindrome is 10,001 and the largest is 99,999.

• Sum: 10,001 + 99,999 = 1,10,000
• Difference: 99,999 − 10,001 = 89,998

Palindromes on the clock

On a 12-hour clock written like 1:21, a time is a palindrome when it reads the same backwards. For each hour from 1 to 9 there are six: for example 1:01, 1:11, 1:21, 1:31, 1:41 and 1:51. With two-digit hours there are only three: 10:01, 11:11 and 12:21. That makes 57 palindromic times in all.

If the time is 10:01, the next palindromic time is 11:11, 70 minutes later, and the one after that is 12:21, 140 minutes after 10:01.

Palindromic dates

Meghana’s birthday, 11/02/2011, reads the same both ways when written as 11022011. Since the year 2000, written as dd/mm/yyyy, there have been only eight such dates: 10/02/2001, 20/02/2002, 01/02/2010, 11/02/2011, 21/02/2012, 02/02/2020, 12/02/2021 and 22/02/2022. All fall in February, because a year starting with 20 needs the month 02. The next one is 03/02/2030.

Palindromes are just one pattern hiding in numbers. For another, read Kaprekar’s routine and 6174. For the textbook’s questions, see the Chapter 3 NCERT solutions; for the whole chapter, see the Chapter 3 notes.