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# Title

Finding Palindromes:

Problem Statement

A palindrome is a number that reads the same from right to left as it does from left to right. For example, 363, 77, and 24642 are palindromes. How many four-digit palindromes exist?

Problem setup

This problem involves the use of permutations: how many different ways can you arrange members of a set.  Here the set referred to is the set of digits one through nine and zero.

Plans to Solve/Investigate the Problem

As we determine the total number of four-digit palindromes possible, we actually only need consider the first two digits as digits three and four create the palindrome.  The first digit of the set of solutions to the problem can be one of nine different digits (for you cannot begin a four digit with zero).  The solution set for the second digit can be from one of ten digits (for you can have zero as the second digit).

Investigation/Exploration of the Problem

Beginning with one as the first digit, there are ten possible palindromes.

1001

1111

1221

1331

1441

1551

1661

1771

1881

1991

(1111 works as eleven is still eleven reversed).  There are no other possible arrangements of the first two digits while limiting the first digit to one.  Beginning with two through nine, there are also ten palindromes per first digit.  Nine different first digits * ten different palindromes per first digit = ninety different palindromes.

The key to understanding the solution to this problem lies in understanding you actually need only consider the first two digits.  For any two-digit number (not beginning with zero) one can write those two digits in reverse order to create a four-digit palindrome.

Another way to look at the solution:

The first two digits can be all numbers between and including 10 and 99.  That is a total of ninety numbers.  As each of those two digits can represent a different four-digit palindrome, you therefore have ninety palindromes.

# Extensions of the Problem

A good extension to the problem, would be to have students find all the possible license plates in a state that have a sequence of three letters followed by three numbers.  How many license plates could start with MOM or POP?

Author & Contact

Pat Devane

Memorial Middle School

Conyers, GA
pjdevane@rockdale.k12.ga.us