[ March 20, 2008 ]
Blending palindromes and nine- & pandigitals using multiplication by 9
B.S. Rangaswamy (email)

" I happened to decipher my 1997 diary, wherein some strange bond
between certain palindromic numbers and pandigitals were recorded,
which I would like to share with your readers."

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There exist nine thousand 8-digit palindromic numbers. Only a very
few of these, when multiplied by 9, result in ninedigitals, as in :

36022063 * 9 = 324198567

and I was able to arrive at 34 such combinations of 8-digit palindromes
and ninedigitals and the list is given in the table below (left part).

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There are ninety thousand 9-digit palindrome numbers. Only a few
hundreds of these, when multiplied by 9, yield pandigitals, as in :

725434527 * 9 = 6528910743

and I was able to extricate 559 such combinations of 9-digit palindromes
and pandigitals and the complete list is displayed here.

❄ ❄

An interesting phenomenon which I notice is that all the 34 equations
can be transformed into 9-digit palindromes and corresponding pandigitals
by the intervention of an appropriate numeral in the centre of the palindrome,
which automatically interposes 0 in the mid portion of the ninedigital
to transform it into a pandigital ! This is illustrated below :

42055024 * 9 = 378495216 | 420545024 * 9 = 3784905216

In the above equation, the intervention of 4 between two 5's in the palindrome
transforms the product from ninedigital into pandigital. Complete list of such
strange equations is furnished hereunder.

With a little change in equation 20 we can produce the following
unexpected gem that will be of immense interest to 'digit' lovers !