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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

The goal of this problem is to find how many four-digit numbers are the same, from left to right and from right to left.  This problem is similar in form to problems that deal with patterns.  The pattern in this case is a series of numbers that are four digits AND the same front to back and vice versa. 

 

Plans to Solve/Investigate the Problem

I must admit that I immediately dove into the problem with pencil and paper.   I have not had prior experience with palindromic numbers so I decided that some research would be wise to make certain that I properly defined any necessary specifics.  

 

The first thing I did was  to gain an understanding of what a palindromic number is and discover any technical rules that may apply to this form of number.  I completed an internet search of a four-digit palindromic number specifically.  Discuss your initial plans/strategies/technologies toward the solution of the problem.

 

Investigation/Exploration of the Problem

The first thing that I discovered with pencil and paper was that since I was asked to define only a four-digit palindrome, the first and fourth digits must be the same.  Secondly, I discovered that the second and third digits must be the same in order for the number to read identically left to right and right to left.  My paper and pencil calculations yielded 100 possible solutions.  Following is a display of my calculations:

 

0000

1001

2002

0110

1111

2112

0220

1221

2222

0330

1331

2332

0440

1441

2442

0550

1551

2552

0660

1661

2662

0770

1771

2772

0880

1881

2882

0990

1991

2992

 

Each end digit, if defined between 0 and 9, patterned in this way would yield 10 x 10 or 100 solutions.  But my web based research resulted in only 90 possible solutions.  Where did the difference occur in the solutions?  I was unsuccessful in tracing my steps back to a particular website that I found extremely helpful, but the site defined four digit palindromes in this way:  "abba where a is between 1 and 9 and b is between 0 and 9."  It totally eliminated 10 of my solutions that had the number zero in the first and fourth positions in the number.  The key was in properly defining the digits that were eligible in a four digit palindrome.

 

Extensions of the Problem

How would the number of possible palindromes be affected if the number of digits of the number increases? decreases?

Author & Contact
Angela Gilliam
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Link(s) to resources, references, lesson plans, and/or other materials
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