Palindromes: How Many 3-5 Digit Palindromes Exist?

After exploring the wonderful words that are palindromes, we turned to numbers that read the same both forwards and backwards.

aibohphobia alula cammac civic deified deleveled detartrated devoved dewed evitative Hannah 

kayak kinnikinnik lemel level madam Malayalam minim murdrum peeweep racecar radar redder 

refer reifier repaper reviver rotator rotavator rotor sagas solos sexes stats tenet terret testset

It all started when my odometer on my car read 38783. Odometers record the number of miles that my car has traveled. It will be 100 miles until my next palindrome at 38883.

I had each Mathlete complete a set of blanks that would complete the palindromes like:

23_ _      345_ _ 2332  34543

Then we explored three-digit palindromes and attacked a challenge that would determine the total number of 3-digit palindromes. They figured that using an organized system of numbers starting with the smallest, 101, until reaching the largest, 999.

Many of the Mathletes started with 101, 111,121,131,141,151,161,171,181,191,202....

Some realized that there are ten numbers for each group of hundreds. There are 9 hundreds so 10x9=90 3-digit palindromes.

Then they tackled the challenge of 4-digit palindromes 1001, 1111, 1221, 1331, 1441, 1551,1661, 1771, 1881, 1991, 2002.....  Surprisingly, the same pattern as 3-digits so a total of 90 numbers.

Then they were able to explore 5-digit palindromes starting with 10001, 10101, 10201, etc. Some noticed that there are additional 3-digit palindromes imbedded inside the 5-digits that start and end with zero, with a total of 100, but there are 9 separate thousands for each group of 100 paindromes, so a total of 900.

So looking at this incredible sequence of numbers, for the following number of digits:

1-digit palindromes (1,2,3,4,5,6,7,8,9) 9

2-digit "     (11,22,33,44,55,66,77,88,99) 9

3-digit " 90

4-digit " 90

5-digit " 900

6-digit " 900

7-digit " 9000

8-digit " 9000

9-digit " 90000

The attached document will allow you to organize 3-5 digit palindromes.

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