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When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

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[ March 1, 2015 ]
Nine- & pandigitals equal to the sum of two squares Some statistics and curios

This topic is a continuation from wonplate193

Statistics and curios for the ninedigital variation

In total there are 65795 ninedigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 18% of the 9! possible ninedigitals. Here is the distribution list :

 (*_1) 28763 (*_2) 25968 (*_3) 1464 (*_4) 7513 (*_5) 55 (*_6) 848 (*_7) 0 (*_8) 901 (*_9) 27 (*_10) 34 (*_11) 0 (*_12) 161 (*_13) 0 (*_14) 0 (*_15) 2           469738125 & 879463125 (*_16) 44 (*_17) 0 (*_18) 4           346978125, 647193825, 783961425 & 968417325 (*_19) 0 (*_20) 2           763498125 & 796843125 (*_21) 0 (*_22) 0 (*_23) 0 (*_24) 8 (*_25) 0 (*_26) 0 (*_27) 0 (*_28) 0 (*_29) 0 (*_30) 0 (*_31) 0 (*_32) 1           439817625

In total there are exactly 100 ninedigitals expressible as a sum of two squares whereby
the concatenation of its basenumbers forms a ninedigital (77 in total) or a pandigital (23 in total).

As a coincidence there are also 23 ninedigitals whereby the basenumbers A and B are the same.
They show up on the right side of the table.

One ninedigital stands out from the rest namely 317928645 because it is the
only one that has more than one solution. Beware this is a unique case!
This couple are the first two from an eightfold (*_8) solution for this ninedigital.
Note: this curio was first observed by Peter Kogel already in 2005.

 317928645    = 25382 + 176492 29432 + 175862

Here is the complete list of all the hundred ninedigital solutions.
What is under construction is the list regarding the pandigitals. B.S. Rangaswamy sent me already one example
to wet your appetite. See at the bottom of the table.

 1 175236849  =  34952 + 127682 612859437  =  137492 + 205892 143752968 = 2 * 84782 2 179684325  =  36542 + 128972 618542937  =  136592 + 207842 145897362 = 2 * 85412 3 189237465  =  45362 + 129872 689537412  =  109742 + 238562 162973458 = 2 * 90272 4 197485632  =  53762 + 129842 756928314  =  149672 + 230852 164275938 = 2 * 90632 5 218367945  =  54692 + 137282 783691245  =  185732 + 209462 178945362 = 2 * 94592 6 231649785  =  65282 + 137492 786425193  =  159482 + 230672 183974562 = 2 * 95912 7 234169785  =  76592 + 132482 814593672  =  158942 + 237062 219367458 = 2 * 104732 8 234718965  =  87932 + 125462 817439625  =  137492 + 250682 346581792 = 2 * 131642 9 234971685  =  56792 + 142382 824691537  =  103592 + 267842 423579618 = 2 * 145532 10 237916845  =  89732 + 125462 841769325  =  170582 + 234692 453968712 = 2 * 150662 11 238176549  =  69452 + 137822 867154293  =  169532 + 240782 461593728 = 2 * 151922 12 238971465  =  76592 + 134282 874932516  =  174962 + 238502 497638152 = 2 * 157742 13 243956817  =  27842 + 153692 891746325  =  137852 + 264902 537198642 = 2 * 163892 14 249813657  =  82592 + 134762 912645873  =  164972 + 253082 571963842 = 2 * 169112 15 251649873  =  53672 + 149282 916782345  =  185072 + 239642 618534792 = 2 * 175862 16 256847193  =  49682 + 152372 921847653  =  160982 + 257432 637459218 = 2 * 178532 17 268371954  =  69752 + 148232 925781634  =  143972 + 268052 639174258 = 2 * 178772 18 283974165  =  43592 + 162782 927814653  =  157982 + 260432 654279138 = 2 * 180872 19 286753194  =  82952 + 147632 934687125  =  143702 + 269852 654713298 = 2 * 180932 20 312897645  =  89342 + 152672 943725186  =  187052 + 243692 765421938 = 2 * 195632 21 317928645  =  25382 + 176492 =  29432 + 175862 948571236  =  136802 + 275942 913524768 = 2 * 213722 22 326785149  =  49652 + 173822 972354861  =  174692 + 258302 943256178 = 2 * 217172 23 328459617  =  54362 + 172892 973416285  =  174062 + 258932 958431762 = 2 * 218912 24 341978265  =  64592 + 173282 25 345172689  =  87452 + 163922 26 346297185  =  79532 + 168242 27 362794185  =  54962 + 182372 28 365984721  =  24362 + 189752 29 368529417  =  53762 + 184292 30 369471825  =  59642 + 182732 31 374921865  =  83522 + 174692 32 378129645  =  83492 + 175622 33 385267914  =  23672 + 194852 34 413629578  =  64532 + 192872 35 435869712  =  78242 + 193562 36 436521978  =  96272 + 185432 37 478192653  =  46982 + 213572 38 481379265  =  48572 + 213962 39 489731265  =  56972 + 213842 40 497163825  =  48962 + 217532 41 497231865  =  59762 + 214832 42 514926873  =  57632 + 219482 43 523687194  =  63452 + 219872 44 523861794  =  78632 + 214952 45 532978641  =  87962 + 213452 46 592876413  =  61982 + 235472 47 598314762  =  64712 + 235892 48 613259874  =  18752 + 246932 49 613478925  =  74582 + 236192 50 614829357  =  78692 + 235142 51 619423785  =  15962 + 248372 52 631548297  =  87962 + 235412 53 635127849  =  91682 + 234752 54 639218457  =  73562 + 241892 55 648912537  =  38642 + 251792 56 651429378  =  68132 + 245972 57 674921853  =  41972 + 256382 58 682143597  =  61892 + 253742 59 694725138  =  96872 + 245132 60 712839546  =  97352 + 248612 61 726389145  =  89762 + 254132 62 726439185  =  71642 + 259832 63 729563481  =  39842 + 267152 64 758394621  =  43952 + 271862 65 789631245  =  97412 + 263582 66 812596473  =  56132 + 279482 67 817453962  =  65492 + 278312 68 823415697  =  35162 + 284792 69 825697314  =  39152 + 284672 70 839147625  =  91562 + 274832 71 846352197  =  73592 + 281462 72 865972341  =  38462 + 291752 73 872156394  =  16352 + 294872 74 914367285  =  64712 + 295382 75 931427586  =  68312 + 297452 76 934816725  =  74312 + 296582 77 951246738  =  85172 + 296432

Another ninedigital that stands out from the rest is 934167285 because it is the only
one that can be written as a sum of two squares in two different ways such that both
their basenumbers forms a ninedigital when multiplied together.
A nice unique case!

 934167285    = 95132 + 290462 and   9513 * 29046 = 276314598 206012 + 225782 and 20601 * 22578 = 465129378

In total there are 219 ninedigitals expressible in this way.
The smallest is 248635917 = 106142 + 116612 and 10614 * 11661 = 123769854
The largest is   987431562 = 215192 + 228992 and 21519 * 22899 = 492763581

Statistics and curios for the pandigital variation

In total there are 568801 pandigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 17,416 % of the 9*9! possible pandigitals. Here is the distribution list :

 (*_1)  (23112)1 + (23193)2 + (23072)3 + (22132)4 + (24904)5 + (21085)6 + (22492)7 + (21296)8 + (20600)9 ( 201886 )tot (*_2)  (26329)1 + (26858)2 + (26881)3 + (25208)4 + (25865)5 + (24452)6 + (26628)7 + (24705)8 + (24742)9 ( 231668 )tot (*_3)  (1840)1 + (919)2 + (1592)3 + (1372)4 + (511)5 + (1434)6 + (1526)7 + (1324)8 + (1471)9 ( 11989 )tot (*_4)  (10254)1 + (10559)2 + (11129)3 + (10170)4 + (8929)5 + (9983)6 + (11204)7 + (10259)8 + (10737)9 ( 93224 )tot (*_5)  (60)1 + (0)2 + (110)3 + (116)4 + (5)5 + (57)6 + (105)7 + (96)8 + (107)9 ( 656 )tot (*_6)  (1408)1 + (740)2 + (1164)3 + (1127)4 + (377)5 + (1185)6 + (1257)7 + (1081)8 + (1156)9 ( 9495 )tot (*_7)  (5)1 + (0)2 + (7)3 + (7)4 + (0)5 + (1)6 + (5)7 + (5)8 + (4)9 ( 34 )tot (*_8)  (1601)1 + (1606)2 + (1864)3 + (1748)4 + (1197)5 + (1739)6 + (2088)7 + (1915)8 + (1992)9 ( 15750 )tot (*_9)  (32)1 + (19)2 + (25)3 + (17)4 + (5)5 + (23)6 + (30)7 + (29)8 + (25)9 ( 205 )tot (*_10)  (30)1 + (2)2 + (42)3 + (90)4 + (0)5 + (40)6 + (82)7 + (53)8 + (72)9 ( 411 )tot (*_11)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_12)  (251)1 + (133)2 + (281)3 + (266)4 + (73)5 + (270)6 + (297)7 + (290)8 + (294)9 ( 2155 )tot (*_13)  (1)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 2 )tot (*_14)  (0)1 + (0)2 + (1)3 + (4)4 + (0)5 + (4)6 + (3)7 + (1)8 + (7)9 ( 20 )tot (*_15)  (2)1 + (0)2 + (4)3 + (6)4 + (0)5 + (0)6 + (5)7 + (1)8 + (2)9 ( 20 )tot (*_16)  (88)1 + (71)2 + (111)3 + (114)4 + (49)5 + (128)6 + (142)7 + (156)8 + (153)9 ( 1012 )tot (*_17)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_18)  (9)1 + (3)2 + (11)3 + (14)4 + (2)5 + (7)6 + (9)7 + (14)8 + (10)9 ( 79 )tot (*_19)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_20)  (1)1 + (0)2 + (4)3 + (6)4 + (0)5 + (7)6 + (17)7 + (3)8 + (8)9 ( 46 )tot (*_21)  (1)1 + (0)2 + (1)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 3 )tot (*_22)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_23)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_24)  (11)1 + (9)2 + (14)3 + (12)4 + (3)5 + (12)6 + (23)7 + (8)8 + (22)9 ( 114 )tot (*_25)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_26)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_27)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_28)  (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (1)8 + (1)9 ( 3 )tot (*_29)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_30)  (0)1 + (0)2 + (2)3 + (2)4 + (0)5 + (0)6 + (1)7 + (1)8 + (0)9 ( 6 )tot (*_31)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_32)  (2)1 + (0)2 + (3)3 + (1)4 + (0)5 + (3)6 + (2)7 + (2)8 + (3)9 ( 16 )tot (*_33)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_34)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_35)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_36)  (0)1 + (0)2 + (1)3 + (0)4 + (0)5 + (1)6 + (1)7 + (0)8 + (2)9 ( 5 )tot       3049186725, 6137904825, 7261938450, 9132768450 & 9702436185 (*_37)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_38)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_39)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_40)  (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 1 )tot       4398176250 (*_41)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_42)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_43)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_44)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_45)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_46)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_47)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot (*_48)  (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 1 )tot       7316984025 (*_subt)  (65037)1 + (64112)2 + (66319)3 + (62415)4 + (61920)5 + (60431)6 + (65919)7 + (61240)8 + (61408)9 ( 568801 )total

 1 1023458976  =  194762 + 253802 1 1024397685  =  57692 + 314822 2 1024537896  =  193862 + 254702 2 1063789245  =  78422 + 316592 3 1024736985  =  165392 + 274082 3 1073982645  =  89462 + 315272 4 1032768549  =  146702 + 285932 4 1074893265  =  15692 + 327482 5 1038249657  =  185042 + 263792 5 1236785940  =  51962 + 347822 6 1038259476  =  198602 + 253742 6 1237694085  =  56822 + 347192 7 1058267349  =  184502 + 267932 7 1258047369  =  67952 + 348122 8 1063275498  =  159032 + 284672 8 1270365489  =  81752 + 346922 9 1073982645  =  198542 + 260732 9 1294037685  =  92582 + 347612 10 1075348296  =  145862 + 293702 10 1324958760  =  71942 + 356822 11 °  =  ° + ° 11 °  =  ° + ° 12 °  =  ° + ° 12 °  =  ° + °

One lovely pandigital already popped up on my screen 1073982645
It combines in an elegant way the two numberformats i.e. ninedigital and pandigital !
This couple are the middle two from a fourfold (*_4) solution for this pandigital.

 1073982645    = 89462 + 315272 198542 + 260732

Then three more pandigitals with double solutions popped up later on !
They are fully pandigital solutions throughout. Enjoy them!

 7095281346    = 207452 + 816392 398612 + 742052

 7125094386    = 312692 + 784052 497312 + 682052

 7465102389    = 369422 + 781052 469832 + 725102

Two pandigitals that stand out from the rest are 5921803476 and 8097452136 because
they are the only ones that can be written as a sum of two squares such that both their
basenumbers form a pandigital when concatenated as well as when multiplied together.
A nice couplet !

 5921803476   =   357902 + 681242 and 35790 * 68124 = 2438157960

 8097452136   =   251702 + 863942 and 25170 * 86394 = 2174536980

All the pandigitals (41) equal to 2 x A2 are A =
22887, 23124, 24957, 25941, 26409, 26733, 27276, 29685, 31389, 35367,
39036, 39147, 39432, 39702, 40293, 41997, 42843, 43059, 44922, 45258,
45624, 46464, 49059, 50889, 53568, 54354, 57321, 59268, 59727, 60984,
61098, 61611, 61866, 62634, 65436, 68823, 68982, 69087, 69696, 69732,
69798.
Note that two of them (see underlined) are palindromic !

While tracing for long(er) chains of pandigitals I stumbled across the following unique and
unexpectedly double expression. A thing of beauty !
Note that A and B are in descending order here. I didn't find a case whereby A and B are ascending.

 7983056241   =    75129 2 +  48360 2 and the concatenation of A | B gives us  7512948360   =    75306 2 +  42918 2 a similar all_pandigital expression !

Can you find longer chains of pandigital expressions using perhaps other operations than concatenation ?

[ April 27, 2014 ]
Order out of chaos using the ninedigits

Numberphile has released a video file that relates to the ninedigits.
Near the end of the video the zero comes in as well!

And here is a video explaining the mysterious math from above
which is in fact the Erdös-Szekeres theorem.

[ December 15, 2013 ]
Nearing the end of the year 2013

There is a unique ninedigital number that when divided by the year 2013 delivers a palindrome.
Ain't that nice!

127495368 = 2013 * 63336

Who likes to do this exercise for the coming years 2014, 2015, ...
And what about the pandigital version of this topic ?

[ January 1, 2013 ]
The ninedigital primes version of WONplate 181

The following ninedigital numbers are all (probable (3-PRP!)) prime.
Note that the 5-digit displacements are also prime !
 1325! + 87649 1462! + 59387 1475! + 96823 1547! + 68239 1685! + 29437 2351! + 74869 2354! + 86197 2468! + 75193 2486! + 15973 2615! + 84793 2876! + 15493 3218! + 45697 3845! + 76129 4618! + 32579 4895! + 76213 4981! + 62753 8153! + 72649 8573! + 41269 9625! + 34781 9824! + 61357 1438! - 57269 1478! - 52369 1486! - 59273 1576! - 28493 1849! - 67523 2471! - 63589 2536! - 84179 3469! - 28751 4285! - 71963 4562! - 93871 4691! - 52837 5239! - 16487 5417! - 86923 5462! - 98317 5743! - 68219 5749! - 26183 7948! - 23561 7982! - 45361 8123! - 47569 9836! - 25147

[ December 30, 2010 ]
From a posting to [SeqFan]
by Eric Angelini.

"As an afterthought, here's one I like (because of
the symmetry in the operations) that was appropriate
for the countdown on Friday night { can you find out the exact year? }:

10 + 9 * 8 * 7 / 6 * 5 * 4 + 321

Happy New Year! "

[ April 2008 ]
Fractions using the same digits as their decimal representation
A webpage by Christian Boyer.

An example using all the nine digits from
http://www.christianboyer.com/fractiondigits/ is

124983 / 576 = 216.984375

The above link came from a reply in the SeqFan mailing list where
Alexander R. Povolotsky's topic was about approximating Pi
using just nine- and pandigitals. Here is his *best* combination !

689725314(0) / 219546387(0) = 3.141592642...

[ March 31, 2008 ]
Blending palindromes with nine- & pandigitals using multiplication by 9
by B.S. Rangaswamy

This topic is a continuation from wonplate 173.

Pandigitals (Kmil = 1000 million) total = 559

`Palindrome	PandigitalP9		P9 * 9Palindrome	NinedigitalP8		P8 * 9     `
`Palindrome	PandigitalP9		P9 * 9Palindrome	NinedigitalP8		P8 * 9     `
`Palindrome	PandigitalP9		P9 * 9Palindrome	NinedigitalP8		P8 * 9     `
`Palindrome	PandigitalP9		P9 * 9Palindrome	NinedigitalP8		P8 * 9     `
```1 Kmil  # 81
2 Kmil  # 46

134050431	1206453879
134060431	1206543879
136717631	1230458679
136727631	1230548679
137606731	1238460579
137626731	1238640579
137818731	1240368579
137848731	1240638579
138929831	1250368479
138959831	1250638479
139828931	1258460379
139848931	1258640379
143050341	1287453069
143060341	1287543069
145030541	1305274869
145080541	1305724869
145272541	1307452869
145282541	1307542869
146717641	1320458769
146727641	1320548769
148919841	1340278569
148969841	1340728569
152676251	1374086259
152898251	1376084259
154030451	1386274059
154080451	1386724059
156252651	1406273859
156303651	1406732859
156373651	1407362859
157818751	1420368759
157848751	1420638759
158707851	1428370659
158747851	1428730659
158919851	1430278659
158969851	1430728659
167030761	1503276849
167080761	1503726849
167363761	1506273849
167414761	1506732849
167484761	1507362849
168929861	1520368749
168959861	1520638749
169818961	1528370649
169858961	1528730649
174898471	1574086239
176030671	1584276039
176080671	1584726039
176252671	1586274039
178050871	1602457839
178060871	1602547839
178252871	1604275839
178363871	1605274839
182565281	1643087529
183676381	1653087429
187050781	1683457029
187060781	1683547029
187303781	1685734029
187494781	1687453029
189272981	1703456829
189282981	1703546829
189373981	1704365829
189484981	1705364829
189515981	1705643829
215030512	1935274608
215080512	1935724608
215272512	1937452608
215282512	1937542608
216252612	1946273508
216303612	1946732508
217030712	1953276408
217080712	1953726408
217414712	1956732408
217484712	1957362408
218050812	1962457308
218060812	1962547308
218363812	1965274308
219272912	1973456208
219282912	1973546208
219373912	1974365208
219484912	1975364208
219515912	1975643208

224050422	2016453798
224060422	2016543798
225717522	2031457698
225727522	2031547698
226404622	2037641598
226818622	2041367598
226848622	2041637598
227959722	2051637498
240545042	2164905378
240656042	2165904378
255717552	2301457968
255727552	2301547968
264010462	2376094158
264404462	2379640158
266818662	2401367958
266848662	2401637958
268121862	2413096758
268454862	2416093758
277929772	2501367948
277959772	2501637948
279232972	2513096748
279565972	2516093748
286010682	2574096138
286232682	2576094138
286606682	2579460138
286626682	2579640138
315010513	2835094617
315494513 [Obs]	2839450617
315595513	2840359617
315989513	2843905617
316595613	2849360517
317010713	2853096417
318232813	2864095317
318343813	2865094317
326020623	2934185607
326090623	2934815607
326131623	2935184607
326494623	2938451607
327020723	2943186507
327090723	2943816507
327353723	2946183507
327595723	2948361507
328464823	2956183407
329050923	2961458307
329060923	2961548307
329464923	2965184307
```
```3 Kmil  # 74
4 Kmil  # 90

336020633	3024185697
336090633	3024815697
336131633	3025184697
336494633	3028451697
341050143	3069451287
341060143	3069541287
344717443	3102456987
344727443	3102546987
347121743	3124095687
347232743	3125094687
360121063	3241089567
360212063	3241908567
360989063	3248901567
361202163	3250819467
361232163	3251089467
361323163	3251908467
364505463	3280549167
364989463	3284905167
365101563	3285914067
365494563	3289451067
365717563	3291458067
365727563	3291548067
378101873	3402916857
378545873	3406912857
389212983	3502916847
389656983	3506912847
412050214	3708451926
412060214	3708541926
412676214	3714085926
412787214	3715084926
413272314	3719450826
413282314	3719540826
416454614	3748091526
416545614	3748910526
416575614	3749180526
417565714	3758091426
417656714	3758910426
417686714	3759180426
420545024	3784905216
420656024	3785904216
422050224	3798452016
422060224	3798542016
423050324	3807452916
423060324	3807542916
425010524	3825094716
425494524	3829450716
427454724	3847092516
427696724	3849270516
428010824	3852097416
428565824	3857092416
431050134	3879451206
431060134	3879541206
432505234	3892547106
432747234	3894725106
432808234	3895274106
432858234	3895724106
435030534	3915274806
435080534	3915724806
435272534	3917452806
435282534	3917542806
435717534	3921457806
435727534	3921547806
436020634	3924185706
436090634	3924815706
436131634	3925184706
436494634	3928451706
438020834	3942187506
438090834	3942817506
438575834	3947182506
438696834	3948271506
439030934	3951278406
439080934	3951728406
439131934	3952187406
439686934	3957182406

445919544	4013275896
446202644	4015823796
448020844	4032187596
448090844	4032817596
448575844	4037182596
448646844	4037821596
448696844	4038271596
451030154	4059271386
451080154	4059721386
452373254	4071359286
452393254	4071539286
455919554	4103275986
455969554	4103725986
459121954	4132097586
459676954	4137092586
462010264	4158092376
462202264	4159820376
467101764	4203915876
480212084	4321908756
480989084	4328901756
485676584	4371089256
485767584	4371908256
486454684	4378092156
486646684	4379820156
486989684	4382907156
487545784	4387912056
487919784	4391278056
487969784	4391728056
512030215	4608271935
512080215	4608721935
512303215	4610728935
513010315	4617092835
513252315	4619270835
513353315	4620179835
513545315	4621907835
514232415	4628091735
514323415	4628910735
514353415	4629180735
519010915	4671098235
519787915	4678091235
519878915	4678910235
521454125	4693087125
521898125	4697083125
523181325	4708631925
523404325	4710638925
523676325	4713086925
524010425	4716093825
524373425	4719360825
526454625	4738091625
526545625	4738910625
526575625	4739180625
529010925	4761098325
529787925	4768091325
529878925	4768910325
531545135	4783906215
531878135	4786903215
533151335	4798362015
533181335	4798632015
534030435	4806273915
534080435	4806723915
534151435	4807362915
534181435	4807632915
536232635	4826093715
536595635	4829360715
536696635	4830269715
536989635	4832906715
537454735	4837092615
537696735	4839270615
541030145	4869271305
541080145	4869721305
541252145	4871269305
541292145	4871629305
542151245	4879361205
542181245	4879631205
543626345	4892637105
543747345	4893726105
546252645	4916273805
546303645	4916732805
546373645	4917362805
546818645	4921367805
546848645	4921637805
547020745	4923186705
547090745	4923816705
547353745	4926183705
547595745	4928361705
548020845	4932187605
548090845	4932817605
548575845	4937182605
548646845	4937821605
548696845	4938271605
```
```5 Kmil  # 64
6 Kmil  # 78

623020326	5607182934
623090326	5607812934
623141326	5608271934
623191326	5608721934
623414326	5610728934
623565326	5612087934
624121426	5617092834
624363426	5619270834
624464426	5620179834
624656426	5621907834
625343526	5628091734
625434526	5628910734
625464526	5629180734
630121036	5671089324
630212036	5671908324
630989036	5678901324
632565236	5693087124
633020336	5697183024
633090336	5697813024
634020436	5706183924
634090436	5706813924
634262436	5708361924
634292436	5708631924
634515436	5710638924
634787436	5713086924
635121536	5716093824
635484536	5719360824
637565736	5738091624
637656736	5738910624
637686736	5739180624
642656246	5783906214
642989246	5786903214
644262446	5798362014
644292446	5798632014
645141546	5806273914
645191546	5806723914
645262546	5807362914
645292546	5807632914
647010746	5823096714
647343746	5826093714
648010846	5832097614
648565846	5837092614
652141256	5869271304
652191256	5869721304
652363256	5871269304
653262356	5879361204
653292356	5879631204
654707456	5892367104
654737456	5892637104
654808456	5893276104
654858456	5893726104
657030756	5913276804
657080756	5913726804
657363756	5916273804
657414756	5916732804
657484756	5917362804
657929756	5921367804
657959756	5921637804
658131856	5923186704
658464856	5926183704
659030956	5931278604
659080956	5931728604
659131956	5932187604
659686956	5937182604

668202866	6013825794
668424866	6015823794
671030176	6039271584
671080176	6039721584
673101376	6057912384
673252376	6059271384
674595476	6071359284
682010286	6138092574
682202286	6139820574
684232486	6158092374
684424486	6159820374
689323986	6203915874
689545986	6205913874
712030217	6408271953
712080217	6408721953
712303217	6410728953
712454217	6412087953
713010317	6417092853
713252317	6419270853
713353317	6420179853
713545317	6421907853
714232417	6428091753
714323417	6428910753
714353417	6429180753
719010917	6471098253
719878917	6478910253
723020327	6507182943
723090327	6507812943
723141327	6508271943
723191327	6508721943
723414327	6510728943
723565327	6512087943
724121427	6517092843
724363427	6519270843
724464427	6520179843
724656427	6521907843
725343527	6528091743
725434527	6528910743
725464527	6529180743
745020547	6705184923
745090547	6705814923
745383547	6708451923
745393547	6708541923
745606547	6710458923
745616547	6710548923
746010647	6714095823
746121647	6715094823
749010947	6741098523
749787947	6748091523
749878947	6748910523
753878357	6784905213
753989357	6785904213
755383557	6798452013
755393557	6798542013
756030657	6804275913
756080657	6804725913
756141657	6805274913
756191657	6805724913
756383657	6807452913
756393657	6807542913
758232857	6824095713
758343857	6825094713
761030167	6849271503
761080167	6849721503
762141267	6859271403
762191267	6859721403
764383467	6879451203
764393467	6879541203
765828567	6892457103
765838567	6892547103
768050867	6912457803
768060867	6912547803
768252867	6914275803
768363867	6915274803
769050967	6921458703
769060967	6921548703
769353967	6924185703
769464967	6925184703
```
```7 Kmil  # 69
8 Kmil  # 57

781050187	7029451683
781060187	7029541683
783474387	7051269483
785101587	7065914283
785494587	7069451283
812050218	7308451962
812060218	7308541962
812676218	7314085962
812787218	7315084962
816454618	7348091562
816575618	7349180562
817565718	7358091462
817656718	7358910462
821565128	7394086152
821787128	7396084152
823151328	7408361952
823181328	7408631952
824010428	7416093852
824373428	7419360852
826454628	7438091652
826545628	7438910652
826575628	7439180652
829010928	7461098352
829787928	7468091352
829878928	7468910352
834020438	7506183942
834090438	7506813942
834262438	7508361942
834292438	7508631942
834515438	7510638942
834787438	7513086942
835121538	7516093842
835484538	7519360842
837565738	7538091642
837656738	7538910642
837686738	7539180642
840121048	7561089432
840212048	7561908432
840989048	7568901432
843787348	7594086132
844020448	7596184032
844090448	7596814032
845020548	7605184932
845090548	7605814932
845383548	7608451932
845393548	7608541932
845606548	7610458932
845616548	7610548932
846010648	7614095832
846121648	7615094832
849010948	7641098532
849787948	7648091532
849878948	7648910532
867050768	7803456912
867060768	7803546912
867151768	7804365912
867181768	7804635912
867383768	7806453912
867393768	7806543912
871050178	7839451602
871060178	7839541602
871262178	7841359602
873262378	7859361402
873292378	7859631402
874383478	7869451302
874393478	7869541302
879272978	7913456802
879282978	7913546802
879515978	7915643802

915010519	8235094671
915595519	8240359671
915989519	8243905671
916595619	8249360571
917010719	8253096471
917343719	8256093471
918232819	8264095371
918343819	8265094371
923050329	8307452961
923060329	8307542961
925010529	8325094761
925494529	8329450761
927454729	8347092561
927696729	8349270561
928010829	8352097461
928565829	8357092461
934030439	8406273951
934080439	8406723951
934151439	8407362951
934181439	8407632951
936232639	8426093751
936595639	8429360751
936696639	8430269751
936989639	8432906751
937454739	8437092651
937696739	8439270651
945141549	8506273941
945191549	8506723941
945262549	8507362941
945292549	8507632941
947010749	8523096741
947343749	8526093741
948010849	8532097641
948565849	8537092641
951434159	8562907431
951989159	8567902431
955141559	8596274031
955191559	8596724031
956030659	8604275931
956080659	8604725931
956141659	8605274931
956383659	8607452931
956393659	8607542931
958232859	8624095731
958343859	8625094731
963878369	8674905321
963989369	8675904321
966383669	8697453021
966393669	8697543021
967050769	8703456921
967060769	8703546921
967151769	8704365921
967181769	8704635921
967262769	8705364921
967292769	8705634921
967383769	8706453921
967393769	8706543921

```

[ June 14, 2014 ]
Observation by Heine Wanderlust (email)

315494513 (a prime by the way) * 9 = the pandigital 2839450617.
But multiply by 18 and you get a pandigital with a nice permutation: 5678901234.
Also in both these pandigitals the differences between each
successive digit creates a palindrome: 656515656, 111191111

[ September 15, 2014 ]
Observation 2 by Heine Wanderlust (email)

Following on from my June 14 2014 post, I've found another Rangaswamy palindrome
yielding a pandigital when multiplied by 18 as well as by 9 :
548696845 * 18 = 9876543210
548696845 * 9 = 4938271605
This type of result, whereby the "countdown" pandigital in a base b can be divided by 2(b–1)
to get a palindrome, occurs in at least two other number bases:
base 8 where 76543210 / 16 = 4364634, and in base 6 where 543210 / 14 = 32423.

[ January 22, 2006 ]
Pandigital... throughout

5-digit factors which taken together form a pandigital number :

 54981 * 62037

equals

 3410856297

as you can see the result of the multiplication is pandigital as well.
And now let us take the square of this pandigital 34108562972

 11633940678784552209

... an order_2 pandigital emerges since all the digits from 0 to 9
occur exactly two times !

Discover more of these gems at wonplate 167 !

[ October 26, 2005 ]
From Palindromic Squares to Pandigitals

264 is a very interesting number since it is the 12th basenumber of a palindromic square.
The square itself is this nice palindrome 69696.
Note the presence of the number of the Beast ! 69696 .

Did you know that when we power up 264 with two exponents and add them up
that we arrive at a pandigital number... in two different ways !

2643 + 2644 = 4875932160
2644 + 2644 = 9715064832

The second equation can be written as a palindromic expression itself using
69696 squared and then doubled

2 * 69696 ^ 2
(There exists another 5-digit palindrome with this property. Can you discover it ?)

B.S. Rangaswamy [ August 11, 2006 ] was enthused by this presentation
and found the following solutions :

232322 *   8 = 4317806592 = 464642 * 2
232322 * 18 = 9715064832 = 696962 * 2
423242 *   4 = 7165283904

One can play along with other than our 264 number namely 2016.
I found the next equation of some interest.

20162 + 20163 = 8197604352

Or the next one with two consecutive integers

233 * 244 = 4036718592

Is there more to discover ?

Yes, see Claudio Meller's contribution at WONplate 183

[ October 2005 ]
Figure this out : 4-1-4

It is a four digit number multiplied by a one digit number to equal another four digit number
and only the nine digits from 1 to 9 can be used once ?

Two solutions to this 9-digit problem can be found.

1738 * 4 = 6952
1963 * 4 = 7852

[ October 2006 ]
Figure this out : 4-1-5

It is a four digit number multiplied by a one digit number to equal a five digit number
and only the ten digits from 0 to 9 can be used once ?

Thirteen solutions to this pandigital problem can be found.

3094 * 7 = 21658
3907 * 4 = 15628
4093 * 7 = 28651
5694 * 3 = 17082
5817 * 6 = 34902
6819 * 3 = 20457
6918 * 3 = 20754
7039 * 4 = 28156
8169 * 3 = 24507
9127 * 4 = 36508
9168 * 3 = 27504
9304 * 7 = 65128
9403 * 7 = 65821

[ September 4, 2005 ]
Generating Pandigitals from Palindromes through Fibonacci iteration
by B.S. Rangaswamy

" I got inspired by your presentation of the derivation of 68 ninedigit numbers (with all numerals from 1 to 9)
from palindromes through Fibonacci iteration. I have developed it further by arriving at a dozen 10 digit numbers
(with all numerals from 0 to 9) from 2 to 10 digit palindromes. Some of the 10 digit numbers arrived at
together with their mother palindromes are :

 75257 9135748206 799997 2067193845 8055508 4913860257 42944924 2361970854

It is interesting to note that 6300036 leads to pandigital 1467908532, which matches identically
with each stage of iteration of 630036 to 146798532. I came across another curio as well :

 630036   146798532 6300036 1467908532 9530359   123894675 95300359 1238904675

Following closely resembling palindromes lead to closely matching pandigitals :

 592070295 2960351478 952070259 4760351298

This task began at my son’s residence in Florida US and was completed at Bangalore India.
I was thrilled at the discovery of each of these pandigitals.

I am grateful to you for your encouragement and guidance in this venture.
B.S.Rangaswamy "

In total there are 117 palindromes that yield pandigital numbers
The smallest one is 75257 and the largest one is 4376006734 ¬

1. Typical generation of pandigitals (10 digits) from palindromes through Fibonacci iteration
1
75257
75258
150515
225773
376288
602061
978349
1580410
2558759
4139169
6697928
10837097
17535025
28372122
45907147
74279269
120186416
194465685
314652101
509117786
823769887
1332887673
2156657560
3489545233
5646202793
9135748026
1
799997
799998
1599995
2399993
3999988
6399981
10399969
16799950
27199919
43999869
71199788
115199657
186399445
301599102
487998547
789597649
1277596196
2067193845
1
6431346
6431347
12862693
19294040
32156733
51450773
83607506
135058279
218665785
353724064
572389849
926113913
1498503762
1
408676804
408676805
817353609
1226030414
2043384023
3269414437
5312798460
1
561262165
561262166
1122524331
1683786497
2806310828
4490097325
7296408153
1
760474067
760474068
1520948135
2281422203
3802370338
6083792541
1
93811839
93811840
187623679
281435519
469059198
750494717
1219553915
1970048632
3189602547
1
1206776021
1206776022
2413552043
3620328065
6033880108
9654208173
1
3067447603
3067447604
6134895207
1
4068228604
4068228605
8136457209
117
in total

2. Above are 117 palindromes in ascending order, leading to pandigitals (10 digits)
75257
799997
2877782
4364634
4689864
5068605
6300036
6431346
6881886
8055508
15844851
42944924
54777745
93811839
95300359
146353641
177858771
185121581
185222581
187333781
207313702
238242832
245363542
295747592
310717013
314878413
324151423
348616843
350878053
354010453
358070853
370464073
370797073
390474093
394070493
395313593
395727593
407525704
408676804
460363064
473303374
475686574
506939605
527121725
527454725
530878035
536454635
547252745
561262165
590272095
592070295
629171926
642676246
655191556
695525596
740838047
760474067
782484287
841303148
914272419
950272059
952070259
1089229801
1097337901
1098228901
1206776021
1289009821
1295335921
1298008921
1395225931
1397007931
2068448602
2069339602
2158448512
2159339512
2359119532
2365445632
2369009632
2456336542
2458118542
2465335642
2468008642
3067447603
3069229603
3076446703
3079119703
3157447513
3159229513
3175445713
3179009713
3256446523
3259119523
3265445623
3269009623
3456226543
3457117543
3465225643
3467007643
3475115743
3476006743
4067337604
4068228604
4076336704
4078118704
4086226804
4087117804
4158228514
4175335714
4178008714
4185225814
4187007814
4258118524
4268008624
4286006824
4365225634
4367007634
4376006734

3. Queen and King combinations of
palindromes vs. pandigitals
PalindromeNinedigital PalindromePandigital
QUEEN KING
630036
9530359
128909821
129808921
139707931
236909632
246808642
317909713
326909623
346707643
347606743
417808714
418707814
426808624
428606824
436707634
437606734
146798532
123894675
257819643
259617843
279415863
473819265
493617285
635819427
653819247
693415287
695213487
835617429
837415629
853617249
857213649
873415269
875213469
6300036
95300359
1289009821
1298008921
1397007931
2369009632
2468008642
3179009713
3269009623
3467007643
3476006743
4178008714
4187007814
4268008624
4286006824
4367007634
4376006734
1467908532
1238904675
2578019643
2596017843
2794015863
4738019265
4936017285
6358019427
6538019247
6934015287
6952013487
8356017429
8374015629
8536017249
8572013649
8734015269
8752013469

4. Scintillating Grid
one pair each of digits 0, 3, 4, 6 & 7
PalindromePandigital
0346707643
0347606743
0436707634
0437606734
3067447603
3076446703
3467007643
3476006743
4067337604
4076336704
4367007634
4376006734
0693415287
0695213487
0873415269
0875213469
6134895207
6152893407
6934015287
6952013487
8134675209
8152673409
8734015269
8752013469

```

```

Contributions

B.S. Rangaswamy (email) - go to topic 1

B.S. Rangaswamy (email) - go to topic 2

```

```