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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).

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 = 5416239875393 + 4623 + 7183 = 625348179

The above is what I call a 'pandigital amicable pair' (see below).

 274 + 354 + 684 + 1494 = 516297843274 + 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

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 :

 A4C25816I0
 A4C25876I0
 A8C65432I0
 A8C65472I0

Recall that fragment ABC must be a multiple of 3. In choice , 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 .

For choice , the only digits available are 1, 3, and 9. No combination
of these will make ABC a multiple of 3, so scratch choice .

For choice , 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 , 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 :

 1896543270
 9816543270
 1836547290
 3816547290
 1896547230
 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 :

 1896543 divided by 7 leaves remainder 5. nope.
 9816543 divided by 7 leaves remainder 2. nope.
 1836547 divided by 7 leaves remainder 6. nope.
 3816547 divided by 7 leaves remainder 0.  YES !
 1896547 divided by 7 leaves remainder 2. nope.
 9816547 divided by 7 leaves remainder 6. nope.

Aha! Only choice  leaves no remainder when its fragment ABCDEFG is
divided by 7.

Finally we found the mystery number ! It is

 3 8 1 6 5 4 7 2 9 0

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://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.

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

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
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 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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Patrick De Geest - Belgium - Short Bio - Some Pictures