[ 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 !
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
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.
[ 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!
[ March 21, 2021 ] Integral Triangles with Ninedigital Areas by Daniel Hardisky
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 "
Example: 413829576 (not the answer)
/\ / \ a / \ b / Area \ /____________\ c
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 !
[ 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.
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.
[ 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.
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 !
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 :
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!)."
[ March 11, 2021 ] A ninedigit problem from Daniel Hardisky
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 ?
4. What is the smallest prime that cannot divide any ninedigital or pandigital number ?
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
But what makes this equation beautiful is that left and right of the central 9 we unveil two curious pandigitals
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.
The story is not at its end. What about that middle 9 ?
[ 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 ?
What can we find in ninedigitals raised to the power 2
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.
2 NOV_substrings with power 3 → 2 solutions
{ 3 NOV_substrings with power 4, 5, 6, 7, 8 & 9 → none found }
{ 4 NOV_substrings with power 11, 12, 13, 14, 15, 16 & 17 → none found }
4 NOV_substrings with power 18 → 1 unique solution.
{ 5 NOV_substrings with power 19 upto 53 → none found }
5 NOV_substrings with power 54 → 1 unique solution.
{ 6 NOV_substrings with power 55 upto 85 → none found }
6 NOV_substrings with power 86 → 1 unique solution.
{ 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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