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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).

Seventh Page

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

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

The result in both cases is the same $$\large\bbox[white,3px,border:1px blue solid]{2025}$$

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

 a s 1 438769251 362859174 2 529468713 459723816 3 547198362 869315742 4 594872163 819235764 5 873569412 149265738 6 917258463 865719342 7 945672831 895713426 8 963574821 895431276

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

 1 9 terms 249758361 22 + 34 + 59 + 77 + 115 + 138 + 173 + 196 + 231 865719324 497218365 24 + 39 + 57 + 72 + 111 + 138 + 173 + 196 + 235 869315742 769318542 27 + 36 + 59 + 73 + 111 + 138 + 175 + 194 + 232 819235764 829415763 28 + 32 + 59 + 74 + 111 + 135 + 177 + 196 + 233 459723816 893745216 28 + 39 + 53 + 77 + 114 + 135 + 172 + 191 + 236 149265738 968234571 29 + 36 + 58 + 72 + 113 + 134 + 175 + 197 + 231 895713426 972185436 29 + 37 + 52 + 71 + 118 + 135 + 174 + 193 + 236 362859174 983642571 29 + 38 + 53 + 76 + 114 + 132 + 175 + 197 + 231 895431276

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)(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

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 .

No. Ninedigital Multiplicand Palindrome Progr. substrings 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)); ) } 1 826453719 11 9090990909 9090990909 2 826453719 28 23140704132 23140704132 3 826453719 65 53719491735 53719491735 4 826453719 209 172728827271 172728827271 5 826453719 308 254547745452 254547745452 6 826453719 407 336366663633 336366663633 7 826453719 506 418185581814 418185581814

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.

Palindrome 1-ninedigital Palindrome 2-ninedigital Palindrome 3-ninedigital 5 digits 9 digits 12 digits 18 digits 20 digits 27 digits 90909 826453719 172728827271 826453719926453719 53719491788719491735 118915239766738494647823255 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

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.

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#NinedigitalNdigitdxEightdigitalE=PalindromeP
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

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