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[ October 15, 2021 ]
Daniel Hardisky has some new ninedigital fractions to share.
(Many others can be found f.i. in Page 3 of this ninedigits section)

The equivalent fractions given here are quite interesting.
Each one of them uses all the digits from 1 to 9 one !

2
6
 = 3
9
 = 58
174

2
4
 = 3
6
 = 79
158

 


Arrange all the digits (1 to 9), each once to form the following fractions :
Examples
3 * 4
5 + 7
= 2 * 8
1 + 6 + 9
1

1 * 8
6 + 9
= 2 * 4
3 + 5 + 7
8
15

5 * 8
3 + 7
= 4 * 9
1 + 2 + 6
4

What integers are possible ?                  
1, 2, 4, 6
What fractions are possible ?
 1/2, 2/7, 8/15, 3/4, 4/5, 8/7, 8/5

 


And finally a challenge for our readers!
Arrange the digits from 1 to 9, each once
to form fractions as shown:
6 + 9
7 + 8
 = 3 + 4
2 + 5
 = 1

5 + 8
4 + 9
 = 3 + 6
2 + 7
 = 1

Can you find other combinations
with the last ratio > 1 ?

 





[ April 4, 2021 ]
Separate ninedigitals in three groups of 2, 3 and 4 digits and multiply them together.
When the product is also ninedigital it should be one of the 205 entries of the table below.
presented by Alexandru Dan Petrescu

[by PDG] A thing of beauty pops up with entry 125. And it is unique !
Enjoy this trio of ninedigitals.

724135698 
+
72 * 413 * 5698 = 169435728
= 
893571426



[ March 25, 2021 ]
Separate ninedigitals in three threedigit groups and multiply them together.
When the product is also ninedigital it should be one of the 39 entries of the table below.
presented by Alexandru Dan Petrescu

The ninedigital in line 34 is the largest product and was already noted by Janean Wilson.
Reference : Wonplate 95 - A ninedigital divertimento (second case)
At least we now have all 39 solutions!

indexabcdefghi
(sorted)
abc * def * ghiABCDEFGHI
1163827945163 * 827 * 945127386945
2234561987234 * 561 * 987129567438
3237618954237 * 618 * 954139728564
4243691875243 * 691 * 875146923875
5248751963248 * 751 * 963179356824
6251738964251 * 738 * 964178569432
7256743891256 * 743 * 891169475328
8261538947261 * 538 * 947132975846
9261594837261 * 594 * 837129763458
10263871945263 * 871 * 945216473985
11281547936281 * 547 * 936143869752
12291534867291 * 534 * 867134726598
13312564897312 * 564 * 897157843296
14319572846319 * 572 * 846154367928
15324659871324 * 659 * 871185972436
16329576841329 * 576 * 841159372864
17342671958342 * 671 * 958219843756
18364581792364 * 581 * 792167495328
19381657942381 * 657 * 942235798614
20387641952387 * 641 * 952236159784
21413568927413 * 568 * 927217459368
22418756923418 * 756 * 923291675384
23423581796423 * 581 * 796195627348
24428657913428 * 657 * 913256731948
25432571869432 * 571 * 869214357968
26432597618432 * 597 * 618159384672
27438516729438 * 516 * 729164759832
28452871936452 * 871 * 936368495712
29463581927463 * 581 * 927249365781
30472518693472 * 518 * 693169435728
31495681723495 * 681 * 723243719685
32513872946513 * 872 * 946423179856
33531768924531 * 768 * 924376814592
34531876942531 * 876 * 942438176952
35536841927536 * 841 * 927417869352
36567843912567 * 843 * 912435918672
37579612843579 * 612 * 843298715364
38612743958612 * 743 * 958435617928
39639725841639 * 725 * 841389614275



[ March 21, 2021 ]
Integral Triangles with Ninedigital Areas by Daniel Hardisky

" I have
a few of these integral triangles with ninedigital areas. These are oblique.
integral right triangles with ninedigital areas.
integral oblique triangles with two sides which are ninedigital numbers AND integer area.

So far I cannot find ninedigital numbers on all three sides of any triangle AND integer area.
We are working on this on my math page.

Checked with Wolfram Alpha for accuracy.

Regards, Daniel Hardisky "

Find integral oblique triangles with the
Areas = 9 digit integers, using all digits
1 to 9 (re-arranged).

Example: 413829576 (not the answer)

              /\
            /   \
        a /      \ b
        /   Area  \
      /____________\
             c
Daniel Hardisky
  Triangles with rational cosines:
a = 2q(p–k), Cosϑ = p/q
b = q2 – k2
c = q2 + k2 – 2pk
–q < k < p
Let q = m2 + n2, p = m2 – n2
n2 = 2mn
or p = 2mn and n2 = m2 – n2
Then Sinϑ = (n2)/q
Area = 1/2abSinϑ = 1/2ab(n2)/q


[ March 23, 2021 ]
" Hi Patrick,
I send you in addition the table for Integral Triangles (oblique) with ninedigital areas by Daniel Hardisky.
Best regards, Alexandru Dan Petrescu
"
[Alignment optimized for Google Chrome]



And here the solutions for the Integral Right Triangles with ninedigital areas
as proposed by Daniel Hardisky [ March 26, 2021 ].

Note that triangles 30 and 31 produce the same ninedigital area ( highlighted ). Just click on the table to see !

[Alignment optimized for Google Chrome]



[ March 18, 2021 ]
Nine- and pandigitals with 5 multiples from Alexandru Dan Petrescu

We checked for ninedigitals and pandigitals having the greatest number of multiples
that are also ninedigitals and pandigitals.

For ninedigital numbers there is only one solution with 5 multiples.

Ninedigital
123456789
x2246913578
x4493827156
x5617283945
x7864197523
x8987654312

For pandigital numbers there are two solutions with 5 multiples.
One of them being the trivial case 1234567890 which is the above ninedigital x 10.
The other one being 1098765432.

Pandigital
1098765432
x22197530864
x44395061728
x55493827160
x77691358024
x88790123456

[ March 21, 2021 ] Addendum by Patrick De Geest
The above found multiples return in an investigation I started some years ago.
The goal was to find all  Products of ninedigitals with pandigitals that result in squares .
The first few solutions are identical with the multiples from Petrescu's table.

123456789 x 1975308624 = 4938271562
123456789 x 3086419725 = 6172839452
123456879 x 3086421975 = 6172843952
- - -

And here I stopped the search at the time in favor of other projects.
Of course there is a relation that I will highlight here.
The relation shows up when variables are put in place !

123456789 [n] x 1975308624 [16n] = 4938271562 [4n or sqrt(16n2)]
123456789 [n] x 3086419725 [25n] = 6172839452 [5n or sqrt(25n2)]
123456879 [n] x 3086421975 [25n] = 6172843952 [5n or sqrt(25n2)]

Some logical questions that arise are for instance
1. Can you extend and/or complete the list ?
2. Are solutions abundant or rare ?
3. Exist there non nine- or pandigital squares ?
   Must the nine- and pandigital numbers share their factors?

Variations on the theme could be :

The product of two ninedigitals is a square.
123456789 [n] x 493827156 [4n] = 2469135782 [2n or sqrt(4n2)]
The multiplier need not be a nine- or pandigital
123456789 [n] x 6049382661 [49n] = 8641975232 [7n or sqrt(49n2)]
123456789 [n] x 7901234496 [64n] = 9876543122 [8n or sqrt(64n2)]

Ps. note that there are some sporadic solutions given by Peter Kogel. Study this page first if you're interested.
See my webpage The Nine Digits Page 2 under 'digital diversions'

Soon after [ March 22, 2021 ] Alexandru Dan Petrescu wrote

" Relating to products of ninedigitals with pandigitals resulting in squares I extended/completed your list.
There are 8 solution for x4 and 512 solution for x5.
Interesting, powers of 2, and for x5 exponent is 9 (again 9!).
"

[ 1 ] 123456789 [n] * 1975308624 [16n] = 4938271562 [4n or sqrt(16n2)]
[ 2 ] 129465573 [n] * 2071453968 [16n] = 5178634922 [4n or sqrt(16n2)]
[ 3 ] 158729463 [n] * 2539671408 [16n] = 6349178522 [4n or sqrt(16n2)]
[ 4 ] 158794623 [n] * 2540713968 [16n] = 6351784922 [4n or sqrt(16n2)]
[ 5 ] 184573629 [n] * 2953178064 [16n] = 7382945162 [4n or sqrt(16n2)]
[ 6 ] 237841956 [n] * 3805471296 [16n] = 9513678242 [4n or sqrt(16n2)]
[ 7 ] 237946581 [n] * 3807145296 [16n] = 9517863242 [4n or sqrt(16n2)]
[ 8 ] 246913578 [n] * 3950617248 [16n] = 9876543122 [4n or sqrt(16n2)]



[ March 11, 2021 ]
A ninedigit problem from Daniel Hardisky

Re-arrange the digits 1 to 9 to make
a 9 digit number. These numbers are always divisible by 9 since
the sum of the digits = 45 and is also divisible by 9.
Example: 619428753 / 9 = 68825417 (not the answer)

1. What is the largest prime found after dividing one of these 9
digit numbers by 9 ?
2.Which of these 9 digit numbers has the greatest number of
divisors ?

Daniel Hardisky

1. 987654231 / 9 = 109739359
2. 769152384 768 divisors

Of course more questions can be posed around this ninedigital and prime topic.

3. Can you find the pandigital equivalent for the above problems 1 and 2 ?

Solution by Alexandru Dan Petrescu [ March 12, 2021 ]
The pandigital equivalent of the two questions proposed by Daniel Hardisky.
P1) What is the largest prime found after dividing one of the pandigital numbers by 9 ?
A1) 9876541023 = 9 x 1097393447. (1097393447 being prime)
P2) Which of pandigital number has the greatest number of divisors ?
A2) 7691523840 has 1728 divisors. Factorization: 7691523840 = 2^8 x 3^2 x 5 x 7 x 11 x 13 x 23 x 29

In PARI/GP the number of divisors can be reproduced with the command
n=7691523840;
length(divisors(n));

4. What is the smallest prime that cannot divide any ninedigital or pandigital number ?

Solution by Alexandru Dan Petrescu [ March 13, 2021 ]
The smallest prime that cannot divide any ninedigital number is
44449
The smallest prime that cannot divide any pandigital number is
111119

In hindsight, now that we know Alexandru's solutions, we see that this problem
was already discussed in the past. Here is the source
Puzzle 926. pandigital and prime numbers
Nevertheless, a second opinion can't do any harm :)
Note also that both solutions belong to the same OEIS sequence A090148. What a coincidence!

Both numbers were also already registered in the Prime Curios! database.
Prime Curios! 44449
Prime Curios! 111119


[ October 9, 2020 ]
From my collection of palindromic quasipronic numbers of the form n*(n+5)
may I present a remarkable repdigital number (see Index Nr 24).

Multiply this eleven digit repdigit 22222222222 with 22222222227
and we get the following palindrome with 21 digits

493827160595061728394

But what makes this equation beautiful is that left and right of the central 9
we unveil two curious pandigitals

4938271605 (9) 5061728394 = L (9) R

Moreover there is a surprising order in the arrangement of the ten digits
From left to right we start with 4 downto 0 intertwined with, also from left to right, the sequence from 9 downto 5.

4938271605 (9) 5061728394

The story is not at its end. What about that middle 9 ?

Divide the pandigital L = 4938271605 with our 9 and see what happens...
Indeed a new palindrome pops up nl. 548696845

And if we add the two pandigitals together we have
L + R = 99999999999, an eleven digit repdigit.

Have you noticed that L * 2 equals the largest pandigital number !
L * 2 =  9876543210 

Now, if we divide R by 2 surprisingly we see another pandigital number popping up !
R / 2 = 5061728394 / 2 =  2530864197 

Remember we started with 22222222222 and 22222222227 with a difference of 5.
Can we bring that number 5 into the game? Sure, we can. Here is how: divide L by 5
L / 5 = 4938271605 / 5 =  987654321 
and we end up with the largest ninedigital number!

But wait, have you tried R – L ?
We get the very first ninedigital  123456789  !

So many topics with a wow factor come together in this story.
Yet, I feel not every chapter is written... can you add more ?


[ December 26, 2016 ]
An astonishing e_quation using just our familiar nine digits.

Incredible Formula - Numberphile



When we put those nine digits in a row we get the number 194673285.
Anyone there who can turn this ninedigital into another curio ?



[ July 23, 2015 ]
Finding one or more ninedigitals as a substring in the decimal expansion
of some ninedigital raised to a power p.

What can we find in ninedigitals raised to the power 2

There are a lot of them but I will concentrate on those with the highest number of ninedigital substrings.
In the case of the power 2 this maximum is with 3 substrings.
162978354 2 =
26561943872549316
26561943872549316
26561943872549316

267453981 2 =
71531631952748361
71531631952748361
71531631952748361

294137658 2 =
86516961853724964
86516961853724964
86516961853724964

418739652 2 =
175342896157081104
175342896157081104
175342896157081104

981425736 2 =
963196475283141696
963196475283141696
963196475283141696v

Let us continue with minimal four ninedigital substrings. I found one with power 3.
It is a nice four in a row solution.
896134527 3 =
719647185932647507781421183
719647185932647507781421183
719647185932647507781421183
719647185932647507781421183

Now, looking for at least 5 ninedigital substrings we have to go to power 7 already.
Powers 4, 5 & 6 yield no records.
351724698 7 =
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472

614925783 7 =
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727

Two nice stepladder_5 solutions !

Let me thicken the plot at this point and leave behind us these rather trivial overlapping solutions.
Instead let me try to hunt for strictly NON_OVERLAPPING [ further on referred as NOV ] ninedigitals substrings.
(Written in Ubasic - program name 'ssninx.ub')
Let us find all smallest solutions from 2 to 9 ninedigital substrings.

{ 2 NOV_substrings with power 2 → none found }

2 NOV_substrings with power 3 → 2 solutions

297146853 3 =
26236953487129018576392477
368571429 3 =
50068548279613994372186589

{ 3 NOV_substrings with power 4, 5, 6, 7, 8 & 9 → none found }

3 NOV_substrings with power 10 → 2 solutions.
It is only at this power 10 that three separated ninedigital substrings appear !

271593864 10 =
2183733009396738195247169455080723624976528132747281610186189679076884521963776638976
479635182 10 =
644332915723468747863172845933720713156733737801139368957793052380596182495377326490624

{ 4 NOV_substrings with power 11, 12, 13, 14, 15, 16 & 17 → none found }

4 NOV_substrings with power 18 → 1 unique solution.

126593784 18 =
69730514093015917842362635285515966162390979853386533698312687378594850347845
437769987132465705319852613947799401827653080762942141372458699063296

{ 5 NOV_substrings with power 19 upto 53 → none found }

5 NOV_substrings with power 54 → 1 unique solution.

457621839 54 =
4649048423815679016881693761707994251367846137627148837809849977509112339115
0893624664041607059823169958996145557237067013400497642008728700778227180271
6619828591635274809169528631549679091382639154700948359979571366641842161148
3357873801497413641003354898905326654648841937365068268075968566615198849564
1847508401637028241230305400402087323352464727855808613518043631524098987819
7644865743219148203100219738559091235979416283395521378773569037854028732788
952004958241

{ 6 NOV_substrings with power 55 upto 85 → none found }

6 NOV_substrings with power 86 → 1 unique solution.

483259761 86 =
691089971277284621583973839908321146944631601201133003854843772671615448200671
262685915616836442117193245682799504223144629651743867521615002321668923138270
732687951714743751378782961329254342951134325410090163829860183409176492183553
310746630772058428151699835838580691150692415388449686925020318868688964370912
798936026644796788897118723452353835177251420939765727091581345769228796322211
232349371258688904374257677622623182596912041405433457744567399420686374957586
747787843855865628218707234366023809207264616321153936468531652523622910413452
658685093044935968826596483821451367475207986095872295275022536365973380099994
536865981060519036886188996252316309481443530912678465755435071179141665451617
176067797792325538546657974609472790542707361

{ 7 NOV_substrings with power 87 upto 119 → none found }

From power 120 and above OVERFLOW occurred and marked the end of the game for me.















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E-mail address : pdg@worldofnumbers.com