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[ 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,

❄ ❄

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 !

For reference goals and easy searching all the nine- & pandigitals
implicitly displayed in these topics are listed here.

Topic → 3784905216
240545042, 2164905378, 240656042, 2165904378, 315989513, 2843905617, 360212063, 3241908567,
360989063, 3248901567, 361323163, 3251908467, 364989463, 3284905167, 420545024, 3784905216,
420656024, 3785904216, 480212084, 4321908756, 480989084, 4328901756, 485767584, 4371908256,
486989684, 4382907156, 513545315, 4621907835, 531545135, 4783906215, 531878135, 4786903215,
536989635, 4832906715, 624656426, 5621907834, 630212036, 5671908324, 630989036, 5678901324,
642656246, 5783906214, 642989246, 5786903214, 713545317, 6421907853, 724656427, 6521907843,
753878357, 6784905213, 753989357, 6785904213, 840212048, 7561908432, 840989048, 7568901432,
915989519, 8243905671, 936989639, 8432906751, 951434159, 8562907431, 951989159, 8567902431,
963878369, 8674905321, 963989369, 8675904321
630989036, 5678901324, 5678901234

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