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The Nine Digits Page 7
with some Ten Digits (pandigital) exceptions
rood Page 1 rood Page 2 rood Page 3 rood Page 4 rood Page 5 rood Page 6

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).


Seventh Page

  Topic 7.15   [ March 30, 2025 ]
Ten different pandigitals added together giving another pandigital.

From "The Little Giant Encyclopedia of Puzzles" (The diagram Group; Sterling Publishing 1996) → Puzzle 189

Be amazed by the following calculation! What is surprising about the total?

\( \begin{align} \color{blue}{1975308624}\\ \color{blue}{1728395046}\\ \color{blue}{1604938257}\\ \color{blue}{1234567980}\\ \color{blue}{0987654312}\\ \color{blue}{0864197523}\\ \color{blue}{0617283945}\\ \color{blue}{0493827156}\\ \color{blue}{0246913578}\\ \color{blue}{0123456789}\\ +\\ \color{blue}{9876543210} \end{align} \)

1. Is it possible to redo the exercise with nine pandigitals without leading zero's ?

2. Can you do the exercise with only ninedigitals (nine or ten) ?

2. Some nice and/or interesting variations on the topic...


  Topic 7.14   [ August 26, 2024 ]
Ninedigital triplet sums, their reversals and sums of their squares.

Here are four solutions of ninedigital triplets that when reversed their sums are both equal to the same value, for all four equations!

\(\bbox[#defade,3px,border:1px solid]{618+294+753}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{357+492+816}=1665\)

\(\bbox[#defade,3px,border:1px solid]{639+174+852}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{258+471+936}=1665\)

\(\bbox[#defade,3px,border:1px solid]{159+834+672}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{276+438+951}=1665\)

\(\bbox[#defade,3px,border:1px solid]{879+654+132}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{231+456+978}=1665\)

But that is not all. Let us square the triplets on both sides and note that each equation add to the same value, yet the sums are different for each equations.

\(\bbox[#defade,3px,border:1px solid]{618^2+294^2+753^2}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{357^2+492^2+816^2}=1035369\)

\(\bbox[#defade,3px,border:1px solid]{639^2+174^2+852^2}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{258^2+471^2+936^2}=1164501\)

\(\bbox[#defade,3px,border:1px solid]{159^2+834^2+672^2}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{276^2+438^2+951^2}=1172421\)

\(\bbox[#defade,3px,border:1px solid]{879^2+654^2+132^2}\Leftarrow\) reversal of \(\Rightarrow\bbox[lavender,3px,border:1px solid]{231^2+456^2+978^2}=1217781\)

Ain't that surprisingly beautiful!

Beyond the ninedigital aspect there is more to discover in the image at → Autour du nombre 159


But there is more to discover. Multiplying the full concatenated ninedigital triplets of the first and third line and the multiplication of their reversals

yield the same results!

\(\Large{\bbox[#defade,3px,border:1px solid]{618294753}*\bbox[#defade,3px,border:1px solid]{159834672} \Leftarrow}\) reversals of \(\Large{\Rightarrow\bbox[lavender,3px,border:1px solid]{357492816}*\bbox[lavender,3px,border:1px solid]{276438951} = 98824939045076016}\)

Let that sink in for a while and you'll enjoy it all the more!


  Topic 7.13   [ August 14, 2024 ]
Ninedigitals 123456789 and 987654321 hiding
in the decimal expansion of squares.

Here is the smallest example for substring \(123456789\)

\(95516788^2=9{\color{blue}{123456789}}836944\)

and a few larger examples (there is no 'largest' one)

\(11111111105^2={\color{blue}{123456789}}987654321025\)
\(10061539042^2=10{\color{blue}{123456789}}3690277764\)
\(26388295198^2=696342{\color{blue}{123456789}}859204\)
\(31416128397^2=986973{\color{blue}{123456789}}789609\)

A much longer list can be found in the webarchive at https://web.archive.org/web/20040909153906/http://blue.kakiko.com/mmrmmr/htm/eqtn19.html
Note that some entries are doubled. Can someone find out who was the creator of these webpages.

Here is the smallest example for substring \(987654321\)

\(44583117^2=1{\color{blue}{987654321}}435689\)

and a few larger examples (there is no 'largest' one)

\(611111111^2=373456789{\color{blue}{987654321}}\)
\(10247438102^2=105009{\color{blue}{987654321}}362404\)
\(20493600326^2=419{\color{blue}{987654321}}827306276\)
\(31619357639^2=999783777502{\color{blue}{987654321}}\)

A much longer list can be found in the webarchive at https://web.archive.org/web/20040909153910/http://blue.kakiko.com/mmrmmr/htm/eqtn20.html
Note that some entries are doubled. Can someone find out who was the creator of these webpages.

Can you find such a square where the root also contains \(123456789\) or \(987654321\) in its decimal expansion ?
Can you find such a square where both ninedigitals (\(123456789\) & \(987654321\)) are present in the decimal expansion without overlapping?


  Topic 7.12   [ February 28, 2024 ]
Ninedigits Squared and Ninedigits Cubed

\(\large\bbox[white,5px,border:1px blue solid]{(1+2+3+4+5+6+7+8+9)^2}\mathbf{\color{blue}{\;=\;}}\bbox[white,5px,border:1px blue solid]{1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3}\)

The result in both cases is the same \(\large\bbox[white,5px,border:1px blue solid]{2025}\)

This is in fact an application of Nicomachus's Theorem stating that

$$\color{blue}{\Large{{\Huge{(}}\sum_{i=1}^n i{\Huge{)}}^2 = \sum_{i=1}^{n} i^3}}$$

  Topic 7.11   [ September 2, 2022 ]
A follow up from WONplate 26 (some properties of 69696)
by Alexandru Petrescu (email)

In a contribution to WONplate 26 \(\bbox[#defade,5px,border:2px solid green]{69696 * palindrome = {nine-}~or~pandigital}\) I showed the next relation:

69696 * 6996 = 487593216 (ninedigital)

Now I went looking for a similar pattern \(\bbox[#defade,5px,border:2px solid green]{abcba * abba = ninedigital}\) and I found only one other number, namely:

65556 * 6556 = 429785136 (ninedigital)

Another pattern \(\bbox[#defade,5px,border:2px solid green]{abcba * abba = palindrome}\) produces 31 solutions.

1	10001 * 1001 = 10011001
2	10101 * 1001 = 10111101
3	10201 * 1001 = 10211201
4	10301 * 1001 = 10311301
5	10401 * 1001 = 10411401
6	10501 * 1001 = 10511501
7	10601 * 1001 = 10611601
8	10701 * 1001 = 10711701
9	10801 * 1001 = 10811801
10	10901 * 1001 = 10911901
11	11011 * 1111 = 12233221
12	11111 * 1111 = 12344321
13	11211 * 1111 = 12455421
14	11311 * 1111 = 12566521
15	11411 * 1111 = 12677621
16	11511 * 1111 = 12788721
17	11611 * 1111 = 12899821
18	12021 * 1221 = 14677641
19	12121 * 1221 = 14799741
20	20002 * 2002 = 40044004
21	20102 * 2002 = 40244204
22	20202 * 2002 = 40444404
23	20302 * 2002 = 40644604
24	20402 * 2002 = 40844804
25	21012 * 2112 = 44377344
26	21112 * 2112 = 44588544
27	21212 * 2112 = 44799744
28	30003 * 3003 = 90099009
29	30103 * 3003 = 90399309
30	30203 * 3003 = 90699609
31	30303 * 3003 = 90999909

[ September 6, 2022 ]
Alexandru explored many more patterns. Hereunder is an example
whereby a long multiplication with binary looking palindromic numbers leads to a palindrome where all ten digits are present.

1000010100001 * 10000100001 * 100000001 * 1000001 * 10001 * 101 = 1010122323456667799998999977666543232210101

The sequence is generated by abcdefgfedcba ➜ abcdefedcba ➜ abcdedcba ➜ abcdcba ➜ abcba ➜ aba.
We obtain the next term by deleting in the previous term the digits marked in red until we arrive at 'aba'.


  Topic 7.10   [ August 28, 2022 ]
An interesting relation between four pandigital numbers
by Alexander R. Povolotsky (email)

Alexander likes to share the following relation he found between these four pandigitals :

\( {(\ 9876543120 \ *\ 9876543210\ ) \over (\ 1234567890 \ *\ 7901234568\ )} = \color{yellow}{10} \)

This is quite remarkable! The outcome even wants to remind us that each pandigital has ten digits.
But part of the magic disappears when you write it out using their resp. primefactors.
You will notice that almost all factors cancel each other out just to leave 2 * 5.

I wonder if there exists a ninedigital analogue formula with a quotient of  9 of course ?

Alexander replies [ August 29, 2022 ]
“Speaking regarding one pair of ninedigits, there's no pair, which ratio yields 9 but I could offer you the pair, which ratio gives 8.

\({ 9876543\color{darkgreen}{12} \over 123456789 } = 8\)

Perhaps someone could write a program to see whether there are existing (single or multiple) pairs, which ratios yield integer outcome...”

Related topic WONplate 114

Alexandru Petrescu found the four ninedigitals analogue formula with a quotient of  9 [ August 31, 2022 ]

I felt that there are many solutions involving two pairs of ninedigital (d1,d2) and (d3,d4).
I tried to find the pairs for which:

d1 / d2 = \(6\) and d3 / d4 = \(\Large\frac{3}{2}\) so (d1 * d3) / (d2 * d4) = \(9\)

I found 162 pairs for (d1,d2) respectively 2212 pairs for (d3,d4).
Combining them we have 358344 solutions. One of them being:

\( {(\ 741293856 \ *\ 185243679\ ) \over (\ 123548976 \ *\ 123495786\ )} = \color{yellow}{9} \)

Applying the same idea for pandigitals, using:

d1 / d2 = \(4\) and d3 / d4 = \(\Large\frac{5}{2}\) so (d1 * d3) / (d2 * d4) = \(10\)

produces 6480 pairs for (d1,d2) respectively 7312 pairs for (d3,d4).
Combining them we obtain 47381760 solutions. One of them being:

\( {(\ 4093827156 \ *\ 2561839740\ ) \over (\ 1023456789 \ *\ 1024735896\ )} = \color{yellow}{10} \)

So Povolotsky's relation is not at all unique...


  Topic 7.9   [ August 22, 2022 ]
Route 1 from ninedigitals to palindromes via sum of 'powers of primes'
Route 2 from ninedigitals to ninedigitals via sum of 'powers of primes'
by Alexandru Petrescu (email)

In both cases the vehicle is this formula \(s = \sum_{i=1}^9(prime(v_i)^i) \)
Let a=v1v2v3v4v5v6v7v8v9 a ninedigital.
Note that prime(1)=2, prime(2)=3, prime(3)=5, prime(4)=7, prime(5)=11, prime(6)=13, prime(7)=17, prime(8)=19 and prime(9)=23.

ROUTE 1 (8 ninedigitals found) Worked Out Example
359678124 51 + 112 + 233 + 134 + 175 + 196 + 27 + 38 + 79 5 + 121 + 12167 + 28561 + 1419857 + 47045881 + 128 + 6561 + 40353607 88866888

This link shows all eight solutions
Palindromes from Consecutive Primes 2 to 23 and the Nine Digits Anagrams
They were already found by Carlos Rivera in 1999.

ROUTE 2 (8 ninedigitals found) Worked Out Example
438769251 71 + 52 + 193 + 174 + 135 + 236 + 37 + 118 + 29 7 + 25 + 6859 + 83521 + 371293 + 148035889 + 2187 + 214358881 + 512 362859174

 as
1438769251362859174
2529468713459723816
3547198362869315742
4594872163819235764
5873569412149265738
6917258463865719342
7945672831895713426
8963574821895431276


  Topic 7.8   [ May 5, 2022 ]
Expressing ninedigitals using 9 consecutive primes
by Alexandru Petrescu

Starting from a ninedigital number N = abcdefghi we construct the following sum
S = 2a + 3b + 5c + 7d + 11e + 13f + 17g + 19h + 23i

bases being the first nine primes in ascending order and
exponents being the digits of N, from left to right
We are looking for numbers N for which S is another ninedigital.
Eight solutions emerged

1249758361 22 + 34 + 59 + 77 + 115 + 138 + 173 + 196 + 2319 terms
8657193249
2497218365 24 + 39 + 57 + 72 + 111 + 138 + 173 + 196 + 2359 terms
8693157429
3769318542 27 + 36 + 59 + 73 + 111 + 138 + 175 + 194 + 2329 terms
8192357649
4829415763 28 + 32 + 59 + 74 + 111 + 135 + 177 + 196 + 2339 terms
4597238168
5893745216 28 + 39 + 53 + 77 + 114 + 135 + 172 + 191 + 2369 terms
1492657389
6968234571 29 + 36 + 58 + 72 + 113 + 134 + 175 + 197 + 2319 terms
8957134269
7972185436 29 + 37 + 52 + 71 + 118 + 135 + 174 + 193 + 2369 terms
3628591749
8983642571 29 + 38 + 53 + 76 + 114 + 132 + 175 + 197 + 2319 terms
8954312769


  Topic 7.7   [ December 9, 2021 ]
Finding record “jumbled” meta_ninedigital primes

A meta_ninedigital number is defined as a normal ninedigital number
with one or more of the nine digits repeated (ad infinitum)
but must remain together in groups.
We distinguish three orders. In short they are AM9, DM9 & JM9.

Classification

Ascending meta_ninedigital (AM9) numbers
numbers of form 1(A)2(B)3(C)4(D)5(E)6(F)7(G)8(H)9(I)

Descending meta_ninedigital (DM9) numbers
numbers of form 9(A)8(B)7(C)6(D)5(E)4(F)3(G)2(H)1(I)

Jumbled meta_ninedigital (JM9) numbers
numbers of form a(A)b(B)c(C)d(D)e(E)f(F)g(G)h(H)i(I)

with {a,b,c,d,e,f,g,h,i} ∈ {1,2,3,4,5,6,7,8,9}

Visit my WONpage 211 to familiarise yourself with the topic
but please do come back here for another challenge.

An informal notation for a jumbled meta_ninedigital number is

A(A)N(B)Y(C)9 (D)D(E) I(F)G(G) I(H)T(I)

Your task now is to find ever larger prime jumbled meta_ninedigitals !
Note that probable primes are fine too for very large constructions.
Just give me the starter ninedigital and the values of A up to I for shorthand display.
Maybe the values A through I make up for a beautiful sequence on its own...

Here is already an example of such a JM9 prime:

Starter ninedigital 987654123 [A=3,B=3,C=3,D=3,E=3,F=3,G=3,H=1,I=1]

9(3)8(3)7(3)6(3)5(3)4(3)1(3)2(1)3(1)

or prime 99988877766655544411123 of length 23


  Topic 7.6   [ December 20, 2021 ]
Reviving an old puzzle (TYCMJ 263) by Charles W. Trigg
from the Two Year College Mathematics Journal

Source 1 Index to Mathematical Problems, 1980-1984 by Stanley Rabinowitz — 1992
Source 2 http://www.mathpropress.com/cmj/pages/page181.html

Question: Find a nonagonal number that is the sum of three
three-digit primes which (concatenated) together contain the nine nonzero
digits once each.
(A nonagonal number is a number of the form n(7n – 5)/2 A001106).

Subquestion: Is there more than one solution ?

Who can provide me with a slick program that spews out the answer(s) ?
Or if you are not a programmer but can reach to the solution by another
method please send your answer(s) as well.

Redo the exercice but with other polygonal number formats.
And what if you use multiplication instead of the sum...

Solution by Alexandru Petrescu [ 23 december 2021 ]

The ninedigital is 149257683 and five permutations of the three three-digit primes
(149, 257, 683). The nonagonal number is 1089, for n = 18.

For multiplication instead of summation, no solution.

A natural generalization of this puzzle
Under the same conditions find polygonal numbers. I impose a condition of
increasing sequence of three 3-digit primes to avoid repeated solution
obtained by permutation of three 3-digit primes. In the table I present
ninedigital, sum of three 3-digit primes, r-gonal and rank of r-gonal sum.

NinedigitalSumr-gonalnth
1492576831089433
1492576831089918
25146798317011021
25146798317011318
25746198317011021
25746198317011318
2574916831431353
2574916831431627
28134756911971514
28146795317011021
28146795317011318
2815936471521439
2834576911431353
2834576911431627
2835476911521439
2935876411521439
38946752113771217
4216738591953362
4796538211953362
48752369117011021
48752369117011318
52164783920071518
52186394723311321
5417698232133827
54182396723311321
54763182920071518
56382194723311321
56382794123311321
5697438212133827


  Topic 7.5   [ December 28, 2021 ]
Observations around ninedigital 826453719
Alexandru Petrescu

Checking for multiples of ninedigital which become palindromic, I obtained some very interesting results.

The ninedigital with this property having the  most  multiplicands m (m < 1000) is  826453719 .

Pari/GP (©_pdg)

 {
nd=826453719;
cnt=0;
for(m=1,10000000,
  pal=nd*m;
  r=digits(pal);
  if(Vecrev(r)==r, cnt+=1; print(cnt," ",nd," * ",m," = ",pal));
)
}
No.NinedigitalMultiplicandPalindromeProgr. substrings
18264537191190909909099090990909
2826453719282314070413223140704132
3826453719655371949173553719491735
4826453719209172728827271172728827271
5826453719308254547745452254547745452
6826453719407336366663633336366663633
7826453719506418185581814418185581814

Progression A ( 209,308,407,506 ) has a common ratio of 99 !
and in the decimal expansion of the palindrome we discover three more progressions
progression B ( 17,25,33,41 ), progression C ( 27,45,63,81 ) & progression D ( 28,47,66,85 )
to be found in the 2 digit substrings of the resp. palindromes.

Checking for the same ninedigital number multiplicands (m > 1000)
I observed a similar pattern between the following index nrs.:

No.NinedigitalMultiplicandPalindromeProgr. substrings
12826453719130791080918819080110809188190801
17826453719250582070927729070220709277290702
21826453719370373060936639060330609366390603
25826453719490164050945549050440509455490504
28826453719609955040654459040550406544590405
31826453719729746030963369030660309633690306
33826453719849537020972279020770209722790207
35826453719969328010981189010880109811890108
368264537191089119000990099000990009900990009

Here the multiplicands are in arithmetic progression (common ratio = 11979 or 99*112)
and the progressions are ( 10,20,30,40,50,60,70,80,90 ), ( 80,70,60,50,40,30,20,10,00 ), ( 91,92,93,94,95,96,97,98,99 ) & ( 8,7,6,5,4,3,2,1,0 ).

Patrick observed that when we add the eight multiplicands together (from No. 12 to No. 35)
we get a palindrome namely 440044.
That is not all because the difference between the above palindrome and the last multiplicand (No. 36)
is again a palindrome namely 331133.

And now the icing on the 826453719_cake !

Definition
k-ninedigital number is a number having 9k digits, every digit from 1 to 9 appears exactly k times.
(1-ninedigital is our familiar ninedigital).

We are looking for sequences with alternating palindromes and k-ninedigital numbers (increasing k, starting from 1),
each term being integer multiple of precedent term.

No better way than to visualize it in the next table.

   
Palindrome1-ninedigitalPalindrome2-ninedigitalPalindrome3-ninedigital
5 digits9 digits12 digits18 digits20 digits27 digits
9090982645371917272882727182645371992645371953719491788719491735118915239766738494647823255
Multiplicand🡖  9091  🡑🡖  209  🡑🡖  4784689  🡑🡖  65  🡑🡖  2213633  🡑
   

It is a thing of beauty ! Thank you Alexandru.


  Topic 7.4   [ January 30, 2022 ]
Looking for patterns in palindromes and ninedigitals
A double submission by Alexandru Petrescu

❶ Firstly Alexandru found 22 cases of a ninedigital divisible by a nine-digit palindrome.
This is consistent with the data I calculated in WONplate 114 [ August 17, 2003 ].

IndexPalindrome * m = Ninedigital
1. 156090651 * 3 = 468271953
2. 157232751 * 3 = 471698253
3. 165090561 * 3 = 495271683
4. 165717561 * 3 = 497152683
5. 175616571 * 3 = 526849713
6. 132969231 * 4 = 531876924
7. 189060981 * 3 = 567182943
8. 198060891 * 3 = 594182673
9. 246171642 * 3 = 738514926
10. 123090321 * 6 = 738541926
11. 123969321 * 6 = 743815926
12. 249717942 * 3 = 749153826
13. 264171462 * 3 = 792514386
14. 132090231 * 6 = 792541386
15. 264717462 * 3 = 794152386
16. 275131572 * 3 = 825394716
17. 279141972 * 3 = 837425916
18. 145232541 * 6 = 871395246
19. 145898541 * 6 = 875391246
20. 297141792 * 3 = 891425376
21. 132636231 * 7 = 928453617
22. 136232631 * 7 = 953628417

Note that in lines (7. & 8.) and (21. & 22.) the palindromes are almost equal
except for a swap between two successive digits.

❷ Secondly Alexandru looked for more complex patterns like this

Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1 (concatenated)
Ninedigital|Ninedigital1 x factor3 = Palindrome3
with greatest Palindrome1.

There are only 10 ninedigitals (some of them having variants) which fulfil this pattern.

Many concepts blend together in this topic  Ninedigitals ,  Palindromes ,  Factors , Concatenations, Multiplications

Hoover Under Me
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Palindrome1 x factor1 = Ninedigital1
Ninedigital1 x factor2 = Palindrome2
Palindrome2 x factor3 = Ninedigital|Ninedigital1
Ninedigital|Ninedigital1 x factor3 = Palindrome3
Hoover Above Me
 Hoover Under Me
1. 1811 x 11814751 = 2138469751
2138469751 x 2472 = 528202028252
528202028252 x 40485833 = 213846975|2138469751
213846975|2138469751 x 2473 = 528202028778202028253
2. 999991 x 26591 = 2658973411
2658973411 x 1332 = 353643463532
353643463532 x 75187973 = 265897341|2658973411
265897341|2658973411 x 1333 = 353643463883643463533
3A. 214224121 x 131 = 2784913561
2784913561 x 772 = 214438344122
214438344122 x 129870133 = 278491356|2784913561
278491356|2784913561 x 773 = 214438344334438344123
3B. 214224121 x 131 = 2784913561
2784913561 x 772 = 214438344122
214438344122 x 129870133 = 278491356|2784913561
278491356|2784913561 x 769230773 = 214224120428448240214224123
3C. 214224121 x 131 = 2784913561
2784913561 x 769230772 = 214224120214224122
214224120214224122 x 133 = 278491356|2784913561
278491356|2784913561 x 773 = 214438344334438344123
3D. 214224121 x 131 = 2784913561
2784913561 x 769230772 = 214224120214224122
214224120214224122 x 133 = 278491356|2784913561
278491356|2784913561 x 769230773 = 214224120428448240214224123
4A. 329231 x 89821 = 2957143861
2957143861 x 72 = 20700007022
20700007022 x 1428571433 = 295714386|2957143861
295714386|2957143861 x 73 = 20700007040700007023
4B. 329231 x 89821 = 2957143861
2957143861 x 72 = 20700007022
20700007022 x 1428571433 = 295714386|2957143861
295714386|2957143861 x 773 = 227700077447700077223
4C. 329231 x 89821 = 2957143861
2957143861 x 772 = 227700077222
227700077222 x 129870133 = 295714386|2957143861
295714386|2957143861 x 73 = 20700007040700007023
4D. 329231 x 89821 = 2957143861
2957143861 x 772 = 227700077222
227700077222 x 129870133 = 295714386|2957143861
295714386|2957143861 x 773 = 227700077447700077223
5. 9991 x 3152421 = 3149267581
3149267581 x 1332 = 418852588142
418852588142 x 75187973 = 314926758|3149267581
314926758|3149267581 x 1333 = 41885258855885258143
6A. 991 x 38961041 = 3857142961
3857142961 x 72 = 27000000722
27000000722 x 1428571433 = 385714296|3857142961
385714296|3857142961 x 73 = 27000000747000000723
6B. 991 x 38961041 = 3857142961
3857142961 x 772 = 297000007922
297000007922 x 129870133 = 385714296|3857142961
385714296|3857142961 x 73 = 27000000747000000723
7. 249421 x 173161 = 4318956721
4318956721 x 14632 = 6318633681362
6318633681362 x 6835273 = 431895672|4318956721
431895672|4318956721 x 14633 = 6318633687678633681363
8A. 1711 x 41861191 = 7158263491
7158263491 x 192 = 136007006312
136007006312 x 526315793 = 715826349|7158263491
715826349|7158263491 x 193 = 136007006446007006313
8B. 1711 x 41861191 = 7158263491
7158263491 x 2092 = 1496077069412
1496077069412 x 47846893 = 715826349|7158263491
715826349|7158263491 x 193 = 136007006446007006313
9A. 1326362311 x 71 = 9284536171
9284536171 x 1432 = 1327688672312
1327688672312 x 69930073 = 928453617|9284536171
928453617|9284536171 x 1433 = 1327688673637688672313
9B. 1326362311 x 71 = 9284536171
9284536171 x 1428571432 = 1326362311326362312
1326362311326362312 x 73 = 928453617|9284536171
928453617|9284536171 x 1433 = 1327688673637688672313
10A. 1362326311 x 71 = 9536284171
9536284171 x 1432 = 1363688636312
1363688636312 x 69930073 = 953628417|9536284171
953628417|9536284171 x 1433 = 1363688637673688636313
10B. 1362326311 x 71 = 9536284171
9536284171 x 1428571432 = 1362326311362326312
1362326311362326312 x 73 = 953628417|9536284171
953628417|9536284171 x 1433 = 1363688637673688636313
 Hoover Above Me
Don't HM
Resumé of the 10 ninedigitals
213846975
265897341
278491356 (4 variants)
295714386 (4 variants)
314926758
385714296 (2 variants)
431895672
715826349 (2 variants)
928453617 (2 variants)
953628417 (2 variants)
at all


  Topic 7.3   [ March 3, 2022 ]
Product of two ninedigitals equal to the product of their reversals
A submission by Alexandru Petrescu

Inspired by the recreational work of science writer Yakov Perelman (1882-1942)
with e.g. equation 46 x 96 = 4416 = 64 x 69 (note 64 reversal of 46 and 69 reversal of 96)
Alexandru proposes a similar problem but now with ninedigital numbers

Solve equation
A x B = Rev(A) x Rev(B)
where Rev(A) means reversal of ninedigital A, the four numbers being distinct.

He found 6 solutions

A B Rev(A) Rev(B) Product
124578963  x  639784512  =  369875421  x  215487936  =  79703691048421056
145697283  x  619487532  =  382796541  x  235784916  =  90257650264775556
146735982  x  647591823  =  289537641  x  328195746  =  95025022083075186
146983572  x  615947823  =  275389431  x  328749516  =  90534211190163756
157346982  x  647829153  =  289643751  x  351928746  =  101933962076166246
159834672  x  618294753  =  276438951  x  357492816  =  98824939045076016


  Topic 7.2   [ March 8, 2022 ]
Dropping/Removing a digit d from a ninedigital N and multiplying
this digit with the remaining eight digits renders a palindrome P
Alexandru Petrescu found 126 solutions !

Here is the list ordered from smallest ninedigital to highest.
The only digits d that occur are 4, 6, 7 and 8.







Index
#		Ninedigital
N		digit
d		xEightdigital
E		=Palindrome
P


12897613548x29761354=238090832


22976135488x29761354=238090832


32976135848x29761354=238090832


42976138548x29761354=238090832


52976183548x29761354=238090832


62976813548x29761354=238090832


72978613548x29761354=238090832


82987613548x29761354=238090832


93165278948x31652794=253222352


103165279488x31652794=253222352


113165279848x31652794=253222352


123165287948x31652794=253222352


133165827948x31652794=253222352


143168527948x31652794=253222352


153186527948x31652794=253222352


163671589248x36715924=293727392


173671592488x36715924=293727392


183671592848x36715924=293727392


193671598248x36715924=293727392


203671859248x36715924=293727392


213678159248x36715924=293727392


223687159248x36715924=293727392


233816527948x31652794=253222352


243867159248x36715924=293727392


254231789567x42318956=296232692


264231879567x42318956=296232692


274231895677x42318956=296232692


284231895767x42318956=296232692


294231897567x42318956=296232692


304237189567x42318956=296232692


314273189567x42318956=296232692


324378615297x43861529=307030703


334386152797x43861529=307030703


344386152977x43861529=307030703


354386157297x43861529=307030703


364386175297x43861529=307030703


374386715297x43861529=307030703


384387615297x43861529=307030703


394539671284x53967128=215868512


404623517897x46235189=323646323


414623518797x46235189=323646323


424623518977x46235189=323646323


434623571897x46235189=323646323


444623751897x46235189=323646323


454627351897x46235189=323646323


464672351897x46235189=323646323


474689157326x48915732=293494392


484695732184x69573218=278292872


494723189567x42318956=296232692


504738615297x43861529=307030703


514762351897x46235189=323646323


524869157326x48915732=293494392


534891567326x48915732=293494392


544891573266x48915732=293494392


554891573626x48915732=293494392


564891576326x48915732=293494392


574891657326x48915732=293494392


584896157326x48915732=293494392


595278614397x52861439=370030073


605286143797x52861439=370030073


615286143977x52861439=370030073


625286147397x52861439=370030073


635286174397x52861439=370030073


645286714397x52861439=370030073


655287614397x52861439=370030073


665349671284x53967128=215868512


675394671284x53967128=215868512


685396471284x53967128=215868512


695396712484x53967128=215868512


705396712844x53967128=215868512


715396714284x53967128=215868512


725396741284x53967128=215868512


735439671284x53967128=215868512


745621897438x56219743=449757944


755621974388x56219743=449757944


765621974838x56219743=449757944


775621978438x56219743=449757944


785621987438x56219743=449757944


795628197438x56219743=449757944


805682197438x56219743=449757944


815728614397x52861439=370030073


825862197438x56219743=449757944


836489157326x48915732=293494392


846495732184x69573218=278292872


856732154896x73215489=439292934


866794358127x69435812=486050684


876943578127x69435812=486050684


886943581277x69435812=486050684


896943581727x69435812=486050684


906943587127x69435812=486050684


916943758127x69435812=486050684


926945732184x69573218=278292872


936947358127x69435812=486050684


946954732184x69573218=278292872


956957321484x69573218=278292872


966957321844x69573218=278292872


976957324184x69573218=278292872


986957342184x69573218=278292872


996957432184x69573218=278292872


1006974358127x69435812=486050684


1017321546896x73215489=439292934


1027321548696x73215489=439292934


1037321548966x73215489=439292934


1047321564896x73215489=439292934


1057321654896x73215489=439292934


1067326154896x73215489=439292934


1077362154896x73215489=439292934


1087423189567x42318956=296232692


1097438615297x43861529=307030703


1107462351897x46235189=323646323


1117528614397x52861439=370030073


1127632154896x73215489=439292934


1137694358127x69435812=486050684


1147956421387x95642138=669494966


1158297613548x29761354=238090832


1168316527948x31652794=253222352


1178367159248x36715924=293727392


1188562197438x56219743=449757944


1199564213787x95642138=669494966


1209564213877x95642138=669494966


1219564217387x95642138=669494966


1229564271387x95642138=669494966


1239564721387x95642138=669494966


1249567421387x95642138=669494966


1259576421387x95642138=669494966


1269756421387x95642138=669494966


  Topic 7.1   [ March 18, 2022 ]
Two equations with ninedigitals and palindromes, nine- and/or pandigitals
Alexandru Petrescu presents four solutions !

 Variant 1 

Let ABCDEFGHI be a ninedigital numbers.
I propose 3 kinds of equations :

ABC + DEF + GHI = {palindrome (1)
ninedigital (2)
pandigital (3)

where AB, DE, GH are twodigit numbers, AB < DE < GH

There are 4 solutions for equation (1), namely

286 + 435 + 791 = 628898826
391 + 584 + 762 = 11322311
563 + 742 + 891 = 181181
583 + 741 + 962 = 204402

No solutions exist for equations (2) and (3).

(PDG) From WONplate 22 I am not sure but might it be the start of an emerging pattern?

27 + 36 + 45 = 1881
563 + 742 + 891 = 181181

Alexandru came up with

8742 + 165 + 93 = 1813181

Not bad at all, but I was more thinking in the direction of a palindromic pattern
using only digits 1 and 8 on the right side of the equation...

 Variant 2 

Let ABCDEFGHI be a ninedigital numbers.
I propose 3 kinds of equations :

ABC * DEF * GHI = {palindrome (1)
ninedigital (2)
pandigital (3)

where ABC, DEF, GHI are threedigit numbers, ABC < DEF < GHI

There are 12 solutions for equation (1), namely

128 * 364 * 957 = 44588544
136 * 248 * 759 = 25599552
138 * 527 * 649 = 47199174
158 * 429 * 637 = 43177134 (°)
182 * 364 * 957 = 63399336
185 * 429 * 637 = 50555505
192 * 563 * 748 = 80855808
194 * 352 * 678 = 46299264
215 * 387 * 649 = 54000045
254 * 396 * 817 = 82177128
512 * 847 * 936 = 405909504
531 * 847 * 962 = 432666234 (°°)

No solutions exist for equations (2) and (3).

(°) From WONplate 163 43177134 is also the area of the Pythagorean triangle
with sides 3476, 24843, 25085 (PDG).

(°°) The last one is also interesting
The Beast Number appears three times

1. as middle part of 432666234
2. as sum of left and right parts 432 + 234 = 666
3. as a curious product 432666234 = 666 * 649649, the yellow term being a tautonym.


For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.

Topic 7.14 → 159834672, 231456978, 258471936, 276438951, 357492816, 618294753, 639174852, 879654132

Topic 7.2 → 289761354, 297613548, 297613584, 297613854, 297618354, 297681354, 297861354, 298761354, 316527894, 316527948, 316527984, 316528794, 316582794, 316852794, 318652794, 367158924, 367159248, 367159284, 367159824, 367185924, 367815924, 368715924, 381652794, 386715924, 423178956, 423187956, 423189567, 423189576, 423189756, 423718956, 427318956, 437861529, 438615279, 438615297, 438615729, 438617529, 438671529, 438761529, 453967128, 462351789, 462351879, 462351897, 462357189, 462375189, 462735189, 467235189, 468915732, 469573218, 472318956, 473861529, 476235189, 486915732, 489156732, 489157326, 489157362, 489157632, 489165732, 489615732, 527861439, 528614379, 528614397, 528614739, 528617439, 528671439, 528761439, 534967128, 539467128, 539647128, 539671248, 539671284, 539671428, 539674128, 543967128, 562189743, 562197438, 562197483, 562197843, 562198743, 562819743, 568219743, 572861439, 586219743, 648915732, 649573218, 673215489, 679435812, 694357812, 694358127, 694358172, 694358712, 694375812, 694573218, 694735812, 695473218, 695732148, 695732184, 695732418, 695734218, 695743218, 697435812, 732154689, 732154869, 732154896, 732156489, 732165489, 732615489, 736215489, 742318956, 743861529, 746235189, 752861439, 763215489, 769435812, 795642138, 829761354, 831652794, 836715924, 856219743, 956421378, 956421387, 956421738, 956427138, 956472138, 956742138, 957642138, 975642138

Topic 7.1 → 286435791, 391584762, 563742891, 583741962, 874216593, 128364957, 136248759, 138527649, 158429637, 182364957, 185429637, 192563748, 194352678, 215387649, 254396817, 512847936, 531847962


Contributions

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