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The Nine Digits Page 2 with some Ten Digits (pandigital) exceptions
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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).

  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.

  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 9-digit 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 10-digits number
[from alt.math.recreational]

Source : “Mathematical Circus”, Martin Gardner, Penguin Books, 1979, p.128 & 135(solution)

In the 10 cells of the above figure inscribe a 10-digit 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).

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 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 9-digits number so that the decimal part of its square root also starts with a 9-digits number is :

135649728^(1/2) = 11646,876319425 736908...

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 e-mail 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

Number_Of_Digits{6666^6666} = 25490

  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 vis-à-vis the basenumbers of the two combinations.

Smallest	12921	18 + 29 + 37 + 46 + 55 + 61 + 74 + 83 + 92
Largest	389909983	15 + 26 + 32 + 41 + 58 + 64 + 73 + 87 + 99
Special	317713	17 + 28 + 32 + 49 + 56 + 61 + 73 + 85 + 94 18 + 26 + 39 + 41 + 54 + 67 + 75 + 83 + 92

  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 9-digits numbers
The smallest one is 4004 and the largest one is 437606734 ¬

  Topic 2.13  
Powers of ninedigitals

A small booklet called "Rekenraadsels" from Deltas vrije-tijd-reeks [1981 - ISBN 90-243-2545-5]
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 = 493827156²
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 0-486-23762-1, pp. 156-162]

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 !
335180136² = 112345723568978496

  Topic 2.11  
I've got this from rec.puzzles

There is a 9-digit 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 tile-layers

A tile-layer 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 tile-layer 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 9-Cell 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 !

  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 3-digit 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.

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

  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 ( ₁26874359, 1₂6874359, 126₈74359, 1268₇4359, 12687₄359, 1268743₅9 )
The table below displays all thirteen solutions.

[ 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!

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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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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