Palindrome numbers
A palindrome number reads the same forwards and backwards: 7, 11, 121, 1,331 and 12,321 are all palindromes. From 0 to 999,999 there are exactly 1,999 of them, about 1 in 500.
What is a palindrome number?
A number is a palindrome when its digits read the same in both directions. 4,554 is one: reverse the digits and you get 4,554 again. 4,545 is not, because reversed it reads 5,454. Every one-digit number counts, including 0. Leading zeros are not allowed, so 10 is not a palindrome (it would have to be written 010 to read the same).
Ten examples: 11, 22, 101, 121, 353, 1,001, 1,331, 12,321, 45,654 and 999,999.
How many palindrome numbers are there?
Exactly 1,999 from 0 to 999,999, counted by checking every number. There is also a quick way to count them, because a palindrome is fixed by its first half: choose the first ⌈L/2⌉ digits (the first can’t be 0) and the rest is a mirror image. That gives 9 × 10⌈L/2⌉−1 palindromes with L digits, so each odd length and the even length after it have the same count: 90 with three digits and 90 with four, 900 with five and 900 with six.
| Digits | How many | Smallest | Largest |
|---|---|---|---|
| 1 | 10 | 0 | 9 |
| 2 | 9 | 11 | 99 |
| 3 | 90 | 101 | 999 |
| 4 | 90 | 1,001 | 9,999 |
| 5 | 900 | 10,001 | 99,999 |
| 6 | 900 | 100,001 | 999,999 |
| Total | 1,999 | 0 | 999,999 |
Between palindromes the gaps can be large. The longest stretch below one million with no palindrome is 1,099 numbers, from 100,002 to 101,100; the palindromes on either side are 100,001 and 101,101.
Special palindromes
Palindromic primes. 113 palindromes below one million are prime, starting 2, 3, 5, 7, 11, 101, 131, 151, 181, 191. The largest is 98,689. There are none with 4 or 6 digits, and only one with 2 (11): every palindrome with an even number of digits is divisible by 11.
Palindromic squares. 15 squares below one million are palindromes: 0, 1, 4, 9, 121, 484, 676, 10,201, 12,321, 14,641, 40,804, 44,944, 69,696, 94,249, 698,896. Several come from squaring a palindrome, like 11² = 121 and 111² = 12,321.
Palindromic cubes. Only 5: 0, 1, 8, 343, 1,331. 1,331 is 11³.
Palindromes in Rarebit
In Rarebit, the daily RNG game, a roll of two or more digits that reads the same backwards earns the Mirror Image badge: 1,989 of the 1,000,000 numbers, odds of 1 in 503, worth 90 points. If your roll lands within 99 of a palindrome, Rarebit can show it as a near miss, so you see how close you came. Palindromes with extra patterns score far higher, such as 7,777, 22,222, 99,999 and 999,999.
Palindrome number questions
What is a palindrome number?
A palindrome number reads the same forwards and backwards, like 121 or 12,321. Every one-digit number, including 0, counts as a palindrome.
Is 17 a palindrome?
No. 17 reversed is 71. The nearest palindromes are 11 below it and 22 above it.
What are 10 palindrome numbers?
11, 22, 101, 121, 353, 1,001, 1,331, 12,321, 45,654 and 999,999.
What is the greatest palindrome number?
There isn’t one: you can always write a longer palindrome. The largest below one million is 999,999.
How many palindromes are there from 0 to 999,999?
1,999: 10 with one digit, 9 with two, 90 with three, 90 with four, 900 with five and 900 with six.