 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).
Second Page
Topic 2.25 [ March 1, 2000 ]
Aran Kuntze is studying the occurrences of the nine digits numbers
in the decimal expansion of
So for instance (taken three at random from her list of 41)
The string 165429837 was found at position 10552019
The string 654321987 was found at position 14597746
The string 976543182 was found at position 22314906
Puzzle 1
Aran (from Finland) is also hoping to find a palindromic position but
had no luck so far. Can you give her a helping hand in her search ?
Puzzle 2
I propose Aran or any other puzzler to try to find a solution that occurs
at a nine digits position. Beware that it will be a hard job as one has to have
access to the first 1.000.000.000 digits of to scan through.
The string some nine digits number was found at position some nine digits number ?
Topic 2.24 [ February 14, 2000 ]
Once again G. L. Honaker, Jr. made an interesting discovery
This time it is about 6 primes in Arithmetic Progression
with a common difference of... our pandigital number !
5077
9876548287
19753091497
29629634707
39506177917
49382721127
The common difference is 9876543210.
See Sloane's A058908.
Who can construct such sequences with 7 or more primes in AP ?
Topic 2.23
From Felice Russo's sequence A039667
There exist only four numbers n so that
juxtaposition of n, 2n and 3n forms a nine digits number.
192 384 576 : 192384576
219 438 657 : 219438657
273 546 819 : 273546819
327 654 981 : 327654981
Topic 2.22 [ September 16, 1999 ]
Two new G. L. Honaker, Jr. discoveries !
The sum of the first 10701 consecutive prime numbers is a nine digit anagram.
572469138
in Honaker's words pandigital (excluding the zero)
or in Weisstein's words zeroless pandigital.
The sum of the first 32423 consecutive prime numbers is pandigital.
5897230146
in this case pandigital means including all 10 digits exactly once.
Note that 32423
is the only palindromic prime that produces such a number.
Featured in Prime Curios! 5897230146
Further information
in Sloane's database : A049442, A049443 and A049446.
( see also A050278, A050288, A050289 and A050290 )
in Weisstein's Math Encyclopedia Pandigital Number.
[ November 11, 2022 ]
Alexandru Petrescu explores other paths to reach more nine and pandigital results. Here is his contribution.
He discovered that the sum of the first 59295 (palindromic!) consecutive composites is a ninedigit anagram !
The sum of the first n consecutive prime numbers is a ninedigit anagram   The sum of the first n consecutive composite numbers is a ninedigit anagram   The sum of the first n consecutive prime numbers is a pandigital anagram   The sum of the first n consecutive composite numbers is a pandigital anagram 
n  sum   n  sum   n  sum   n  sum 
8971  394521678   14829  125376498   14353  1063254978   47554  1267389504 
10474  547128369   14847  125678493   18572  1829360475   49237  1358079462 
10701  572469138   16895  162394758   22876  2835410967   49604  1378269450 
 [#3]   21146  253486197   25212  3478029561   53356  1593207684 
   23692  317654892   26799  3954061782   56449  1782046395 
   25867  378159264   27803  4271593608   56645  1794365802 
   28643  462983715   28752  4583260179   57328  1837624590 
   29389  487231695   30510  5190863247   57576  1853460927 
   30418  521694738   32011  5741092638   59295  1965072348 
   33681  638719254   32423  5897230146   62248  2164385079 
   34219  659143728   32515  5932401786   65250  2376809514 
   41292  957361482   35137  6980571324   66989  2504396871 
   41417  963127458   37055  7803615924   67828  2567140389 
   41454  964837512    [#13]   68124  2589460713 
    [#14]      69621  2703816459 
         75626  3187265940 
         79023  3478256901 
         80736  3629784051 
         83062  3840691257 
         83875  3915807462 
         87976  4305791628 
         88174  4325086179 
         90241  4529068317 
         92224  4729156380 
         92294  4736298501 
         95443  5063128749 
         96581  5183906247 
         101059  5672914308 
         104992  6120485379 
         105438  6172308459 
         105974  6234875019 
         119684  7942058361 
         122491  8316924705 
         125974  8794052136 
         127167  8960524713 
         128445  9140586723 
         130042  9368104574 
         131084  9518046237 
         132572  9734215086 
         132614  9740351862 
         132648  9745321086 
          [#41] 

Topic 2.21
An extraordinary find !
An extraordinary find by Carlos Rivera [ Visit 'PP&P' Puzzle 41 ]
He succeeded in amalgamating four concepts :
the Number of the Beast (666) as the quotient,
the nine_digits (the numerators)
the palindromes (the denominators)
the primes  e.g. 1371731 versus the composites  e.g. 396693 
913572846 1371731 
666 
264197538 396693 
Topic 2.20 [ December 12,1998 ]
Add up the values (A = 1, B = 2, C = 3, etc.) of the letters
of the written out numbers (in English)
From Carlos Rivera's Puzzle 33 of his PP&P website
A result from Carlos Rivera's work on adding the letters of written_out numbers
(in English) is that he proved that these two 9digit numbers are equal !
The palindromic construction of the equation is a free bonus.
123456789 = 987654321
ONE HUNDRED TWENTY THREE MILLION FOUR HUNDRED FIFTY SIX THOUSAND SEVEN HUNDRED EIGHTY NINE
O+N+E+H+U+N+D+R+E+D+T+W+E+N+T+Y+T+H+R+E+E+M+I+L+L+I+O+N+F+O+U+R+H+U+N+D+R+E+D+F+I+F+T+Y+S+I+X+
T+H+O+U+S+A+N+D+S+E+V+E+N+H+U+N+D+R+E+D+E+I+G+H+T+Y+N+I+N+E = 964
And this value 964 is also the total of the second nine digits number !
NINE HUNDRED EIGHTY SEVEN MILLION SIX HUNDRED FIFTY FOUR THOUSAND THREE HUNDRED TWENTY ONE
N+I+N+E+H+U+N+D+R+E+D+E+I+G+H+T+Y+S+E+V+E+N+M+I+L+L+I+O+N+S+I+X+H+U+N+D+R+E+D+F+I+F+T+Y+F+O+U+R+
T+H+O+U+S+A+N+D+T+H+R+E+E+H+U+N+D+R+E+D+T+W+E+N+T+Y+O+N+E = 964
[ Quod Erat Demonstrandum ) ]
The same method can be applied to prove that :
964 = 469 273 = 372 etc. but I leave that as an exercice for the curious puzzlers amongst you !
Topic 2.19
What is the 10digits number
[from alt.math.recreational]
Source : “Mathematical Circus”, Martin Gardner, Penguin Books, 1979, p.128 & 135(solution)
WWWLinks :
SelfDescriptive Number
Autobiographical numbers : A046043
A pandigital problem
0  1  2  3  4  5  6  7  8  9 
         
In the 10 cells of the above figure inscribe a 10digit number such that the digit in the first cell indicates the total number
of zeros in the entire number, the digit in the cell marked '1' indicates the total of 1's in the number, and so on to the last cell,
whose digit indicates the total number of 9's in the number. (zero is a digit, of course). The answer is unique.
On simple request I will mail you this unique answer (find my email address at the bottom of this page).
How to arrive at this unique solution ?
Number Problem #1 by QYV
Problem 4 by John Scholes
Topic 2.18 [ November 10, 1998 ]
Puzzle : Juxtaposition of prime factors is a Nine Digits number
The smallest number whereby the juxtaposition (or concatenation if you like)
of its prime factors is a nine digits number is :
2992890 = 2 x 3 x 5 x 67 x 1489 or juxtaposed 235671489.
And here is the largest one :
842696243 = 8641 x 97523 or juxtaposed 864197523.
The smallest pandigital number :
15618090 = 2 x 3 x 5 x 487 x 1069 or juxtaposed 2354871069.
The puzzle questions are now :
[A] Find the largest number whereby the juxtaposition of its prime factors
is a pandigital number (the zero allowed).
[B] What are the smallest and largest juxtaposed nine digits or pandigital numbers ?
[C] What are the smallest and largest numbers for each possible 'n' prime factors whose
juxtaposition is a nine digits or pandigital numbers ?
[D] Are there nine digits or pandigital numbers whose prime factors when juxtaposed yield
another nine digits or pandigital number ?
Topic 2.17 [ October 4, 1998 ]
Puzzle : Root Extracting Nine Digits numbers
The smallest 9digits number so that the decimal part of its square root also starts with a 9digits number is :
135649728^(1/2) = 11646,876319425 736908...
Here are more random solutions. E.g.:
235916748^(1/2) = 15359,581634927 430535...
317625984^(1/3) = 682,294715863 639889...
413786925^(1/4) = 142,624518379 632388...
529638741^(1/5) = 55,564312978 124193...
613824957^(1/6) = 29,152384967 733170...
713625894^(1/7) = 18,398461752 548250...
893257614^(1/8) = 13,148375269 564618...
926374815^(1/9) = 9,915386274 757628...
The puzzle question is now :
Find a root of a 9digits number so that
[A] the same 9digits number reappears (if none exists then the one with most digitmatches)
or so that
[B] the same 9digits number but reversed appears
immediately after the decimal point (the 'comma' for us Europeans) of this root.
Note that you may choose roots other than the square root (of the form '1/n').
[ Don't ask for solutions via email as this is an open puzzle meaning I haven't a solution myself. ]
Happy hunting !
If you want to know the integers such that in the decimal representations of the square root of those integers,
the digits to the immediate right of the decimal point are 123456789 and 987654321 then visit the archives of
the Southwest Missouri State University's Problem Corner  Solution to Problem #14 !
Maybe at this point you would like to know the answer to the following
question posed by Carlos Rivera
Let`s suppose that N is a 9 digits number: what is the minimal
arithmetical test (if it exists) that we need to apply to N for
testing if N is an anagram of the 9digits number 123456789 ?
Carlos explains his best solution for that problem (in UBASIC)
Let's suppose you have a number N of 9 digits, N = d1d2d3d4d5d6d7d9 :
If prm(d1) * prm(d2) * prm(d3) *... * prm(d8) * prm(d9) =
2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 = 223092870
then N is an anagram;
If the product differs from 223092870
then N is not an anagram.
Other smaller productnumbers can easily be calculated for testing anagrams of less than 9 digits numbers. Beautiful!
Carlos, your best solution is most elegant. That way, you also found an original application for the uniqueness of primes.
Topic 2.16
Extracting the Nine Digits from the Number of the Beast 666
This goes as follows :
Sum_Of_Digits{6^6} = 27 Number_Of_Digits = 5
Sum_Of_Digits{66^66} = 531 Number_Of_Digits = 121
Sum_Of_Digits{666^666} = 8649 Number_Of_Digits = 1881 [P. De Geest]
The combination of all three numbers miraculously shows all the nine digits in a row 275318649 !
Note also that each time the numbers of digits of the next result appears to be a palindromic number.
Alas, the pattern stops with the third one because
Number_Of_Digits{6666^6666} = 25490
and
25490 is not palindromic. Nor are the following terms
321591,
3882602,
45492721, etc.
Topic 2.15
From Nine Digit Basenumbers via Nine Digit Powers arriving at Palindromes
Here is one of my recent research projects.
It makes a link between palindromes, the 'nine digits' and powers :
Task : Find a construction of the form :
A^{a} + B^{b} + C^{c} + D^{d} + E^{e} + F^{f} + G^{g} + H^{h} + I^{i} = P
whereby P is a palindromic number.
The letters A to I represent the nine digits (1 to 9) and all the nine digits must be used exactly once.
The order is unimportant.
Idem dito for the exponents a to i.
Some examples (two extremes and a random one) :
1^{1} + 2^{2} + 3^{3} + 4^{4} + 5^{5} + 6^{6} + 7^{7} + 8^{8} + 9^{9} = 405071317
1^{9} + 2^{8} + 3^{7} + 4^{6} + 5^{5} + 6^{4} + 7^{3} + 8^{2} + 9^{1} = 11377
4^{9} + 8^{6} + 6^{7} + 1^{4} + 3^{5} + 2^{3} + 5^{1} + 9^{8} + 7^{2} = 43851251
The following table shows the smallest and the largest of the 223 palindromic combinations I found.
[ In fact the search yielded exactly 211 different palindromes. ]
The third row shows a special and unique palindrome that can be written in exactly two ways
so that all the exponents differ visàvis the basenumbers of the two combinations.
Smallest 
12921 
1^{8} + 2^{9} + 3^{7} + 4^{6} + 5^{5} + 6^{1} + 7^{4} + 8^{3} + 9^{2} 
Largest 
389909983 
1^{5} + 2^{6} + 3^{2} + 4^{1} + 5^{8} + 6^{4} + 7^{3} + 8^{7} + 9^{9} 
Special 
317713 
1^{7} + 2^{8} + 3^{2} + 4^{9} + 5^{6} + 6^{1} + 7^{3} + 8^{5} + 9^{4}
1^{8} + 2^{6} + 3^{9} + 4^{1} + 5^{4} + 6^{7} + 7^{5} + 8^{3} + 9^{2} 
Topic 2.14
From Palindromes via Fibonacci arriving at Nine Digits
As palindromes are my cup of tea allow me to continue this section with them.
What I try to accomplish here is to establish a relationship between three known mathematical concepts.
Via Fibonacci iteration and starting from a palindromic number arriving at a nine digit number !
For the moment I haven't found a palindrome that transforms into 123456789 or its reversal 987654321.
In total there are 68 palindromes that yield 9digits numbers
The smallest one is 4004 and the largest one is 437606734 ¬
1
4004
4005 8009 12014 20023
32037 52060 84097 136157
220254 356411 576665 933076
1509741 2442817 3952558 6395375
10347933 16743308 27091241 43834549
70925790 114760339 185686129 300446468
486132597
= smallest
 1
630036
630037 1260073 1890110
3150183 5040293 8190476
13230769 21421245 34652014
56073259 90725273
146798532
 1
1559551
1559552 3119103 4678655
7797758 12476413 20274171
32750584 53024755 85775339
138800094 224575433 363375527
587950960
951326487
 1
4187814
4187815 8375629 12563444
20939073 33502517 54441590
87944107
142385697
 1
4870784
4870785 9741569 14612354
24353923 38966277 63320200
102286477 165606677
267893154
 1
6097906
6097907 12195813 18293720
30489533 48783253 79272786
128056039 207328825 335384864
542713689

1
6834386
6834387 13668773 20503160
34171933 54675093 88847026
143522119 232369145
375891264

1
9530359
9530360 19060719 28591079
47651798 76242877
123894675

68 in total 
1
82466428
82466429
164932857

1
437606734
437606735
875213469
= largest


Inspired ? I hope that the readers will find more of these transformational RECREATIONAL FACTS.
For instance :
Who can find a palindrome that transforms into another palindrome via Fibonacci iteration ?
Or :
Who can find a startnumber that transforms via Fibonacci iteration into some other
operational result of that startnumber ?
[ September 11, 2005 ]
Investigation and a new challenge by Carlos Rivera
For your Fibonacci and Pandigital page, specially to your statement 'For the moment I haven't found
a palindrome that transforms into 123456789 or its reversal 987654321.' what I can say is that
NO palindrome can touch these numbers. Here are my results with some extra's as well :
15432098
15432099
30864197
46296296
77160493
123456789 
61728394
61728395
123456789 
493827160
493827161
987654321 
110972395
110972396
221944791
332917187
554861978
887779165
1442641143
2330420308
3773061451
6103481759
9876543210 
127932098
127932099
255864197
383796296
639660493
1023456789 
511728394
511728395
1023456789 
The only strange thing is that I was thinking that no two fibonacci series,
starting with two distinct initial numbers, could hit the same number. Was I wrong ?
Perhaps the statement is true for any k term of fibonacci sequence (a0, a1, a2, a3, ...) if k>2, ...
Saludos. Regards.

[ February 18, 2022 ]
Daniel Hardisky (email) would like you to find
“Generalized Fibonacci Sequences” and “Generalized Tribonacci Sequences”
and find the maximum differences in increasing order. Daniel himself was surprised at the large 'gaps' between successive terms.
His collage on a placard hereunder provides more information on the subject.
Are you up for his challenge ?

Topic 2.13
Powers of ninedigitals
A small booklet called "Rekenraadsels" from Deltas vrijetijdreeks [1981  ISBN 9024325455]
inspired me to start this nine digit topic.
The first two numbers multiplied together deliver a result equal to the square of another number
also containing all nine digits
246913578 x 987654312 = 493827156^{2}
Note that 246913578 = 2 * 123456789.
Use the digits from 1 to 9 once to form two numbers, so that one number is twice the other one
6729 x 2 = 13458
There are more solutions then the one given above.
Michael Winckler published the following puzzle a while ago (Puzzle No. 121) :
For each ratio 1/2, 1/3, ..., 1/9 find two integers a and b such that
1. a/b has that ratio and that
2. in forming a and b each digit 1 up to 9 is used exactly once.
All possible solutions to this puzzle were sent in by Carlos Rivera shortly after.
Topic 2.12
0123456789 or pan and ninedigital diversions
Many more digital combinations can be found in “Madachy's Mathematical Recreations” from
Joseph S. Madachy [Dover N.Y., 1979  ISBN 0486237621, pp. 156162]
I'll give a few excerpt to whet your appetite.
291548736 = 8 x 92 x 531 x 746
124367958 = 627 x 198354 = 9 x 26 x 531487
A square that yields all the nine digits twice !
335180136^{2} = 112345723568978496
[ May 11, 2008 ]
Investigation by Peter Kogel
Hi Patrick,
In Joseph Madachy's “Mathematical Recreations” [ on page 159 ] he presents the following result
and asks whether there are more results.
246913578 x 987654312 = 493827156 x 493827156
i.e. A x B = C^{2} where A, B and C are zeroless pandigital numbers.
Not being one to allow such a challenge to pass by, I set about searching for other
9 and 10 digit pandigital examples.
I was a bit disappointed to eventually discover that there are well over 12000 such solutions
because this somehow seemed to dilute the aesthetic nature of the puzzle. The large number of
solutions though does prove to be a veritable gold mine of supplementary results.
Zeroless solutions # 620
Pandigital solutions # 6619
Mixed solutions # 5587
Total solutions # 12826
Zeroless smallest
231597684^{2} = 164938572 x 325196748
Zeroless largest
659418732^{2} = 769321854 x 897542163
Pandigital smallest
1378965042^{2} = 1280467539 x 1485039276
Pandigital largest
8326790451^{2} = 7450286193 x 9306412857
There are 40 examples where the sum of A and B is also zeroless and 404 examples
where the sum is pandigital. E.g. the smallest for each type is :
246913578^{2} = 123456789 x 493827156
617283945 = 123456789 + 493827156
2053914768^{2} = 1026957384 x 4107829536
5134786920 = 1026957384 + 4107829536
Note that the number C is actually the geometric mean of A and B !
I found that there are 36 solutions where the arithmetic mean is also pandigital. E.g.
Geometric Mean ( 1076539482, 4306157928 ) = 2153078964
Arithmetic Mean ( 1076539482, 4306157928 ) = 2691348705
There are no such solutions for zeroless numbers.
There are 19 examples where both A and B are pandigital square numbers and
one example where they are zeroless squares.
35853^{2} = 1285437609 
 23439^{2} = 549386721 
71433^{2} = 5102673489 x 

27273^{2} = 743816529 x 
2561087349^{2}   639251847^{2} 
To the best of my knowledge these results have never been noted before. I find this surprising
considering that the 30 zeroless squares and 87 pandigital squares are very well known.
BTW the first square on the LHS is a palindrome !
I found two solutions where C^{2} is also doubly pandigital; i.e. it contains each digit twice: viz.
3672980514^{2} = 1836490257 x 7345961028
3672980514^{2} = 13490785856223704196
However, the second solution of this type is the 'pièce de résistance' for I was
elated to note that A + B is also pandigital.
1854763209 + 7419052836 = 9273816045
3709526418^{2} = 1854763209 x 7419052836
3709526418^{2} = 13760586245839910724

Regards,
Peter Kogel (alias Peter Pan).
[ January 21, 2009 ]
Reversal investigations by Peter Kogel
Hi P@rick,
The reversal relationship you found for the pan x pan = pal investigation
(see bottom part of the webpage at twopan.htm)
led me to wonder whether I couldn't use the same technique for the
pan x pan = pan^2 project I ran last year (see above).
I was delighted to find 6 solutions; viz:
429731586^{2} = 859463172 x 214865793  ^{ } 

397568412 x 1590273648 = 795136824^{2}

493582716^{2} = 987165432 x 246791358  ^{ } 

853197642 x 3412790568 = 1706395284^{2}

1356902478^{2} = 2713804956 x 678451239  ^{ } 

932154876 x 3728619504 = 1864309752^{2}

1395702684^{2} = 2791405368 x 697851342  ^{ } 

243158796 x 972635184 = 486317592^{2}

1459027368^{2} = 2918054736 x 729513684  ^{ } 

486315927 x 1945263708 = 972631854^{2}

3097521864^{2} = 6195043728 x 1548760932  ^{ } 

2390678451 x 9562713804 = 4781356902^{2} 

There are also 23 solutions where the square itself can be reversed; e.g.
Smallest 
261845739 x 1047382956 = 523691478^{2}  ^{ } 

874196325^{2} = 4370981625 x 174839265 
Largest 
175264389 x 4381609725 = 876321945^{2}  ^{ } 

549123678^{2} = 1098247356 x 274561839 

Undoubtedly best of all is the following remarkable result
where each ninedigital number is reversed (unique case):
Forward   Backward 
159723648   846327951 
x   x 
489153672   276351984 
=   = 
279516384^{2}   483615972^{2} 

Regards,
Pete Kogel

Topic 2.11
I've got this from rec.puzzles
There is a 9digit number in which the digits 1 through 9 appear exactly once.
If you only take the first N digits from the left, the number you're left with is
divisible by that same value N. What is this unique number ?
Visit the
Palindromic Puzzle's page for a quick access to the solution.
Topic 2.10
Ninedigital tilelayers
A tilelayer has exactly 123456789 tiles and has to make a rectangle that best approaches a neat square.
No tiles are fractured or left out. [ Dutch Source : original but dead link = http://www.win.tue.nl/math/dw/ida/h1s4.html#tegelfactor ]
He chooses this rectangle : 11409 x 10821
Can you prove that this is the best solution ?
And what if you gave our man 987654321 tiles to play with ?
[
January 8, 2007 ]
B.S.Rangaswamy (
email) got involved in the above puzzle.
He could give the tilelayer a very narrow straight pathway, measuring
2601 x
379721
(length almost 146 times that of the width) to house
987654321 tiles. There can be no
wider path since
379721 happens to be the largest prime factor of
987654321.
Thanks for providing me an opportunity to get involved in this puzzle and express my view.
Topic 2.9
Also from rec.puzzles
This one from a book by 'L.A. Graham'
Using all the 9 digits from 1 to 9 and only once, what two numbers multiplied
give the largest product ? Such as 12345 times 6789.
Quite promptly the answer came
87531 x 9642 = 843973902
Topic 2.8
Source : All the Math that's Fit to Print
by Keith Devlin [ Chapter 17 p.44 ]
Arrange the digits 1 to 9 into two numbers, one of which is the square of the other.
The two possible solutions are
567^{2} = 321489
and
854^{2} = 729316
Topic 2.7
A Selection of Magic Squares Websources
Magic Squares by Harvey Heinz
Magic Squares  the ultimate database by Mutsumi Suzuki
What is a Magic square ? by Allan Adler
Magic squares  Building a 9Cell Square by Suzanne Alejandre
More than Magic squares by Ivars Peterson
Magic Square by E. Weisstein
Creating magic squares by Zimaths
Construct a Magic Square using all the digits from 1 to 9
Note that all the rows and columns and diagonals add up to 15 (fifteen).
Visit WONplate 9 and WONplate 14 to see two beautiful Magic Squares constructed only with palindromic numbers !
Pity that the magic sums are not palindromic as well !
Topic 2.6
From “All the Math that's Fit to Print”
by Keith Devlin pages 180 and 182
Find a palindrome that when multiplied with 123456789
gives a number that ends with its reversal ...987654321.
Answer : that number is 989010989
The result is 122100120987654321
Topic 2.5
From “Dictionary of Curious and Interesting Numbers (revised edition 1997)”
by David Wells page 149
The minimum sum of ninedigital 3digit primes, 149 + 263 + 587 = 999 and PALINDROMIC.
Can you find more of these fun facts beyond 149263587 ?!
Topic 2.4
MORELAND PI (anagram of PALINDROMES) welcomes the nine digits
Pi search for the nine digits
Consult the 100.000.000 digits of PI and try to locate the nine digits.
Results
The string 123456789 did not occur in the first 100000000 digits of pi after position 0.
The string 987654321 did not occur in the first 100000000 digits of pi after position 0.
This is a subtopic where I need your help. After all, of the 362.880 possible
combinations I only checked the above two...
Martin Gardner wrote a chapter about Random Numbers in his book "Mathematical Carnival".
Here's a small excerpt [ p.167 ]
... It (the brain) acquires its ability to "see" patterns only after years of experience during which
the patterned external world imposes its order on the brain's tabula rasa. It is true, of course,
that one is surprised by a sequence of 123456789 in a series of random digits because such a
sequence is defined by human mathematicians and used in counting, but there is a sense in which
such sequences correspond to the structure of the outside world...
Topic 2.3
Curiously this sum is palindromic and repunital
987654321 + 1 + 123456789 = 1111111111
Topic 2.2
From the book “The Penguin Dictionary of Curious and Interesting Numbers”
Surprising facts by David Wells
The difference between 123456789 and 987654321 is an anagram of the ninedigital : 864197532
The smallest ninedigital square is
11826^{2} = 139854276
The largest ninedigital square is
30384^{2} = 923187456
Albert H. Beiler lists 30 of them (from a total of 83) in his book :
"Recreations in the Theory of Numbers" [ Second Ed. Table 64 p. 148 ].
The squares are composed of the digits 1 up to 9.
11826, 12363, 12543, 14676, 15681, 15963, 18072, 19023, 19377, 19569,
19629, 20316, 22887, 23019, 23178, 23439, 24237, 24276, 24441, 24807,
25059, 25572, 25941, 26409, 26733, 27129, 27273, 29034, 29106, 30384
Here is the complete list of the ninedigital squares
139854276, 152843769, 157326849, 215384976, 245893761, 254817369, 326597184, 361874529, 375468129, 382945761,
385297641, 412739856, 523814769, 529874361, 537219684, 549386721, 587432169, 589324176, 597362481, 615387249,
627953481, 653927184, 672935481, 697435281, 714653289, 735982641, 743816529, 842973156, 847159236, 923187456.
Here are the 53 numbers left out from the table :
These squares are composed of 9 different digits whereby zero is allowed !
10124, 10128, 10136, 10214, 10278, 12582, 12586, 13147, 13268, 13278,
13343, 13434, 13545, 13698, 14098, 14442, 14743, 14766, 15353, 16549,
16854, 17252, 17529, 17778, 17816, 20089, 20513, 20754, 21397, 21439,
21744, 21801, 21877, 21901, 22175, 22456, 23113, 23682, 23728, 23889,
24009, 25279, 26152, 26351, 27105, 27209, 27984, 28171, 28256, 28346,
28582, 28731, 29208
Note that only one of them is palindromic :
28582^{2} = 816930724
The smallest pandigital square is
32043^{2} = 1026753849
The largest pandigital square is
99066^{2} = 9814072356
Albert H. Beiler list 10 of them (from a total of 87) in his book :
"Recreations in the Theory of Numbers" [ Second Ed. Table 65 p. 148 ].
32043, 32286, 33144, 35172, 39147, 45624, 55446, 68763, 83919, 99066
Here are the 77 numbers left out from the table :
35337, 35757, 35853, 37176, 37905, 38772, 39336, 40545, 42744, 43902,
44016, 45567, 46587, 48852, 49314, 49353, 50706, 53976, 54918, 55524,
55581, 55626, 56532, 57321, 58413, 58455, 58554, 59403, 60984, 61575,
61866, 62679, 62961, 63051, 63129, 65634, 65637, 66105, 66276, 67677,
68781, 69513, 71433, 72621, 75759, 76047, 76182, 77346, 78072, 78453,
80361, 80445, 81222, 81945, 84648, 85353, 85743, 85803, 86073, 87639,
88623, 89079, 89145, 89355, 89523, 90144, 90153, 90198, 91248, 91605,
92214, 94695, 95154, 96702, 97779, 98055, 98802
David W. Wilson pointed out to me that three amongst them are palindromic :
35853^{2} = 1285437609
84648^{2} = 7165283904
97779^{2} = 9560732841
Our nine digit number 123456789 can be multiplied with 2, 4, 5, 7 and 8
to become a nine digits anagram !
x 2 = 246913578
x 4 = 493827156
x 5 = 617283945
x 7 = 864197523
x 8 = 987654312 note the reversal of the two last digits !
Our nine digit reversal number 987654321 multiplied with 2, 4, 5, 7 and 8
becomes a pandigital number (The nine familiar digits plus zero !).
x 2 = 1975308642
x 4 = 3950617284
x 5 = 4938271605
x 7 = 6913580247
x 8 = 7901234568
Topic 2.1
Find a nine digit number that gives primes
whenever any one digit is dropped
This is a mathproblem once solved by Carlos Rivera
The number 123456789 is a bad example as it produces only primes when the digits 1 and 4 are dropped.
All the others are composite.
_{1}23456789 and 123_{4}56789 ( 23456789 and 12356789 are prime ! )
The number 987654321 is also a bad candidate as it produces only primes when the digits 2 and 5 are dropped.
All the others are composite.
9876543_{2}1 and 9876_{5}4321
Carlos Rivera came to rescue me [ July 8, 1998 ]. He wrote some code and let it ran for three hours.
As a result of this he could tell me that there are no solutions whereby 9, 8 or 7 primes shows up.
The following ninedigit number is prime '6' times when one of its digits is dropped :
126874359 ( _{1}26874359, 1_{2}6874359, 126_{8}74359, 1268_{7}4359, 12687_{4}359, 1268743_{5}9 )
The table below displays all thirteen solutions.
Ninedigit Number  '6' digits droppings 
126874359  1,2,8,7,4,5 
149682573  1,4,8,2,5,7 
182746953  1,8,2,7,4,5 
192564873  1,2,5,4,8,7 
382476519  8,2,4,7,5,1 
412869537  4,1,2,8,5,7 
428965731  4,2,8,5,7,1 
453827691  4,5,8,2,7,1 
457263819  4,5,7,2,8,1 
564328791  5,4,2,8,7,1 
637154829  7,1,5,4,8,2 
694532871  4,5,2,8,7,1 
864231759  8,4,2,1,7,5 
[ April 17, 2022 ]
Thanks, Alexandru Dan Petrescu for spotting an error in the first entry of the table
whereby the digits 2 and 6 were swapped.
When 3, 6 or 9 is dropped it produces always a multiple of 3, because every ninedigit is a multiple of 9!
For reference goals and easy searching I list here all the nine & pandigitals implicitly displayed in these topics.
Topic 2.19 → 6210001000
Topic 2.18 → 2354871069
Topic 2.15 → 123456789, 987654321, 486132597, 967453182, 897651432, 562184379, 782961354, 869147532
Topic 2.14 → 142768593, 569213874, 129674583, 123456789, 123456798, 123456879, 123456897, 123456978, 123456987, 123457689, 123459876, 123465789, 123498765, 123546789, 123987654, 124356789, 129876543, 132456789, 198765432, 213456789
Topic 2.13 → 672913458
Topic 2.9 → 875319642
Topic 2.8 → 567321489, 854729316
Topic 2.7 → 816357492, 834159672
Topic 2.5 → 149263587
Contributions
David W.Wilson (email)  go to topic  [ Fri, April 10, 1998 ].
Carlos Rivera (email) from Nuevo León, México.
 go to topic 1  [ Wed, July 8, 1998 ].
 go to topic 2  [ Sun, September 11, 2005 ].
Aran Kuntze (email) from Finland  go to topic  [ March 2000 ].
B.S.Rangaswamy (email) from India  go to topic  [ Januari 2007 ].
Peter Kogel (email) from South Africa  go to topic  [ May 11, 2008 ].
Peter Kogel (email) from South Africa  go to topic  [ January 21, 2009 ].
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