generalized Smarandache palindrome

A generalized Smarandache palindrome (GSP) is a concatenated number of the form: a1a2anana2a1, for n1, or a1a2an-1anan-1a2a1, for n2, where all a1,a2,,an are positive integers of various number of digits in a given base b.

Proposed Problem

Find the number of GSP of four digits that are not palindromic numbersMathworldPlanetmath in base 10.

M. Khoshnevisan, Griffith University, Gold Coast, Queensland 9726, Australia.


Before solving the problem, let see some examples:

1) 1235656312 is a GSP because we can group it as (12)(3)(56)(56)(3)(12), i.e. ABCCBA.

2) The number 5675 is also a GSP because it can be written as (5)(67)(5).

3) Obviously, any palindromic number is a GSP number as well.

A palindromic number of four digits has the concatenated form: abba, where a{1,2,,9} and b{0,1,2,,9}. There are 9×10=90 palindromic numbers of four digits. For example, 1551, or 2002 are palindromic (and, of course, GSP too); yet 3753 is not palindromic but it is a GSP for 3753=3(75)3, i.e. of the form ABA; similarly 4646, for it can be organized as (46)(46), i.e. of the form CC. Therefore, a SGP, different from a palindromic number, should have the concatenated forms: 1) ABA, where A{1,2,,9} and B{00,01,02,03,,99}-{00,11,22,33,,99}; 2) or CC, where C{10,11,12,,99}-{11,22,33,,99}. In the first case, one has 9×(100-10)=9×90=810. In the second case, one has 90-9=81. Total: 810+81=891 GSP numbers of four digits which are not palindromic.


1. Charles Ashbacher, Lori Neirynck, smarandache/GeneralizedPalindromes.htmThe Density of Generalized Smarandache Palindromes, Journal of Recreational Mathematics, Vol. 33 (2), 2006

2. G. Gregory, smarandache/GSP.htmGeneralized Smarandache Palindromes

3. M. Khoshnevisan, ”Generalized Smarandache Palindrome”, Mathematics Magazine, Aurora, Canada, 10/2003.

4. M. Khoshnevisan, Proposed Problem 1062 (on Generalized Smarandache Palindrome), The ΠME Epsilon, USA, Vol. 11, No. 9, p. 501, Fall 2003.

5. Mark Evans, Mike Pinter, Carl Libis, Solutions to Problem 1062 (on Generalized Smarandache Palindrome), The ΠME Epsilon, Vol. 12, No. 1, 54-55, Fall 2004.

6. N. Sloane, of Integers, Sequence A082461

7. F. Smarandache, smarandache/Sequences-book.pdfSequences of Numbers Involved in Unsolved Problems, Hexis, 1990, 2006

Title generalized Smarandache palindrome
Canonical name GeneralizedSmarandachePalindrome
Date of creation 2013-03-22 17:03:25
Last modified on 2013-03-22 17:03:25
Owner dankomed (17058)
Last modified by dankomed (17058)
Numerical id 7
Author dankomed (17058)
Entry type Definition
Classification msc 11Z05
Related topic FlorentinSmarandache