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 9-digits number so that
[A] the same 9-digits number reappears (if none exists then the one with most digit-matches)
or so that
[B] the same 9-digits 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 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
- 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 9-digits 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.
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.
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 :
- Aa + Bb + Cc + Dd + Ee + Ff + Gg + Hh + Ii = 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) :
- 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 405071317
- 19 + 28 + 37 + 46 + 55 + 64 + 73 + 82 + 91 = 11377
- 49 + 86 + 67 + 14 + 35 + 23 + 51 + 98 + 72 = 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 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 |
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 ¬
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 ?
|
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
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.
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
|
[ 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 = C2 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
- 2315976842 = 164938572 x 325196748
- Zero-less largest
- 6594187322 = 769321854 x 897542163
- Pandigital smallest
- 13789650422 = 1280467539 x 1485039276
- Pandigital largest
- 83267904512 = 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 :
- 2469135782 = 123456789
x 493827156- 617283945 = 123456789 + 493827156
- 20539147682 = 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.
| 358532 = 1285437609 |
| 234392 = 549386721 |
| 714332 = 5102673489 x |
|
272732 = 743816529 x |
| 25610873492 | | 6392518472 |
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 C2 is also doubly pandigital; i.e. it contains each digit twice: viz.
- 36729805142 = 1836490257 x 7345961028
- 36729805142 = 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
- 37095264182 = 1854763209
x 7419052836
37095264182 = 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:
| 4297315862 = 859463172 x 214865793 | |
 |
397568412 x 1590273648 = 7951368242
|
| 4935827162 = 987165432 x 246791358 | |
| |
853197642 x 3412790568 = 17063952842
|
| 13569024782 = 2713804956 x 678451239 | |
| |
932154876 x 3728619504 = 18643097522
|
| 13957026842 = 2791405368 x 697851342 | |
| |
243158796 x 972635184 = 4863175922
|
| 14590273682 = 2918054736 x 729513684 | |
| |
486315927 x 1945263708 = 9726318542
|
| 30975218642 = 6195043728 x 1548760932 | |
| |
| 2390678451 x 9562713804 = 47813569022 |
|
There are also 23 solutions where the square itself can be reversed; e.g.
| Smallest |
| 261845739 x 1047382956 = 5236914782 | |
| |
| 8741963252 = 4370981625 x 174839265 |
| Largest |
| 175264389 x 4381609725 = 8763219452 | |
| |
| 5491236782 = 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 |
|---|
| = | | = |
| 2795163842 | | 4836159722 |
|---|
|
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 ?
Visit the Palindromic Puzzle's page for a quick access to the solution.
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.
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
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
- 5672 = 321489
- and
- 8542 = 729316
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 !- Pity that the magic sums are not palindromic as well !
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
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 ?!
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...
Curiously this sum is palindromic and repunital
- 987654321 + 1 + 123456789 = 1111111111
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
- 118262 = 139854276
- The largest ninedigital square is
- 303842 = 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 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 :
- 285822 = 816930724
- The smallest pandigital square is
- 320432 = 1026753849
- The largest pandigital square is
- 990662 = 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 :
- 358532 = 1285437609
- 846482 = 7165283904
- 977792 = 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
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.
- 123456789 and 123456789 ( 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.
- 987654321 and 987654321
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.
| 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!
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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