The Nine Digits Page -- 123456789

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The Nine Digits Page 7 with some Ten Digits (pandigital) exceptions
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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 = 2^a + 3^b + 5^c + 7^d + 11^e + 13^f + 17^g + 19^h + 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	249758361	2² + 3⁴ + 5⁹ + 7⁷ + 11⁵ + 13⁸ + 17³ + 19⁶ + 23¹	9 terms
1	249758361	865719324	9
2	497218365	2⁴ + 3⁹ + 5⁷ + 7² + 11¹ + 13⁸ + 17³ + 19⁶ + 23⁵	9 terms
2	497218365	869315742	9
3	769318542	2⁷ + 3⁶ + 5⁹ + 7³ + 11¹ + 13⁸ + 17⁵ + 19⁴ + 23²	9 terms
3	769318542	819235764	9
4	829415763	2⁸ + 3² + 5⁹ + 7⁴ + 11¹ + 13⁵ + 17⁷ + 19⁶ + 23³	9 terms
4	829415763	459723816	8
5	893745216	2⁸ + 3⁹ + 5³ + 7⁷ + 11⁴ + 13⁵ + 17² + 19¹ + 23⁶	9 terms
5	893745216	149265738	9
6	968234571	2⁹ + 3⁶ + 5⁸ + 7² + 11³ + 13⁴ + 17⁵ + 19⁷ + 23¹	9 terms
6	968234571	895713426	9
7	972185436	2⁹ + 3⁷ + 5² + 7¹ + 11⁸ + 13⁵ + 17⁴ + 19³ + 23⁶	9 terms
7	972185436	362859174	9
8	983642571	2⁹ + 3⁸ + 5³ + 7⁶ + 11⁴ + 13² + 17⁵ + 19⁷ + 23¹	9 terms
8	983642571	895431276	9

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.

Ninedigital	Sum	r-gonal	n^th
149257683	1089	4	33
149257683	1089	9	18
251467983	1701	10	21
251467983	1701	13	18
257461983	1701	10	21
257461983	1701	13	18
257491683	1431	3	53
257491683	1431	6	27
281347569	1197	15	14
281467953	1701	10	21
281467953	1701	13	18
281593647	1521	4	39
283457691	1431	3	53
283457691	1431	6	27
283547691	1521	4	39
293587641	1521	4	39
389467521	1377	12	17
421673859	1953	3	62
479653821	1953	3	62
487523691	1701	10	21
487523691	1701	13	18
521647839	2007	15	18
521863947	2331	13	21
541769823	2133	8	27
541823967	2331	13	21
547631829	2007	15	18
563821947	2331	13	21
563827941	2331	13	21
569743821	2133	8	27

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=ndm; 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.	Ninedigital	Multiplicand	Palindrome	Progr. substrings
12	826453719	13079	10809188190801	10809188190801
17	826453719	25058	20709277290702	20709277290702
21	826453719	37037	30609366390603	30609366390603
25	826453719	49016	40509455490504	40509455490504
28	826453719	60995	50406544590405	50406544590405
31	826453719	72974	60309633690306	60309633690306
33	826453719	84953	70209722790207	70209722790207
35	826453719	96932	80109811890108	80109811890108
36	826453719	108911	90009900990009	90009900990009

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

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 ].

Index	Palindrome	*	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

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 x Eightdigital
E = Palindrome
P
1 2₈9761354 8 x 29761354 = 238090832
2 29761354₈ 8 x 29761354 = 238090832
3 2976135₈4 8 x 29761354 = 238090832
4 297613₈54 8 x 29761354 = 238090832
5 29761₈354 8 x 29761354 = 238090832
6 2976₈1354 8 x 29761354 = 238090832
7 297₈61354 8 x 29761354 = 238090832
8 29₈761354 8 x 29761354 = 238090832
9 316527₈94 8 x 31652794 = 253222352
10 31652794₈ 8 x 31652794 = 253222352
11 3165279₈4 8 x 31652794 = 253222352
12 31652₈794 8 x 31652794 = 253222352
13 3165₈2794 8 x 31652794 = 253222352
14 316₈52794 8 x 31652794 = 253222352
15 31₈652794 8 x 31652794 = 253222352
16 36715₈924 8 x 36715924 = 293727392
17 36715924₈ 8 x 36715924 = 293727392
18 3671592₈4 8 x 36715924 = 293727392
19 367159₈24 8 x 36715924 = 293727392
20 3671₈5924 8 x 36715924 = 293727392
21 367₈15924 8 x 36715924 = 293727392
22 36₈715924 8 x 36715924 = 293727392
23 3₈1652794 8 x 31652794 = 253222352
24 3₈6715924 8 x 36715924 = 293727392
25 4231₇8956 7 x 42318956 = 296232692
26 42318₇956 7 x 42318956 = 296232692
27 42318956₇ 7 x 42318956 = 296232692
28 4231895₇6 7 x 42318956 = 296232692
29 423189₇56 7 x 42318956 = 296232692
30 423₇18956 7 x 42318956 = 296232692
31 42₇318956 7 x 42318956 = 296232692
32 43₇861529 7 x 43861529 = 307030703
33 4386152₇9 7 x 43861529 = 307030703
34 43861529₇ 7 x 43861529 = 307030703
35 438615₇29 7 x 43861529 = 307030703
36 43861₇529 7 x 43861529 = 307030703
37 4386₇1529 7 x 43861529 = 307030703
38 438₇61529 7 x 43861529 = 307030703
39 ₄53967128 4 x 53967128 = 215868512
40 462351₇89 7 x 46235189 = 323646323
41 4623518₇9 7 x 46235189 = 323646323
42 46235189₇ 7 x 46235189 = 323646323
43 46235₇189 7 x 46235189 = 323646323
44 4623₇5189 7 x 46235189 = 323646323
45 462₇35189 7 x 46235189 = 323646323
46 46₇235189 7 x 46235189 = 323646323
47 4₆8915732 6 x 48915732 = 293494392
48 ₄69573218 4 x 69573218 = 278292872
49 4₇2318956 7 x 42318956 = 296232692
50 4₇3861529 7 x 43861529 = 307030703
51 4₇6235189 7 x 46235189 = 323646323
52 48₆915732 6 x 48915732 = 293494392
53 48915₆732 6 x 48915732 = 293494392
54 48915732₆ 6 x 48915732 = 293494392
55 4891573₆2 6 x 48915732 = 293494392
56 489157₆32 6 x 48915732 = 293494392
57 4891₆5732 6 x 48915732 = 293494392
58 489₆15732 6 x 48915732 = 293494392
59 52₇861439 7 x 52861439 = 370030073
60 5286143₇9 7 x 52861439 = 370030073
61 52861439₇ 7 x 52861439 = 370030073
62 528614₇39 7 x 52861439 = 370030073
63 52861₇439 7 x 52861439 = 370030073
64 5286₇1439 7 x 52861439 = 370030073
65 528₇61439 7 x 52861439 = 370030073
66 53₄967128 4 x 53967128 = 215868512
67 539₄67128 4 x 53967128 = 215868512
68 5396₄7128 4 x 53967128 = 215868512
69 5396712₄8 4 x 53967128 = 215868512
70 53967128₄ 4 x 53967128 = 215868512
71 539671₄28 4 x 53967128 = 215868512
72 53967₄128 4 x 53967128 = 215868512
73 5₄3967128 4 x 53967128 = 215868512
74 5621₈9743 8 x 56219743 = 449757944
75 56219743₈ 8 x 56219743 = 449757944
76 5621974₈3 8 x 56219743 = 449757944
77 562197₈43 8 x 56219743 = 449757944
78 56219₈743 8 x 56219743 = 449757944
79 562₈19743 8 x 56219743 = 449757944
80 56₈219743 8 x 56219743 = 449757944
81 5₇2861439 7 x 52861439 = 370030073
82 5₈6219743 8 x 56219743 = 449757944
83 ₆48915732 6 x 48915732 = 293494392
84 6₄9573218 4 x 69573218 = 278292872
85 ₆73215489 6 x 73215489 = 439292934
86 6₇9435812 7 x 69435812 = 486050684
87 69435₇812 7 x 69435812 = 486050684
88 69435812₇ 7 x 69435812 = 486050684
89 6943581₇2 7 x 69435812 = 486050684
90 694358₇12 7 x 69435812 = 486050684
91 6943₇5812 7 x 69435812 = 486050684
92 69₄573218 4 x 69573218 = 278292872
93 694₇35812 7 x 69435812 = 486050684
94 695₄73218 4 x 69573218 = 278292872
95 6957321₄8 4 x 69573218 = 278292872
96 69573218₄ 4 x 69573218 = 278292872
97 695732₄18 4 x 69573218 = 278292872
98 69573₄218 4 x 69573218 = 278292872
99 6957₄3218 4 x 69573218 = 278292872
100 69₇435812 7 x 69435812 = 486050684
101 732154₆89 6 x 73215489 = 439292934
102 7321548₆9 6 x 73215489 = 439292934
103 73215489₆ 6 x 73215489 = 439292934
104 73215₆489 6 x 73215489 = 439292934
105 7321₆5489 6 x 73215489 = 439292934
106 732₆15489 6 x 73215489 = 439292934
107 73₆215489 6 x 73215489 = 439292934
108 ₇42318956 7 x 42318956 = 296232692
109 ₇43861529 7 x 43861529 = 307030703
110 ₇46235189 7 x 46235189 = 323646323
111 ₇52861439 7 x 52861439 = 370030073
112 7₆3215489 6 x 73215489 = 439292934
113 ₇69435812 7 x 69435812 = 486050684
114 ₇95642138 7 x 95642138 = 669494966
115 ₈29761354 8 x 29761354 = 238090832
116 ₈31652794 8 x 31652794 = 253222352
117 ₈36715924 8 x 36715924 = 293727392
118 ₈56219743 8 x 56219743 = 449757944
119 9564213₇8 7 x 95642138 = 669494966
120 95642138₇ 7 x 95642138 = 669494966
121 956421₇38 7 x 95642138 = 669494966
122 95642₇138 7 x 95642138 = 669494966
123 9564₇2138 7 x 95642138 = 669494966
124 956₇42138 7 x 95642138 = 669494966
125 95₇642138 7 x 95642138 = 669494966
126 9₇5642138 7 x 95642138 = 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 :

AB^C + DE^F + GH^I =

{

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

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com