[ March 1, 2000 ] Aran Kuntze is studying the occurrences of the nine digits numbers in the decimal expansion of .
So for instance (taken at random from her list) ¬
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 ?
[ 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 ?
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.
[ September 16, 1999 ] Two new G. L. Honaker, Jr. discoveries !
The sum of the first 10701 consecutive prime numbers is a nine digit anagram.
The sum of the first 32423 consecutive prime numbers is pandigital.
Further information in Sloane's database : A049442, A049443 and A049446. ( see also A050278, A050288, A050289 and A050290 ) in Weisstein's Math Encyclopedia Pandigital Number.
An extraordinary find !
[ December 12,1998 ] From Carlos Rivera's Puzzle 33 of his PP&P website. Add up the values (A = 1, B = 2, C = 3, etc.) of the letters of the written out numbers (in English) .
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 !
What is the 10-Digits number [from alt.math.recreational].
Source : "Mathematical Circus", Martin Gardner, Penguin Books, 1979, p.128 & 135(solution).
WWW-Links :
How to arrive at this unique solution ?
[ 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 :
The smallest pandigital 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 whosejuxtaposition is a nine digits or pandigital numbers ? [D] Are there nine digits or pandigital numbers whose prime factors when juxtaposed yieldanother nine digits or pandigital number ?
[ 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 :
[ 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.
Extracting the Nine Digits from the Number of the Beast 666 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
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 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}
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 ¬
Inspired ? I hope that the readers will find more of these transformational RECREATIONAL FACTS. For instance :
[ 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.
123456789
987654321
9876543210
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.
Powers of nine digits
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 = 493827156^{2} 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
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
0123456789 or digital 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 ! 335180136^{2} = 112345723568978496
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 zero-less 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. Zero-less solutions # 620 Pandigital solutions # 6619 Mixed solutions # 5587 Total solutions # 12826 Zero-less smallest 231597684^{2} = 164938572 x 325196748 Zero-less 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 zero-less 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 zero-less numbers. There are 19 examples where both A and B are pandigital square numbers and one example where they are zero-less squares. 35853^{2} = 1285437609 23439^{2} = 549386721 71433^{2} = 5102673489 x27273^{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 zero-less 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).
In Joseph Madachy’s "Mathematical Recreations" [ on page 159 ] he presents the following result and asks whether there are more results.
i.e. A x B = C^{2} where A, B and C are zero-less 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.
Total solutions # 12826
Zero-less smallest
Pandigital smallest
There are 40 examples where the sum of A and B is also zero-less and 404 examples where the sum is pandigital. E.g. the smallest for each type is :
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.
There are no such solutions for zero-less numbers.
There are 19 examples where both A and B are pandigital square numbers and one example where they are zero-less squares.
To the best of my knowledge these results have never been noted before. I find this surprising considering that the 30 zero-less 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.
However, the second solution of this type is the 'pièce de résistance' for I was elated to note that A + B is also pandigital.
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 159723648846327951 x x 489153672276351984 = = 279516384^{2}483615972^{2} Regards, Pete Kogel
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:
There are also 23 solutions where the square itself can be reversed; e.g.
Undoubtedly best of all is the following remarkable result where each ninedigital number is reversed (unique case):
Regards, Pete Kogel
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 ?
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 broken 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 ?
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.
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 ¬
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 ¬
Visit WONplate 9 and WONplate 14 to see two beautiful Magic Squares constructed only with palindromic numbers !
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.
From "Dictionary of Curious and Interesting Numbers" 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 ?!
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...
Here are some interesting pandigital sequences : 0123456789 : from 17.387.594.880-th of pi 9876543210 : from 21.981.157.633-th of pi
Curiously this sum is palindromic and repunital ¬
Surprising facts appeared in David Wells' book "The Penguin Dictionary of Curious and Interesting Numbers" ¬
This is another mathproblem recently solved by Carlos Rivera. Find a nine digit number that gives primes whenever any one digit is dropped.
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, 12_{6}874359, 126_{8}74359, 1268_{7}4359, 12687_{4}359, 1268743_{5}9 ) The table below displays all thirteen solutions.
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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