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

❄ ❄

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 !