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

" I happened to decipher my 1997 diary, wherein some strange bond
between certain palindromic numbers and pandigitals were recorded,

❄ ❄

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).

❄ ❄

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.

Palindromes to nine- & pandigitals
SI    Palindrome
P8
Ninedigital
P8 * 9
Palindrome
P9
Pandigital
P9 * 9
1240550422164953782405_4_504221649_0_5378
2240660422165943782406_5_604221659_0_4378
3315995132843956173159_8_951328439_0_5617
4360220633241985673602_1_206332419_0_8567
5360990633248915673609_8_906332489_0_1567
6361331633251984673613_2_316332519_0_8467
7364994633284951673649_8_946332849_0_5167
8420550243784952164205_4_502437849_0_5216
9420660243785942164206_5_602437859_0_4216
10480220844321987564802_1_208443219_0_8756
11480990844328917564809_8_908443289_0_1756
12485775844371982564857_6_758443719_0_8256
13486996844382971564869_8_968443829_0_7156
14513553154621978355135_4_531546219_0_7835
15531551354783962155315_4_513547839_0_6215
16531881354786932155318_7_813547869_0_3215
17536996354832967155369_8_963548329_0_6715
18624664265621978346246_5_642656219_0_7834
19630220365671983246302_1_203656719_0_8324
20630990365678913246309_8_903656789_0_1324
21642662465783962146426_5_624657839_0_6214
22642992465786932146429_8_924657869_0_3214
23713553176421978537135_4_531764219_0_7853
24724664276521978437246_5_642765219_0_7843
25753883576784952137538_7_835767849_0_5213
26753993576785942137539_8_935767859_0_4213
27840220487561984328402_1_204875619_0_8432
28840990487568914328409_8_904875689_0_1432
29915995198243956719159_8_951982439_0_5671
30936996398432967519369_8_963984329_0_6751
31951441598562974319514_3_415985629_0_7431
32951991598567924319519_8_915985679_0_2431
33963883698674953219638_7_836986749_0_5321
34963993698675943219639_8_936986759_0_4321

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

SI    Palindrome
P8
Ninedigital
P8 * 9
Palindrome
P9
Pandigital
P9 * 9
20630990365678913246309_8_903656789_0_1324

Now change the last but one digit in the palindromes from 3 into a 2
and observe that both equations still produce valid nine- & pandigitals
where only the digits 2 and 3 are swapped. Ain't that glittering !

SI    Palindrome
P8
Ninedigital
P8 * 9
Palindrome
P9
Pandigital
P9 * 9
20630990265678912346309_8_902656789_01234

The digit groups from 0 (1) to 4 and from 5 to 9 are swapped
but now the digits in each group are always in ascending order !

A000173 Prime Curios! Prime Puzzle
Wikipedia 173 Le nombre 173
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