- 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).
Fourth Page
[ July 4, 2005 ]
More Ninedigital Powers
by Peter Kogel
" Hi Patrick,
Herewith is another pandigital 'project' that I investigated some while ago.
Notable results include:
213452 + 87962 = 532978641 |
The first term contains 1...5 and the second term contains 6...9.
25382 + 176492 = 29432 + 175862 = 317928645 |
Only example where two sums are equal.
5393 + 2463 + 7183 = 541623987 5393 + 4623 + 7183 = 625348179 |
The above is what I call a 'pandigital amicable pair' (see below).
274 + 354 + 684 + 1494 = 516297843 274 + 354 + 864 + 1494 = 549617283 |
Only one term on the left hand side is different.
95 + 175 + 265 + 435 + 585 = 816725493 |
Only example for 5th power.
Kind regards,
Pete Kogel. "
Pandigital diversions concerning
a1P + a2P + ... + apP = N [ P ⩾ 2 ]
P = 2
A2 + B2 = N
E.g.:
15962 + 248372 = 619423785
In the above, the numbers A & B combined and also the result on the right hand side
contain the digits 1...9 (zero-less pandigital).
There are 78 such solutions ¬
A B | A B | A B | A B |
1596 24837
1635 29487
1875 24693
2367 19485
2436 18975
2538 17649
2784 15369
2943 17586
3495 12768
3516 28479
3654 12897
3846 29175
3864 25179
3915 28467
3984 26715
4197 25638
4359 16278
4395 27186
4536 12987
4698 21357 |
4857 21396
4896 21753
4965 17382
4968 15237
5367 14928
5376 12984
5376 18429
5436 17289
5469 13728
5496 18237
5613 27948
5679 14238
5697 21384
5763 21948
5964 18273
5976 21483
6189 25374
6198 23547
6345 21987
6453 19287 |
6459 17328
6471 23589
6471 29538
6528 13749
6549 27831
6813 24597
6831 29745
6945 13782
6975 14823
7164 25983
7356 24189
7359 28146
7431 29658
7458 23619
7659 13248
7659 13428
7824 19356
7863 21495
7869 23514
7953 16824 |
8259 13476
8295 14763
8349 17562
8352 17469
8517 29643
8745 16392
8793 12546
8796 21345
8796 23541
8934 15267
8973 12546
8976 25413
9156 27483
9168 23475
9627 18543
9687 24513
9735 24861
9741 26358 |
Observations
There are only 5 examples where A or B appears in more than one solution: viz ¬
53762 + 129842 = 197485632
53762 + 184292 = 368529417
64712 + 235892 = 598314762
64712 + 295382 = 914367285
76592 + 132482 = 234169785
76592 + 134282 = 238971465
87962 + 213452 = 532978641
87962 + 235412 = 631548297
87932 + 125462 = 234718965
89732 + 125462 = 237916845
There are several examples where simply swapping two digits in A leads to a member of another solution ¬
E.g.:
4359
4395
5469
5496
5679
5697
Etc.
213452 + 87962 = 532978641
In the solution above, A contains the digits 1...5 and B contains the digits 6...9
25382 + 176492 = 29432 + 175862 = 317928645
The above is the only example where N is duplicated.
P = 3
A3 + B3 + C3 = N
E.g.:
1233 + 4653 + 8973 = 824139765
A, B & C combined and N are zero-less pandigital.
There are 101 solutions; viz ¬
A B C | A B C | A B C | A B C |
123 465 897
124 389 576
124 635 789
124 639 875
126 593 784
128 634 759
129 375 468
129 365 847
132 479 865
135 297 684
136 582 749
137 286 954
138 259 746
138 264 795
139 576 824
142 396 857
145 297 863
145 296 873
145 693 728
146 397 825
147 265 983
149 258 637
153 469 728
156 492 738
159 423 768
162 354 789
|
165 324 789
165 234 978
167 532 894
172 348 596
175 629 843
176 549 832
178 426 593
182 379 564
184 527 936
186 395 742
189 246 537
192 548 763
192 356 847
193 468 572
194 285 736
194 376 825
194 627 835
196 273 458
196 342 785
197 364 825
198 243 675
213 467 589
213 578 649
215 378 496
216 397 485
216 387 495
|
219 438 567
219 436 587
231 456 789
234 619 857
237 654 819
246 539 718
249 381 675
251 486 739
253 419 768
254 391 768
258 314 697
259 641 837
261 457 938
267 384 951
269 537 841
271 398 546
271 458 639
274 591 863
278 351 496
279 413 856
312 479 658
317 489 526
318 547 692
321 498 576
321 564 897
328 519 674
|
342 596 871
351 468 792
362 459 718
372 519 864
374 526 918
378 512 649
381 469 572
386 519 724
394 561 728
397 458 621
398 415 672
418 693 725
429 576 831
429 671 835
462 539 718
483 597 621
487 521 693
489 613 725
493 581 627
493 512 786
514 639 782
574 639 812
591 624 738
|
While looking through the above list I noticed that some values occur in more than one solution
and I wondered whether I could chain several such solutions together; wherein the last number
becomes the first number in the next link: e.g. ¬
(123, 465, 897)
(897, 321, 564)
(564, 182, 379)
A slightly longer chain would be ¬
(193, 468, 572)
(572, 381, 469)
(469, 153, 728)
(728, 693, 145)
(145, 296, 873)
The longest chain that I found consists of 24 members ¬
(172, 348, 596) , (596, 871, 342) , (342, 785, 196) , (196, 273, 458) , (458, 271, 639) ,
(639, 875, 124) , (124, 389, 576) , (576, 831, 429) , (429, 671, 835) , (835, 194, 627) ,
(627, 581, 493) , (493, 786, 512) , (512, 649, 378) , (378, 215, 496) , (496, 278, 351) ,
(351, 792, 468) , (468, 193, 572) , (572, 381, 469) , (469, 153, 728) , (728, 145, 693) ,
(693, 418, 725) , (725, 613, 489) , (489, 317, 526) , (526, 918, 374)
This chain was found by 'hand' so it is possible that I may have missed a longer chain.
While searching for the above I came across the following nine-member 'pandigital sociable group'
i.e. the last member can be chained with the very first member.
 |
(124, 389, 576)
(576, 831, 429)
(429, 671, 835)
(835, 627, 194)
(194, 376, 825)
(825, 146, 397)
(397, 621, 458)
(458, 271, 639)
(639, 875, 124)
|
---|
There is also one example of a 'pandigital amicable pair'.
(539, 246, 718)
(718, 462, 539)
Other notable curiosities
(123, 564, 879) & (124, 639, 875)
| | | | | |
(231, 465, 897) & (142, 396, 857)
The digits in
A,
B &
C are permuted.
(231, 456, 789)
B &
C contain the series 456_789 in order.
(394, 561, 728) = (394, 394+167, 394+167+167)
A,
B,
C are in arithmetic progression (difference = 167).
P = 4
A4 + B4 + C4 + D4 = N
E.g.:
244 + 694 + 784 + 1354 = 824139765
A, B, C & D combined and N are zero-less pandigital.
There are only 5 solutions; viz ¬
244 + 694 + 784 + 1354 = 392164578
274 + 354 + 684 + 1494 = 516297843
274 + 354 + 864 + 1494 = 549617283
484 + 574 + 964 + 1234 = 329685714
644 + 724 + 854 + 1394 = 469152738
The second and third solutions shown above are particularly noteworthy (another 'pandigital amicable pair').
P = 5
A5 + B5 + C5 + D5 + E5 = N
95 + 175 + 265 + 435 + 585 = 816725493
The above is the only solution.
P ⩾ 6
AP + BP + CP + DP + EP + FP + ... = N
No solutions.
More Topics
[ March 24, 2005 ]
The Unique Pandigital 3816547290
by John Morse
" In the header of this message is 3,816,547,290 which is the only 10-digit
pandigital number whose first N digits are a multiple of N. I first discovered
that in The Dictionary of Curious and Interesting Numbers by David Wells,
and I wrote an explanation of how one could figure out that is the number.
John Morse
Albany New York USA "
Mathematics Puzzle
Find the one and only 10-digit number such that :
1) All digits are different, and
2) the number formed by the first N digits is divisible by N.
This explanation is my own, but I first learned about the amazing
mystery number (381,654,729) in an entertaining book, one of my ten
favorite math-themed books of all time :
Wells, David. "The Penguin Book of Curious and Interesting Numbers"
(New York, New York: Penguin Putnam, Inc., 1997, page 185).
Procedure
Let the letters A thru J stand for the digits of the number. Then :
A B C D E F G H I J is the mystery number, whereas :
AB is evenly divisible by 2,
ABC is evenly divisible by 3,
ABCD is evenly divisible by 4,
ABCDE is evenly divisible by 5,
ABCDEF is evenly divisible by 6,
ABCDEFG is evenly divisible by 7,
ABCDEFGH is evenly divisible by 8,
ABCDEFGHI is evenly divisible by 9,
ABCDEFGHIJ is evenly divisible by 10.
A number is evenly divisible by 10 only if its last digit is zero.
Hence, we have found right away that J must equal zero, since the
ten-digit number ends in that digit J.
What about divisibility by 9 ? This particular number takes care of that
matter because its nine-digit 'fragment' ABCDEFGHI must use one each of
digits 1 thru 9, and the sum of the nine digits is 45, which is a multiple of 9.
A number is evenly divisible by 9 if the sum of its digits is 9 or a
multiple of 9 itself. Hence, ANY arrangement of the digits 1 thru 9
(or 0 thru 9), each digit used exactly once, will always result in the
number being a multiple of 9.
A number is evenly divisible by 5 only if its last digit is a five or
a zero.
Since all the digits of ABCDEFGHIJ are different, we cannot use zero
since J already has that value.
The first five digits of the entire number are ABCDE, hence digit E
must be 5.
So far, the mystery number looks like this :
A B C D 5 F G H I 0
Even numbers only are evenly divisible by even divisors, e.g., if you
want to divide some number N by 2 or 4 or 6, etc., that number N must
be even since dividing an odd number by an even number will always leave
(an odd) remainder.
Hence, consider the mystery number's even-numbered digits. They
correspond to letters B, D, F, and H. Since zero is being used for J, the
even digits 2, 4, 6, and 8 must correspond to those other four letters.
Therefore, the remaining letters in the number - A, C, G, and I - must
stand for odd digits.
Because the odd digit 5 is already in use via the letter E, A can only
be 1, 3, 7, or 9. The same restrictions apply to C, G, and I.
There are many ways that AB is evenly divisible by 2 : merely let B
equal 2 or 4 or 6 or 8. Hence, there are too many possible ways so
far to assign digits for A and B. How about ABC, which must be a
multiple of 3 ?
The digits A + B + C must add up to a multiple of 3, and in this
instance, B is the only even digit. Hence, the following trios are possible
for fragment ABC :
123 129 147 183 189 321 327 369 381
387 723 729 741 783 789 921 927 963
Eighteen possible permutations exist for ABC; it looks like a lot of
trial and error lies ahead if we test all these 3-digit groups in
combination with the remaining digits for D, F, G, and H. (It does not
matter yet what digit I must be, since the entire 9-digit number up to I
is already divisible by 9.)
However, note that the four-digit fragment ABCD must be a multiple of 4,
and this is possible only if the number formed by the last two digits -
here, CD - is a multiple of 4.
Since C is odd, D can only be 2 or 6. If we try 4 or 8 instead for D, then we
get numbers such as 18, 34, 78, or 94 - none of which are evenly divisible by 4.
A similar situation arises for the 8-digit fragment ABCDEFGH. Since G
is odd, H can only be 2 or 6. A number is divisible by 8 only if its
last three digits (here, FGH) is evenly divisible by 8.
So, between them, D and H use up the digits 2 and 6. We are not sure
which letter will have which digit, but we can eliminate the 3-digit
fragments above where B happens to be 2 or 6, leaving these possible
permutations for the digit group ABC :
147 183 189 381 387 741 783 789
Good. Number of permutations is now only eight instead of eighteen.
Since B and F must be even, they must use the only remaining even
digits - 4 and 8 - between them. Let's take a look at the mystery
number where we assign the digits 2 and 6 to D and H and the digits
4 and 8 to B and F :
A4C258G6I0
A4C658G2I0
A8C254G6I0
A8C654G2I0
Consider the six-digit fragment ABCDEF. Its digits must add up to a multiple
of 3. All multiples of six are even, but we already know that F must be even.
Since the digits ABC add up to a multiple of 3, so must those in fragment
DEF. Since E is 5, only two possibilities exist for digits in DEF :
258 654
So, we can eliminate two of the four 10-digit numbers above, leaving :
A4C258G6I0
A8C654G2I0
Let's take another look at the 8-digit fragment ABCDEFGH and see if we
can determine whether F must be 4 or 8.
If D=2, then B must be 4, F must be 8, and H must be 6. Since FGH must
be a multiple of 8, it could only end in the following three digits :
816 836 876 896
Of these four 3-digit numbers, 836 and 876 are NOT multiples of 8, so
eliminate them, leaving only 816 and 896 for FGH.
If D=6, then B must be 8, F must be 4, and H must be 2. Since FGH must
be a multiple of 8, it could only end in the following three digits :
412 432 472 492
Of these four 3-digit numbers, 412 and 492 are NOT multiples of 8, so
eliminate them, leaving only 432 and 472 for FGH.
Hence, possible arrangements of digits in the mystery number could be :
[1] A4C25816I0
[2] A4C25876I0
[3] A8C65432I0
[4] A8C65472I0
Recall that fragment ABC must be a multiple of 3. In choice [1], the
only available digits for A and C are 3, 7, and 9. No combination of
these digits will enable ABC to be a multiple of 3, so eliminate choice [1].
For choice [2], the only digits available are 1, 3, and 9. No combination
of these will make ABC a multiple of 3, so scratch choice [2].
For choice [3], the only digits left are 1, 7, and 9. The only workable
combination that will make ABC a multiple of 3 is when neither A nor C
is 7, so I must be 7.
For choice [4], the only digits available are 1, 3, and 9. Digits
1 and 3 may represent A and C, in which case I must be 9. If digits
1 and 9 stand in for A and C, then I must be 3.
So, two choices for the mystery number were removed, leaving these
possible configurations for the mystery number :
[1] 1896543270
[2] 9816543270
[3] 1836547290
[4] 3816547290
[5] 1896547230
[6] 9816547230
Almost there! Each of these six numbers is such that the number AB is
divisible by 2, ABC by 3, ABCD by 4, ABCDE by 5, ABCDEF by 6, ABCDEFGH
by 8, ABCDEFGHI by 9, and ABCDEFGHIJ by 10.
What about the fragment ABCDEFG ? Is it divisible by 7 ? Test each of
the six numbers above using their first seven digits :
[1] 1896543 divided by 7 leaves remainder 5. nope.
[2] 9816543 divided by 7 leaves remainder 2. nope.
[3] 1836547 divided by 7 leaves remainder 6. nope.
[4] 3816547 divided by 7 leaves remainder 0.
YES !
[5] 1896547 divided by 7 leaves remainder 2. nope.
[6] 9816547 divided by 7 leaves remainder 6. nope.
Aha! Only choice [4] leaves no remainder when its fragment ABCDEFG is
divided by 7.
Finally we found the mystery number ! It is
Are all the digits different ? Yes. A number with that property is
known as "pandigital". The prefix "pan-" means "all".
How about divisibility of the first N digits by N ? Check and see :
38 divided by 2 = 19, no remainder. Yawn.
381 divided by 3 = 127, no remainder. Okay.
3816 divided by 4 = 954, no remainder. Okay!
38165 divided by 5 = 7633, no remainder. Big deal.
381654 divided by 6 = 63609, no remainder. Good.
3816547 divided by 7 = 545221, no remainder. Good!
38165472 divided by 8 = 4770684, no remainder. Great!
381654729 divided by 9 = 42406081, no remainder. Of course.
3816547290 divided by 10 = 381654729, no remainder. Wonderful!!!
There is no other 10-digit number whose first N digits are evenly
divisible by N - and where every digit occurs once !
Other internet sources discussing this topic :
http://ken.duisenberg.com/potw/archive/arch96/960919sol.html
http://www.nrich.maths.org.uk/public/viewer.php?obj_id=796&part=solution&refpage=viewer.php
http://begghilos2.ath.cx/~jyseto/Academia/Math-Problem-2.php
http://www.rodoval.com/heureka/probsnum.html
... many more when 3816547290 is entered as keyword in GOOGLE for instance !
[ February 27, 2005 ]
Nine and Ten digit squares
by Peter Kogel
" I've had a fascination for numbers and rec maths ever since I first read
Martin Gardner's column in Scientific American many years ago. My particular
fascination is for nine and ten digit number patterns and I've found your
excellent website the source of much inspiration.
This project has been to investigate patterns along the lines of 99066^2
( 9814072356 ) and have found some interesting results that you might like.
Some highlights
77772277772 = 60485271895340361729 |
The square contains each digit 0...9 twice and the square root is a rather
particular palindrome.
42539071862 = 1809572634_7102438596 |
The lefthand side is pandigital and the square on the righthand side contains
the digits 0...9 repeated in each half (there are two other such examples).
Nine and Ten digit squares
It is well known that the square of 11826 ( 139854276 ) contains all the digits 1...9
and that the square of 32043 ( 1026753849 ) contains all the digits 0...9.
Others have investigated cases where the digits in the square are repeated more than once.
I have extended this investigation up to the 6th repeat of the digits. My findings are shown below.
N | D | Comments |
11826 [ A071519 ]
30384
335180136
999390432
10546200195312
31621017808182
333350001269641272
999994443856900365
10540978243301566001337
31622759033293797517068
394589436883062505110868355361 |
1
1
2
2
3
3
4
4
5
5
6 |
Smallest
Largest
Smallest
Largest
Smallest
Largest
Smallest
Largest
Smallest
Largest
Probably not the smallest |
Where D = The number of times the 9 digits (1...9) are repeated in N squared |
N | D | Comments |
32043 [ A054038 ]
99066
3164252736
9994363488
316245509988426
999944387118711
31622952459028694643
99999444387327303945
3162279417919838932896672
9999994444387345066672935
316227783585222352038673081356
> 9.9999994 x 10^29 |
1
1
2
2
3
3
4
4
5
5
6
6
|
Smallest
Largest
Smallest
Largest
Smallest
Largest
Smallest
Largest
Smallest
Largest
Smallest
Largest |
Where D = The number of times the 10 digits (0...9) are repeated in N squared |
I haven't yet found the largest example where all ten digits are repeated 6 times.
Perhaps someone else can find it and perhaps extend the table even further.
While searching for the above I came across the following interesting pair:
A = 3162455132903162 = 100011224676255433788499379856
B = 3162455165903162 = 100011226763475832394584979856
A and B differ by only two digits!
Pandigital squares
It is well known that the squares of A = 57321 and B = 60984 each contain all ten digits
and that 'A & B' combined also contains the ten digits. I wondered whether such results
could be extended such that 'A & B' contain the digits repeated twice. I very soon found
literally hundreds of solutions of which the following type is of interest:
A = 41514322532 = 17234389751248656009
B = 99860760782 = 99721715435603862084
Note that the squares of 'A' & 'B' each contain all ten digits repeated twice,
'A' contains no digit larger than 5 and 'A & B' combined contains all ten digits repeated twice.
There are many other such examples.
There are also many examples where 'A' and 'B' are pan-digital. E.g.:
A = 31754620892 = 10083559478676243921
B = 31758042692 = 10085732754998624361
Similarly the following are pan-9-digital
A = 3459186722 = 119659727638243584
B = 3519876242 = 123895287449165376
While searching for these examples I came across the following remarkable trio:
42539071862 = 1809572634_7102438596
52960318742 = 2804795361_0423951876
64320159872 = 4137082965_7023584169
Notice that the digits 0...9 are contained in each 'half' of the square
on the right hand side. There are no pan-9-digital equivalents.
[ Tom Marlow discovered only the first two squares. He submitted it to Ed Pegg's
Mathpuzzle site. Source
material added 26 January 2003.]
Complete listing of the above squared nine- and pandigitals
with the higher powers as well,
up to the fifth for ninedigitals
up to the sixth for pandigitals
available at http://web.archive.org/web/20080708203024/http://blue.kakiko.com/mmrmmr/htm/eqtn11.html
Palindromes
There are many examples where N is a palindrome.
358532 = 1285437609
846482 = 7165283904
977792 = 9560732841
3819991832 = 145923375812667489
9624942692 = 926395217857844361
44025520442 = 19382464500128577936
62652256262 = 39253052144687091876
77772277772 = 60485271895340361729
92946649292 = 86390796142382575041
117248448427112 = 137471986585646734329829521
118991441998112 = 141589632687895763492435721
125672112765212 = 157934799268716582325863441
185883663885812 = 345527364996127848287193561
189171661719812 = 357859175978342281269464361
239059229509322 = 571493152135897342879668624
240623113260422 = 578994826351369112427385764
242481661842422 = 587973563298617234289114564
260568008650622 = 678956871321495791532263844
278618448168722 = 776282396599457131215864384
287203223027822 = 824856913175677163284939524
3194361316349132 = 102039442193877465886320517569
3195236263259132 = 102095347780461683147295283569
3198000600089132 = 102272078381704355869639441569
Etc.
I have not found any solutions where the square is also a palindrome
and I suspect that none exist.
Higher powers
It is natural to ask whether the above results can be extended to higher powers.
The answer is of course "yes!"
E.g.
7205585 = 194242706843325709850513196768
(digits 0...9 repeated 3 times)
In the following, the notation (N, P, D) means that the digits 1...9 (or 0...9)
are repeated 'D' times in the value of NP.
Nine digit examples:
(N, 3, 2) has 2 solutions | N = 496536, 982617 |
(N, 4, 2) has 1 solution | N = 24267 |
(N, 4, 3) has 8 solutions | N = 3374532, 3928791, 4143474, 4552878, 4714896, 4796571, 4905006, 5577408 |
(N, 5, 4) has 2 solutions | N = 12036828, 12343788 |
Ten digit examples: [ A074205 ]
(N, 3, 2) has 138 solutions | N = 2158479, 2190762, 2205528, 2219322, ... ..., 4631793 |
(N, 4, 2) has 4 solutions | N = 69636, 70215, 77058, 80892 |
(N, 5, 3) has 7 solutions | N = 643905, 680061, 720558, 775113, 840501, 878613, 984927 |
(N, 6, 4) has ? solutions | N = 3470187, ... Source: OEIS |
(N, 7, 4) has 2 solutions | N = 421359, 493107 |
(N, 8, 5) has ? solutions | N = 1472157, ... Source: OEIS |
(N, 9, 5) has 1 solution | N = 320127 |
(N, 9, 6) has 1 solution | N = 3976581 |
(N, 10, 8) has ? solutions | N = 81785058, ... ..., 95927037 |
(N, 11, 8) has 1 solution | N = 15763347 |
(N, 12, 9) has 1 solution | N = 31064268 |
(N, 13,10) has 2 solutions | N = 44626422, 44695491 |
(N, 14, 12) has 2 solutions | N = 330096453, 346527657 |
(N, 15, 12) has 1 solution | N = 85810806 |
The challenge above was to make 'D' as small as possible for any particular 'P'.
I also have constructed a table of smallest and largest of any particular type
up to and including P = 15.
There are also examples where P = D in viz:
Nine digit examples:
(481514667, 3, 3) Smallest | (998782725, 3, 3) Largest |
(590291892, 4, 4) Smallest | (998709318, 4, 4) Largest |
(653813013, 5, 5) Smallest | (989543343, 5, 5) Largest |
(853836318, 6, 6) Smallest | (897702789, 6, 6) Largest |
I have not yet found any example where P = D = 7 and I suspect there are none
because it becomes increasingly unlikely that the power of any number will not
contain a zero when P is large.
Ten digit examples:
(4642110594, 3, 3) Smallest | (9999257781, 3, 3) Largest |
(5623720662, 4, 4) Smallest | (9999112926, 4, 4) Largest |
(6312942339, 5, 5) Smallest | (9995722269, 5, 5) Largest |
(6813614229, 6, 6) Smallest | (9999409158, 6, 6) Largest |
(7197035958, 7, 7) Smallest | (9998033316, 7, 7) Largest |
(7513755246, 8, 8) Smallest | (9993870774, 8, 8) Largest |
(7747685775, 9, 9) Smallest | (9986053188, 9, 9) Largest |
(7961085846, 10, 10) Smallest | (9964052493, 10, 10) Largest |
(8120306331, 11, 11) Smallest | (9975246786, 11, 11) Largest |
(8275283289, 12, 12) Smallest | (9966918135, 12, 12) Largest |
(8393900487, 13, 13) Smallest | (9938689137, 13, 13) Largest |
(8626922994, 14, 14) Smallest | (9998781633, 14, 14) Largest |
(8594070624, 15, 15) Smallest | (9813743148, 15, 15) Largest |
I have not checked any power larger than 15 as yet.
Pandigital powers
There are a few examples where 'N' is pandigital.
Complete listing available at http://web.archive.org/web/20080708203024/http://blue.kakiko.com/mmrmmr/htm/eqtn11.html
Nine digit examples:
(345918672, 2, 2) Smallest | (976825431, 2, 2) Largest | # 28 |
John Morse's list that his computer program generated for
those # 28 order-2 ninedigital squares [ April 4, 2008 ]
345918672^2 = 119659727638243584
351987624^2 = 123895287449165376
359841267^2 = 129485737436165289
394675182^2 = 155768499286733124
429715863^2 = 184655722913834769
439516278^2 = 193174558626973284
487256193^2 = 237418597616853249
527394816^2 = 278145291943673856
527498163^2 = 278254311968374569
528714396^2 = 279538912537644816
572493816^2 = 327749169358241856
592681437^2 = 351271285764384969
729564183^2 = 532263897116457489
746318529^2 = 556991346728723841
749258163^2 = 561387794822134569
754932681^2 = 569923352841847761
759142683^2 = 576297613152438489
759823641^2 = 577331965422496881
762491835^2 = 581393798441667225
783942561^2 = 614565938947238721
784196235^2 = 614963734988175225
845691372^2 = 715193896675242384
891357624^2 = 794518413862925376
914863275^2 = 836974811943725625
915786423^2 = 838664772551134929
923165487^2 = 852234516387947169
928163754^2 = 861487954239372516
976825431^2 = 954187922648335761
|
(516473892, 3, 3) Smallest | (751396842, 3, 3) Largest | # 5 |
(N, 4, 4) No solutions | # 0 |
(961527834, 5, 5) only 1 solution | # 1 |
Ten digit examples:
(3175462089, 2, 2) Smallest | (9876124053, 2, 2) Largest | # 534 |
(4680215379, 3, 3) Smallest | (9863527104, 3, 3) Largest | # 74 |
(5702631489, 4, 4) Smallest | (9846032571, 4, 4) Largest | # 13 |
(7351062489, 5, 5) Smallest | (9847103256, 5, 5) Largest | # 8 |
(7025869314, 6, 6) Smallest | (9247560381, 6, 6) Largest | # 6 |
There do not appear to be any examples for higher powers.
A curious pan-digital example is the following
9758041263 = 929154533715687684842912376
Notice that the left hand side including the index contains the digits 0...9 and that the right hand side
contains the digits 1...9 repeated 3 times. I have found no other such example for any power.
Palindromes
To date I haven't searched for palindrome solutions for cubes and higher powers
but I have no reason to believe that they do not exist. I'll investigate further when
I have a bit more time.
[ May 18, 2005 ]
Nine and Ten digit powers
More results from Peter Kogel
Here are some more results from Peter Kogel which can be added as an addendum :
333333500000125024109149626 squared, contains the digits 1..9 repeated six times (smallest)
111111222222333349447774470014236868657845737074239129
999999944444387333120727861 squared, contains the digits 1..9 repeated six times (largest)
999999888888777752667554514316632321351744642421635321
3162277835852223520386730813562 contains digits 0..9 repeated six times (smallest)
100000011111222223236955438883649634757986945845567794798736
9999999444443873456035583500562 contains digits 0..9 repeated six times (largest)
999999888888777777633214105446041204560215564562321035203136
32541967082 = 10589_7962143580_37264 digits 0..9 contained in the centre and each digit repeated twice overall
85916740232 = 73816_8625174930_04529 ditto
89520631742 = 80139_4350712869_54276 ditto
90761483522 = 82376_4689075123_15904 ditto
96831047522 = 93762_5176382049_81504 ditto
97058431262 = 94203_3907865214_51876 ditto
43259073 = 80952_7368052419_17643 digits 0..9 contained in the centre (only cubic example)
869122976116 contains the digits 0..9 repeated sixteen times (smallest)
997090225216 contains the digits 0..9 repeated sixteen times (largest)
880038967817 contains the digits 0..9 repeated seventeen times (smallest)
974038376717 contains the digits 0..9 repeated seventeen times (largest)
998581978520 =
972018388122736146908354492278598571766247280628254566407260946047011341
594668735384194513571360871003364967097249794743843109249267940157739156
83917180391558815561723889985220630315208530426025390625
Ie. the 200 digit number contains each digit 0..9 repeated 20 times!
This is the largest such example.
" I used Yuji Kida's excellent UBASIC for all my calculations. The
program I wrote is not very elegant (I am only an amateur programmer)
and the algorithm I developed is probably far from optimal. Indeed,
from what I have learned from this project I would probably tackle it
from a completely different angle if I were ever to do it again. "
[ December 29, 2004 ]
Solutions for ABC + DEF = GHI
Linda Rojewski
Linda Rojewski asked for all the combinations of a three digit number added
to a three digit number to equal a three digit number. You can only use digits 1-9,
and can not use any number twice. You can add to carry over a number of which
this number would not count as usage of that number.
Solution:
I (pdg) used the well known UBASIC tool and the result is listed below.
In total there are 336 solutions though not all are unique!
124 + 659 = 783 is listed as well as 659 + 124 = 783.
Question:
From this list it should be easy to extract only the distinct solutions.
How many do you count ?
Send them in and I will highlight them.
1 124659783 124 + 659 = 783
2 125739864 125 + 739 = 864
3 127359486 127 + 359 = 486
4 127368495 127 + 368 = 495
5 128367495 128 + 367 = 495
6 128439567 128 + 439 = 567
7 129357486 129 + 357 = 486
8 129438567 129 + 438 = 567
9 129654783 129 + 654 = 783
10 129735864 129 + 735 = 864
11 134658792 134 + 658 = 792
12 135729864 135 + 729 = 864
13 138429567 138 + 429 = 567
14 138654792 138 + 654 = 792
15 139428567 139 + 428 = 567
16 139725864 139 + 725 = 864
17 142596738 142 + 596 = 738
18 142695837 142 + 695 = 837
19 143586729 143 + 586 = 729
20 145692837 145 + 692 = 837
21 146583729 146 + 583 = 729
22 146592738 146 + 592 = 738
23 152487639 152 + 487 = 639
24 152784936 152 + 784 = 936
25 154629783 154 + 629 = 783
26 154638792 154 + 638 = 792
27 154782936 154 + 782 = 936
28 157329486 157 + 329 = 486
29 157482639 157 + 482 = 639
30 158634792 158 + 634 = 792
31 159327486 159 + 327 = 486
32 159624783 159 + 624 = 783
33 162387549 162 + 387 = 549
34 162783945 162 + 783 = 945
35 163782945 163 + 782 = 945
36 167328495 167 + 328 = 495
37 167382549 167 + 382 = 549
38 168327495 168 + 327 = 495
39 173286459 173 + 286 = 459
40 173295468 173 + 295 = 468
41 175293468 175 + 293 = 468
42 176283459 176 + 283 = 459
43 182367549 182 + 367 = 549
44 182394576 182 + 394 = 576
45 182457639 182 + 457 = 639
46 182493675 182 + 493 = 675
47 182754936 182 + 754 = 936
48 182763945 182 + 763 = 945
49 183276459 183 + 276 = 459
50 183492675 183 + 492 = 675
51 183546729 183 + 546 = 729
52 183762945 183 + 762 = 945
53 184392576 184 + 392 = 576
54 184752936 184 + 752 = 936
55 186273459 186 + 273 = 459
56 186543729 186 + 543 = 729
57 187362549 187 + 362 = 549
58 187452639 187 + 452 = 639
59 192384576 192 + 384 = 576
60 192483675 192 + 483 = 675
61 192546738 192 + 546 = 738
62 192645837 192 + 645 = 837
63 193275468 193 + 275 = 468
64 193482675 193 + 482 = 675
65 194382576 194 + 382 = 576
66 195273468 195 + 273 = 468
67 195642837 195 + 642 = 837
68 196542738 196 + 542 = 738
69 214569783 214 + 569 = 783
70 214659873 214 + 659 = 873
71 215478693 215 + 478 = 693
72 215748963 215 + 748 = 963
73 216378594 216 + 378 = 594
74 216738954 216 + 738 = 954
75 218349567 218 + 349 = 567
76 218376594 218 + 376 = 594
77 218439657 218 + 439 = 657
78 218475693 218 + 475 = 693
79 218736954 218 + 736 = 954
80 218745963 218 + 745 = 963
81 219348567 219 + 348 = 567
82 219438657 219 + 438 = 657
83 219564783 219 + 564 = 783
84 219654873 219 + 654 = 873
85 234657891 234 + 657 = 891
86 235746981 235 + 746 = 981
87 236718954 236 + 718 = 954
88 236745981 236 + 745 = 981
89 237654891 237 + 654 = 891
90 238419657 238 + 419 = 657
91 238716954 238 + 716 = 954
92 239418657 239 + 418 = 657
93 241596837 241 + 596 = 837
94 243576819 243 + 576 = 819
95 243675918 243 + 675 = 918
96 245673918 245 + 673 = 918
97 245718963 245 + 718 = 963
98 245736981 245 + 736 = 981
99 246573819 246 + 573 = 819
100 246591837 246 + 591 = 837
101 246735981 246 + 735 = 981
102 248319567 248 + 319 = 567
103 248715963 248 + 715 = 963
104 249318567 249 + 318 = 567
105 251397648 251 + 397 = 648
106 254619873 254 + 619 = 873
107 254637891 254 + 637 = 891
108 257391648 257 + 391 = 648
109 257634891 257 + 634 = 891
110 259614873 259 + 614 = 873
111 264519783 264 + 519 = 783
112 269514783 269 + 514 = 783 |
113 271593864 271 + 593 = 864
114 271683954 271 + 683 = 954
115 273186459 273 + 186 = 459
116 273195468 273 + 195 = 468
117 273546819 273 + 546 = 819
118 273591864 273 + 591 = 864
119 273645918 273 + 645 = 918
120 273681954 273 + 681 = 954
121 275193468 275 + 193 = 468
122 275418693 275 + 418 = 693
123 275643918 275 + 643 = 918
124 276183459 276 + 183 = 459
125 276318594 276 + 318 = 594
126 276543819 276 + 543 = 819
127 278316594 278 + 316 = 594
128 278415693 278 + 415 = 693
129 281394675 281 + 394 = 675
130 281673954 281 + 673 = 954
131 283176459 283 + 176 = 459
132 283671954 283 + 671 = 954
133 284391675 284 + 391 = 675
134 286173459 286 + 173 = 459
135 291357648 291 + 357 = 648
136 291384675 291 + 384 = 675
137 291546837 291 + 546 = 837
138 291573864 291 + 573 = 864
139 293175468 293 + 175 = 468
140 293571864 293 + 571 = 864
141 294381675 294 + 381 = 675
142 295173468 295 + 173 = 468
143 296541837 296 + 541 = 837
144 297351648 297 + 351 = 648
145 314658972 314 + 658 = 972
146 316278594 316 + 278 = 594
147 317529846 317 + 529 = 846
148 317628945 317 + 628 = 945
149 318249567 318 + 249 = 567
150 318276594 318 + 276 = 594
151 318627945 318 + 627 = 945
152 318654972 318 + 654 = 972
153 319248567 319 + 248 = 567
154 319527846 319 + 527 = 846
155 324567891 324 + 567 = 891
156 324657981 324 + 657 = 981
157 327159486 327 + 159 = 486
158 327168495 327 + 168 = 495
159 327519846 327 + 519 = 846
160 327564891 327 + 564 = 891
161 327618945 327 + 618 = 945
162 327654981 327 + 654 = 981
163 328167495 328 + 167 = 495
164 328617945 328 + 617 = 945
165 329157486 329 + 157 = 486
166 329517846 329 + 517 = 846
167 341586927 341 + 586 = 927
168 342576918 342 + 576 = 918
169 346572918 346 + 572 = 918
170 346581927 346 + 581 = 927
171 348219567 348 + 219 = 567
172 349218567 349 + 218 = 567
173 351297648 351 + 297 = 648
174 352467819 352 + 467 = 819
175 354618972 354 + 618 = 972
176 354627981 354 + 627 = 981
177 357129486 357 + 129 = 486
178 357291648 357 + 291 = 648
179 357462819 357 + 462 = 819
180 357624981 357 + 624 = 981
181 358614972 358 + 614 = 972
182 359127486 359 + 127 = 486
183 362187549 362 + 187 = 549
184 362457819 362 + 457 = 819
185 364527891 364 + 527 = 891
186 367128495 367 + 128 = 495
187 367182549 367 + 182 = 549
188 367452819 367 + 452 = 819
189 367524891 367 + 524 = 891
190 368127495 368 + 127 = 495
191 372546918 372 + 546 = 918
192 376218594 376 + 218 = 594
193 376542918 376 + 542 = 918
194 378216594 378 + 216 = 594
195 381294675 381 + 294 = 675
196 381546927 381 + 546 = 927
197 382167549 382 + 167 = 549
198 382194576 382 + 194 = 576
199 384192576 384 + 192 = 576
200 384291675 384 + 291 = 675
201 386541927 386 + 541 = 927
202 387162549 387 + 162 = 549
203 391257648 391 + 257 = 648
204 391284675 391 + 284 = 675
205 392184576 392 + 184 = 576
206 394182576 394 + 182 = 576
207 394281675 394 + 281 = 675
208 397251648 397 + 251 = 648
209 415278693 415 + 278 = 693
210 418239657 418 + 239 = 657
211 418275693 418 + 275 = 693
212 419238657 419 + 238 = 657
213 428139567 428 + 139 = 567
214 429138567 429 + 138 = 567
215 438129567 438 + 129 = 567
216 438219657 438 + 219 = 657
217 439128567 439 + 128 = 567
218 439218657 439 + 218 = 657
219 452187639 452 + 187 = 639
220 452367819 452 + 367 = 819
221 457182639 457 + 182 = 639
222 457362819 457 + 362 = 819
223 462357819 462 + 357 = 819
224 467352819 467 + 352 = 819 |
225 475218693 475 + 218 = 693
226 478215693 478 + 215 = 693
227 482157639 482 + 157 = 639
228 482193675 482 + 193 = 675
229 483192675 483 + 192 = 675
230 487152639 487 + 152 = 639
231 492183675 492 + 183 = 675
232 493182675 493 + 182 = 675
233 514269783 514 + 269 = 783
234 517329846 517 + 329 = 846
235 519264783 519 + 264 = 783
236 519327846 519 + 327 = 846
237 524367891 524 + 367 = 891
238 527319846 527 + 319 = 846
239 527364891 527 + 364 = 891
240 529317846 529 + 317 = 846
241 541296837 541 + 296 = 837
242 541386927 541 + 386 = 927
243 542196738 542 + 196 = 738
244 542376918 542 + 376 = 918
245 543186729 543 + 186 = 729
246 543276819 543 + 276 = 819
247 546183729 546 + 183 = 729
248 546192738 546 + 192 = 738
249 546273819 546 + 273 = 819
250 546291837 546 + 291 = 837
251 546372918 546 + 372 = 918
252 546381927 546 + 381 = 927
253 564219783 564 + 219 = 783
254 564327891 564 + 327 = 891
255 567324891 567 + 324 = 891
256 569214783 569 + 214 = 783
257 571293864 571 + 293 = 864
258 572346918 572 + 346 = 918
259 573246819 573 + 246 = 819
260 573291864 573 + 291 = 864
261 576243819 576 + 243 = 819
262 576342918 576 + 342 = 918
263 581346927 581 + 346 = 927
264 583146729 583 + 146 = 729
265 586143729 586 + 143 = 729
266 586341927 586 + 341 = 927
267 591246837 591 + 246 = 837
268 591273864 591 + 273 = 864
269 592146738 592 + 146 = 738
270 593271864 593 + 271 = 864
271 596142738 596 + 142 = 738
272 596241837 596 + 241 = 837
273 614259873 614 + 259 = 873
274 614358972 614 + 358 = 972
275 617328945 617 + 328 = 945
276 618327945 618 + 327 = 945
277 618354972 618 + 354 = 972
278 619254873 619 + 254 = 873
279 624159783 624 + 159 = 783
280 624357981 624 + 357 = 981
281 627318945 627 + 318 = 945
282 627354981 627 + 354 = 981
283 628317945 628 + 317 = 945
284 629154783 629 + 154 = 783
285 634158792 634 + 158 = 792
286 634257891 634 + 257 = 891
287 637254891 637 + 254 = 891
288 638154792 638 + 154 = 792
289 642195837 642 + 195 = 837
290 643275918 643 + 275 = 918
291 645192837 645 + 192 = 837
292 645273918 645 + 273 = 918
293 654129783 654 + 129 = 783
294 654138792 654 + 138 = 792
295 654219873 654 + 219 = 873
296 654237891 654 + 237 = 891
297 654318972 654 + 318 = 972
298 654327981 654 + 327 = 981
299 657234891 657 + 234 = 891
300 657324981 657 + 324 = 981
301 658134792 658 + 134 = 792
302 658314972 658 + 314 = 972
303 659124783 659 + 124 = 783
304 659214873 659 + 214 = 873
305 671283954 671 + 283 = 954
306 673245918 673 + 245 = 918
307 673281954 673 + 281 = 954
308 675243918 675 + 243 = 918
309 681273954 681 + 273 = 954
310 683271954 683 + 271 = 954
311 692145837 692 + 145 = 837
312 695142837 695 + 142 = 837
313 715248963 715 + 248 = 963
314 716238954 716 + 238 = 954
315 718236954 718 + 236 = 954
316 718245963 718 + 245 = 963
317 725139864 725 + 139 = 864
318 729135864 729 + 135 = 864
319 735129864 735 + 129 = 864
320 735246981 735 + 246 = 981
321 736218954 736 + 218 = 954
322 736245981 736 + 245 = 981
323 738216954 738 + 216 = 954
324 739125864 739 + 125 = 864
325 745218963 745 + 218 = 963
326 745236981 745 + 236 = 981
327 746235981 746 + 235 = 981
328 748215963 748 + 215 = 963
329 752184936 752 + 184 = 936
330 754182936 754 + 182 = 936
331 762183945 762 + 183 = 945
332 763182945 763 + 182 = 945
333 782154936 782 + 154 = 936
334 782163945 782 + 163 = 945
335 783162945 783 + 162 = 945
336 784152936 784 + 152 = 936
|
Contributions
Linda Rojewski (email) - go to topic
Peter Kogel (email) - go to topic - go to topic addendum - go to more ninedigital powers
John Morse (email) - go to topic
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