[ 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."
❄ ❄
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 |
1 | 24055042 | 216495378 | 2405_4_5042 | 21649_0_5378 |
2 | 24066042 | 216594378 | 2406_5_6042 | 21659_0_4378 |
3 | 31599513 | 284395617 | 3159_8_9513 | 28439_0_5617 |
4 | 36022063 | 324198567 | 3602_1_2063 | 32419_0_8567 |
5 | 36099063 | 324891567 | 3609_8_9063 | 32489_0_1567 |
6 | 36133163 | 325198467 | 3613_2_3163 | 32519_0_8467 |
7 | 36499463 | 328495167 | 3649_8_9463 | 32849_0_5167 |
8 | 42055024 | 378495216 | 4205_4_5024 | 37849_0_5216 |
9 | 42066024 | 378594216 | 4206_5_6024 | 37859_0_4216 |
10 | 48022084 | 432198756 | 4802_1_2084 | 43219_0_8756 |
11 | 48099084 | 432891756 | 4809_8_9084 | 43289_0_1756 |
12 | 48577584 | 437198256 | 4857_6_7584 | 43719_0_8256 |
13 | 48699684 | 438297156 | 4869_8_9684 | 43829_0_7156 |
14 | 51355315 | 462197835 | 5135_4_5315 | 46219_0_7835 |
15 | 53155135 | 478396215 | 5315_4_5135 | 47839_0_6215 |
16 | 53188135 | 478693215 | 5318_7_8135 | 47869_0_3215 |
17 | 53699635 | 483296715 | 5369_8_9635 | 48329_0_6715 |
18 | 62466426 | 562197834 | 6246_5_6426 | 56219_0_7834 |
19 | 63022036 | 567198324 | 6302_1_2036 | 56719_0_8324 |
20 | 63099036 | 567891324 | 6309_8_9036 | 56789_0_1324 |
21 | 64266246 | 578396214 | 6426_5_6246 | 57839_0_6214 |
22 | 64299246 | 578693214 | 6429_8_9246 | 57869_0_3214 |
23 | 71355317 | 642197853 | 7135_4_5317 | 64219_0_7853 |
24 | 72466427 | 652197843 | 7246_5_6427 | 65219_0_7843 |
25 | 75388357 | 678495213 | 7538_7_8357 | 67849_0_5213 |
26 | 75399357 | 678594213 | 7539_8_9357 | 67859_0_4213 |
27 | 84022048 | 756198432 | 8402_1_2048 | 75619_0_8432 |
28 | 84099048 | 756891432 | 8409_8_9048 | 75689_0_1432 |
29 | 91599519 | 824395671 | 9159_8_9519 | 82439_0_5671 |
30 | 93699639 | 843296751 | 9369_8_9639 | 84329_0_6751 |
31 | 95144159 | 856297431 | 9514_3_4159 | 85629_0_7431 |
32 | 95199159 | 856792431 | 9519_8_9159 | 85679_0_2431 |
33 | 96388369 | 867495321 | 9638_7_8369 | 86749_0_5321 |
34 | 96399369 | 867594321 | 9639_8_9369 | 86759_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 |
20 | 63099036 | 567891324 | 6309_8_9036 | 56789_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 |
20 | 63099026 | 567891234 | 6309_8_9026 | 56789_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 !
