- 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
[ 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 leeds 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'.
[ 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.
\({ 987654312 \over 123456789 } = 8\)
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 = \(\frac{3}{2}\) 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 = \(\frac{5}{2}\) 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...
[ 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 |
[ 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 | 249758361 |
22 + 34 + 59 + 77 + 115 + 138 + 173 + 196 + 231 | 9 terms |
865719324 | 9 |
2 | 497218365 |
24 + 39 + 57 + 72 + 111 + 138 + 173 + 196 + 235 | 9 terms |
869315742 | 9 |
3 | 769318542 |
27 + 36 + 59 + 73 + 111 + 138 + 175 + 194 + 232 | 9 terms |
819235764 | 9 |
4 | 829415763 |
28 + 32 + 59 + 74 + 111 + 135 + 177 + 196 + 233 | 9 terms |
459723816 | 8 |
5 | 893745216 |
28 + 39 + 53 + 77 + 114 + 135 + 172 + 191 + 236 | 9 terms |
149265738 | 9 |
6 | 968234571 |
29 + 36 + 58 + 72 + 113 + 134 + 175 + 197 + 231 | 9 terms |
895713426 | 9 |
7 | 972185436 |
29 + 37 + 52 + 71 + 118 + 135 + 174 + 193 + 236 | 9 terms |
362859174 | 9 |
8 | 983642571 |
29 + 38 + 53 + 76 + 114 + 132 + 175 + 197 + 231 | 9 terms |
895431276 | 9 |
[ 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
[ 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 | nth |
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 |
[ 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. | Ninedigital | Multiplicand | Palindrome | Progr. substrings |
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.
[ 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
|
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 |
|
[ 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 |
[ 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 |
|
[ 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.
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