World!Of Numbers  HOME plate WON  


The Nine Digits Page 2 with some Ten Digits (pandigital) exceptions  
Page 1 Page 3 Page 4 Page 5 Page 6 Page 7 
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).
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)
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 !
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.
Topic 2.22 [ September 16, 1999 ]
Two new G. L. Honaker, Jr. discoveries !
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 !

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.
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)
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).
Topic 2.18 [ November 10, 1998 ]
Puzzle : Juxtaposition of prime factors is a Nine Digits number
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...
The puzzle question is now :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...
[ 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
Carlos explains his best solution for that problem (in UBASIC)
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
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 :
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) :
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 
[ September 11, 2005 ] Investigation and a new challenge by Carlos Rivera

[ 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) :
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 531487A square that yields all the nine digits twice !
335180136^{2} = 112345723568978496
[ May 11, 2008 ] Investigation by Peter Kogel
[ January 21, 2009 ]

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.
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.
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
8  1  6 

3  5  7 
4  9  2 
Note that all the rows and columns and diagonals add up to 15 (fifteen).
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.
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
Consult the 100.000.000 digits of PI and try to locate the nine digits.
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...
Recently a NEW RECORD is set in the calculation of PI. Declared record : 51.539.600.000 decimal digits The BrouwerHeyting Sequence Table of current records for the computation of constants Here are some interesting pandigital sequences : 0123456789 : from 17.387.594.880th of pi9876543210 : from 21.981.157.633th of pi 

Topic 2.3
Curiously this sum is palindromic and repunital
Topic 2.2
From the book “The Penguin Dictionary of Curious and Interesting Numbers”
Surprising facts by David Wells
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.
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.
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!
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 → 149263587David 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 ].
[ TOP OF PAGE]