[ September 15, 2003 ] “Ninedigital fractions” equations and their distinct integer solutions ( sometimes generalized to “Ninedigital expressions” )
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by Terry Trotter (email) by Jean Claude Rosa (email) by Patrick De Geest (email) by Jerry Levy (email)
Introduction see Levy Expressions by Terry Trotter (†)
Variation #0 : the original equation A/BC + D/EF + G/HI [L. Mittenzwey, circa 1880]
BC, EF and HI are not digit products but digit concatenations.
Variation #1 : the equation A/(B*C) + D/(E*F) + G/(H*I) [Jerry Levy's suggestion]
Why do I underscore the specification distinct ? (info by J.C.R.)
We can say that, for example, the following 3 solutions are identical : 1/(3*6) + 5/(8*9) + 7/(2*4) = 1 1/(6*3) + 5/(8*9) + 7/(4*2) = 1 5/(9*8) + 1/(6*3) + 7/(2*4) = 1 Likewise for each solution we can produce an extra of 47 but identical ones, meaning 48 in total (including the original solution). Why 48 ? For each triplet, A/(B*C) for example, we have 2 possible dispositions : A/(B*C) and A/(C*B) There are 3 triplets so 8 possible dispositions ( 8 = 2*2*2 ). On the other hand the order in which we write the triplets in the expression has no effect on the final value. Example: N = 1/(3*6) + 5/(8*9) + 7/(2*4) N = 5/(8*9) + 1/(3*6) + 7/(2*4) There are 6 different dispositions among our 3 triplets so in total we have 6*8, meaning 48 different dispositions giving the same result. Samples N = 1 ; 1/(3*6) + 5/(8*9) + 7/(2*4) so 1 distinct solution. N = 2 ; 5/(1*8) + 7/(2*4) + 9/(3*6) N = 2 ; 5/(2*4) + 7/(1*8) + 9/(3*6) so 2 distinct solutions. N = 4 ; 5/(3*4) + 6/(8*9) + 7/(1*2) N = 4 ; 8/(4*9) + 5/(3*6) + 7/(1*2) so 2 distinct solutions. N = 5 ; 5/(3*8) + 7/(4*6) + 9/(1*2) N = 5 ; 5/(4*6) + 7/(3*8) + 9/(1*2) so 2 distinct solutions. Note that we have 7*48 or 336 undistinct solutions.
Note that we have 7*48 or 336 undistinct solutions.
Variation #2 : the equation (A/B)*C + (D/E)*F + (G/H)*I [Jerry Levy's suggestion]
Variation #3 : the equation (A/B)^C + (D/E)^F + (G/H)^I [Jerry Levy's suggestion]
WTM column see Power-full Fractions by Terry Trotter
135 different values - 200 distinct solutions (see subscript values)
Six distinct solutions exist for integer value 672 due to sums of powers of 2 permutations. Six cases with three distinct solutions : values 17575, 59433, 78893, 1953509, 4783257 & 40353895.
Fourteen numbers turned out to be prime. Prime Curios! 241 is the smallest and Prime Curios! 4784009 is the largest of the lot.
Two numbers happen to be palindromic. 2222 is the first which is in fact also a repdigit. 23432 is the second one.
Variation #4 : the equation A/(B+C) + D/(E+F) + G/(H+I) [Jean Claude Rosa's suggestion]
Variation #5 : the equation (A+B)/C + (D+E)/F + (G+H)/I [Jean Claude Rosa's suggestion]
Note that 13 has 13 distinct solutions, ain't we lucky this time !!
Variation #6 : the equation A/(B–C) + D/(E–F) + G/(H–I) [Terry Trotter's suggestion]
Note the symmetry in values and distinct solutions.
All values have composite distinct solutions, except for –9 and 9 which have a prime number of solutions namely 113.
Variation #7 : the equation (A–B)/C + (D–E)/F + (G–H)/I [Terry Trotter's suggestion]
Values –2 and 2 each have the record palindromic number of solutions nl. 212. We have prime number of solutions for the first and last two values.
Variation #8 : the equation (A+B)*C + (D+E)*F + (G+H)*I [Terry Trotter's suggestion]
129 different values - 7560 distinct solutions (see subscript values) This equation always deliver integer solutions whatever ninedigital combination is used. In total we have factorial 9 (or 9! = 362880) undistinct solutions. Values range from 70 to 198 without gaps.
Value 162 yields the record number of distinct solutions nl. 206 Value 70 gives only 1 but unique distinct solution.
Variation #9 : the equation (A+B)^C + (D+E)^F + (G+H)^I [Terry Trotter's suggestion]
6283 different values - 7560 distinct solutions (see subscript values) Here also all possible combinations yield integer results. There are too many integer solutions to display them all. I shall restrict myself to some subgroups and highlighting the curios.
Value 915 is the smallest solution. Value 38443890843 is the largest solution.
Speaking of palindromes... I count 287 primes with 1 distinct solution and I count 76 primes with 2 distinct solutions.
Value 941 (Prime Curios!) is the smallest prime solution. Value 38443418449 (Prime Curios!) is the largest prime solution.
This must be the ninedigital find of the year ! Look at the following unique dual ninedigital equation :
Variation #9 shows no other such ninedigital (or even pandigital) solutions.
Variation #10 : the equation (A–B)*C + (D–E)*F + (G–H)*I [Terry Trotter's suggestion]
205 different values - 60480 distinct solutions (see subscript values) All possible combinations result in integer solution either positive or negative. Values range from –102 to 102 without gaps. The number of negative solutions is equal to the number of positive solutions.
Variation #11 : the equation (A–B)^C + (D–E)^F + (G–H)^I [Terry Trotter's suggestion]
10426 different values - 60480 distinct solutions (see subscript values) All possible combinations result in integer solution either positive or negative. Like variation #9 there are too many integer solutions to display. Therefore I shall restrict myself to some interesting categories and/or highlighting the curios. Values range from –40369992 to 40370007 but not consecutively, there are gaps.
3649 values are negative, 1 value is equal to 0 and 6776 values are positive. Note that this last total is palindromic ! Variation #11 shows for the first time that the negative values do not match the positive values.
The beast enclosed 46664 = (1–4)^2 + (3–9)^6 + (7–8)^5 46664 = ... (46664 has like 666 also 12 distinct solutions !) See also WONplate 154
1111 = (2–6)^4 + (7–5)^9 + (8–1)^3 1111 = (6–2)^4 + (7–5)^9 + (8–1)^3 1111 = (6–9)^4 + (7–3)^5 + (8–2)^1 1111 = (7–3)^5 + (8–2)^1 + (9–6)^4
7777 = (4–7)^1 + (6–8)^2 + (9–3)^5 7777 = (4–7)^1 + (8–6)^2 + (9–3)^5
There is one value having the record number of distinct solutions namely... 1 itself !!
Variation #12 : the equation A^B/C + D^E/F + G^H/I [Jerry Levy's suggestion]
Almost all values (1364) have just 1 distinct solution. Only 64 values show 2 distinct solutions.
Variation #13 : the equation A^(B/C) + D^(E/F) + G^(H/I) [Jerry Levy's suggestion]
259 different values - 468 distinct solutions (see subscript values) Values range from 30 to 134217881 but not consecutively, there are gaps.
Variation #14 : the equation (A+B)*C + (D–E)/F + G^H*I [Terry Trotter's all five operations suggestion]
13253 different values - 45360 distinct solutions (see subscript values) Values range from 24 to 939524151 but not consecutively, there are gaps.
Let me quote Terry himself about the expression (A + B)*C + (D – E)/F + G^H*I :
" I have designed this new variant for our Levy Expressions project -- #14. Since I did add it to my webpage, I thought you might like to know about it, at least. Note that all the previous expressions used but 2 operations in each triad. Now I'm using all five operations, based on the following logic : We (or at least I did) teach in school math the following hierarchy of operations : 1. addition 2. subtraction 3. multiplication 4. division 5. exponentiation This goes from easy to 'hard', for most kids. At least, we teach them in that order. Plus when we study logarithms, we show that to multiply two numbers, we add the logs; when we divide two numbers, we subtract their logs; and when we need to raise a number to a power, i.e. exponentiate, we multiply the base's log by the exponent. Hence, I used #1 & 3 with A, B & C; used #2 & 4 with D, E & F; and used #3 & 5 with G, H & I. Hey, it may be corny, but I'm trying to teach kids to like math & look at it in a new way. :>) And it does review the order of operations with a novel approach. Hope you like it. "
The first thing that catches my attention is that this variation produced a record number of different values nl. 13253. This can makes room for lots of discoveries I hope.
152 palindromes can be detected. The largest one is 327723 = (6+7)*3 + (9–1)/2 + 4^8*5
[ August 31, 2003 ] Pandigitals divisible by 5-digit palprimes by Jean Claude Rosa (email)
Jean Claude continued "WONplate 114 continued..." and researched the case of pandigitals divisible by a palindromic prime or palprime for short and more in particular by a 5-digit palprime (in total there are 93 - from 10301 to 98689). 24 palprimes are concerned in 27 solutions. No results exist for the 'not listed' palprimes !
smallest palprime: 1037625498/12421 = 83538 largest palprime: 8713609542/97579 = 89298
There are some curious palindromes expressable in two ways.
And of course the obligatory Number of the Beast in
Perhaps someone would like to extend the ninedigital variant of this same exercise.
[ August 25, 2003 ] WONplate 114 continued... by Jean Claude Rosa (email)
WONPlate 114 discusses 'Palindromic quotients through pandigital divisions' and only the denominators 2 to 9 were considered. Jean Claude extended the topic and investigated the pandigitals divided by palindromic primes limiting himself to 11 and the fifteen threedigit palprimes from 101 to 929.
Divisor 11 gave 14 solutions.
divisor 11 number of solutions: 14 smallest: 2063784591/11 = 187616781 largest: 8459120637/11 = 769010967
This number of solutions (#14) already predicts how many of the fifteen palprimes will yield results !
divisor 101 number of solutions: nihil
The fourteen remaining palprimes give the following data
divisor 151 number of solutions: 3 smallest: 1374086259/151 = 9099909 largest: 6509781234/151 = 43111134
divisor 181 number of solutions: 3 smallest: 1078324695/181 = 5957595 largest: 1647083529/181 = 9099909
divisor 191 number of solutions: 7 smallest: 1304985726/191 = 6832386 largest: 9781342065/191 = 51211215
divisor 313 number of solutions: 4 smallest: 1078423659/313 = 3445443 largest: 2914386507/313 = 9311139
divisor 353 number of solutions: 3 smallest: 2108749635/353 = 5973795 largest: 4196750283/353 = 11888811
divisor 373 number of solutions: 6 smallest: 1028947356/373 = 2758572 largest: 7219560843/373 = 19355391
divisor 727 number of solutions: 4 smallest: 4069713285/727 = 5597955 largest: 6870549123/727 = 9450549
divisor 757 number of solutions: 5 smallest: 1982705634/757 = 2619162 largest: 5481760239/757 = 7241427
divisor 787 number of solutions: 4 smallest: 2389570461/787 = 3036303 largest: 7129046583/787 = 9058509
divisor 797 number of solutions: 4 smallest: 1930526874/797 = 2422242 largest: 5360841972/797 = 6726276
divisor 919 number of solutions: 6 smallest: 1587362049/919 = 1727271 largest: 6971402583/919 = 7585857
divisor 929 number of solutions: 5 smallest: 1765082349/929 = 1899981 largest: 8095471362/929 = 8714178
In total we have 61 solutions, a nice prime !
As always the Number of the Beast lurks around the corner (two corners to be more precise!).
Perhaps someone would like to extend the ninedigital variant of this same exercise and see if there's anything interesting going on there, like for instance the following unique nine- & pandigital equation :
ps. the Number of the Ninedigital Beast houses here :
[ August 20, 2003 ] Pandigital divisions equal to ratio N/D
Terry Trotter (email) inspired by Puzzle 121 from Michael Winckler's site (Puzzle No. 121), expanded the topic by investigating more ratios N/D and looking not only at ninedigit numbers
but also at the 10-digit pandigital version.
The ninedigit version with ratios 1/2, 1/3 up to 1/9 are already presented in WONplate 107 Now Terry has data for all the fractions in the range of 0 < N < D < 20 for both 9-digit & 10-digit work. For this purpose he asked and got a ubasic program written by Carlos Rivera. The code is short and straightforward so I'll list it here (some minor modifications from myself pdg). (Change in line 20 the values N1 and N2 for other fractions)
10 'trotter.ub 20 B1=1000:B2=99999:N1=4:N2=5:Cc=0 30 FF=6469693230 : 'use FF=3234846615 for 9-digit version 35 ' FF=1:for I=0 to 9:FF*=prm(I+1):next I : 'use for I=1 to 9 for 9-digit version 40 for B=B1 to B2:if B@N2<>0 then goto 70 50 A=B*N1\N2:Z=A*10^alen(B)+B 60 gosub *TEST:if T=1 then inc Cc:print Cc,A;"=";N1;"/";N2;"*";B 70 next B 80 end 90 *TEST:T=0:F=1:L=alen(Z) 100 for I=1 to L:D=val(mid(str(Z),I+1,1)):P=prm(D+1) 110 if F@P=0 then cancel for:return 120 F*=P:next I 130 if F=FF then T=1 140 return
Part 1.
All possible ninedigit solutions (numerator > 1) and (denominator < 10)
4 / 5 = 9876 / 12345 1 unique and beautiful solution
2 / 7 = 3654 / 12789 2 / 7 = 3674 / 12859 2 / 7 = 5342 / 18697 2 / 7 = 7418 / 25963 2 / 7 = 9786 / 34251 2 / 7 = 9862 / 34517 6 solutions
2 / 9 = 3924 / 17658 2 / 9 = 7596 / 34182 2 solutions
(numerator >= 1) and (10 < denominator < 20)
4 / 11 = 6492 / 17853 4 / 11 = 7836 / 21549 2 solutions
5 / 11 = 9765 / 21483 1 unique solution
1 / 12 = 3816 / 45792 1 / 12 = 6129 / 73548 1 / 12 = 7461 / 89532 1 / 12 = 7632 / 91584 4 solutions
5 / 12 = 8235 / 19764 1 unique solution
1 / 13 = 5184 / 67392 1 / 13 = 6273 / 81549 1 / 13 = 7281 / 94653 3 solutions
2 / 13 = 2538 / 16497 2 / 13 = 3876 / 25194 2 / 13 = 3918 / 25467 2 / 13 = 4518 / 29367 2 / 13 = 6972 / 45318 2 / 13 = 7896 / 51324 2 / 13 = 8196 / 53274 7 solutions
5 / 13 = 4765 / 12389 5 / 13 = 8345 / 21697 5 / 13 = 8365 / 21749 3 solutions
1 / 14 = 1839 / 25746 1 / 14 = 1956 / 27384 my birthyear! - pdg 1 / 14 = 2967 / 41538 1 / 14 = 3297 / 46158 1 / 14 = 3678 / 51492 1 / 14 = 3912 / 54768 1 / 14 = 4398 / 61572 1 / 14 = 4713 / 65982 8 solutions
5 / 14 = 4635 / 12978 1 unique solution
1 / 15 = 1863 / 27945 1 / 15 = 6183 / 92745 2 solutions
8 / 15 = 9432 / 17685 1 unique solution
1 / 16 = 2871 / 45936 1 / 16 = 4581 / 73296 1 / 16 = 6147 / 98352 3 solutions
3 / 16 = 3294 / 17568 3 / 16 = 3672 / 19584 3 / 16 = 5967 / 31824 3 solutions
5 / 16 = 9765 / 31248 1 unique solution
7 / 16 = 5796 / 13248 7 / 16 = 9387 / 21456 2 solutions
11 / 16 = 9867 / 14352 1 unique solution
2 / 17 = 8514 / 72369 1 unique solution
3 / 17 = 5184 / 29376 1 unique solution
4 / 17 = 4536 / 19278 4 / 17 = 7428 / 31569 2 solutions
5 / 17 = 6435 / 21879 5 / 17 = 8145 / 27693 2 solutions
1 / 18 = 1593 / 28674 1 unique solution
7 / 18 = 8379 / 21546 1 unique solution
1 / 19 = 2736 / 51984 1 / 19 = 4293 / 81567 2 solutions
2 / 19 = 5136 / 48792 2 / 19 = 5238 / 49761 2 / 19 = 7254 / 68913 2 / 19 = 9132 / 86754 4 solutions
3 / 19 = 4671 / 29583 1 unique solution
4 / 19 = 3492 / 16587 1 unique solution
5 / 19 = 9165 / 34827 1 unique solution
6 / 19 = 4968 / 15732 6 / 19 = 8694 / 27531 2 solutions
7 / 19 = 7938 / 21546 1 unique solution
8 / 19 = 5368 / 12749 8 / 19 = 5864 / 13927 8 / 19 = 5872 / 13946 3 solutions
Part 2.
(numerator >= 1) and (denominator < 20) Listing all possible pandigital solutions would bring us too far. Instead I opt for displaying only the 20 unique pandigital solutions. With the aid of the above ubasic program one should easily find all the other solutions.
5 / 9 = 15930 / 28674
8 / 9 = 47016 / 52893
6 / 11 = 27486 / 50391
7 / 12 = 17829 / 30564
7 / 13 = 32571 / 60489
14 / 15 = 86310 / 92475
5 / 16 = 23490 / 75168
6 / 17 = 19872 / 56304
8 / 17 = 46152 / 98073
12 / 17 = 51084 / 72369
15 / 17 = 46710 / 52938
16 / 17 = 57312 / 60894
11 / 18 = 48312 / 79056
17 / 18 = 19584 / 20736
2 / 19 = 10248 / 97356
4 / 19 = 17460 / 82935
6 / 19 = 25704 / 81396
13 / 19 = 45981 / 67203
14 / 19 = 49518 / 67203
The top 5 pandigital fractions
Part 3.
Arriving at this point, it is time to relax from the serious work. As there is no fractional limit (?) one can investigate beyond the denominator limit that Terry imposed upon himself and go beyond 20 to discover more curios and trivia. Again, feel free to use the above ubasic program.
13 and 666 are well-known numbers for their negative connotation (the unlucky number and the number of the beast). There exist now a single fraction such that brings these two numbers together :
ps. 65934 – 1287 = 64647 almost a palindrome, ain't I unlucky ?! Also no such luck for the pandigital version with that same fraction. The number of the beast must like this number 65934 as I found the following beastly combinatorial equations.
Other numbers than 666 can be linked with "other than 65934" numerators. Do there exist such numbers with more combinatorial solutions than the above 8/16 ?
ps. Add the 8 denominators (13+18+...) from the ninedigital version together. An interesting palindrome arises, isn't it ! See WONplate 104
Part 4.
Ninedigital Fractions equal to Pandigital Fractions
Some 'unique' solutions.
[ December 25, 2001 ] Ninedigital Smith numbers
Smith numbers are composites such that'the sum of their digits' equal 'the sum of the digits of their prime factors'. Shyam Sunder Gupta, editor of the site Number Recreationsinvestigated the ninedigital versions and found that
[ February 10, 2001 ] Pandigital pointing to Ninedigital
The 1234567890^{th} palindrome is 234567891198765432
[ January 13, 2001 ] Amazing Nine Digits Property from sci.math by Preon
where a through i are the unique digits 1 through 9. bc, ef and hi are the concatenations (not the products) of b with c, e with f and h with i. The puzzle seems to have only one solution... as I discovered with a search !
The solution is very easy to search for, Preon tells. It is
" The solution is unique as my program revealed, took maybe 10 minutes to write the code and 1 minute to run and display the only answer."
Paul E. Triulzi, a senior IT specialist (email), noted that the equation from Preon has some interesting symmetry [ August 31, 2001 ]. The numerators are all odd and sequential. The numerator and denominator numbers are sequential (9,1,2 wraps around if 0 is not counted).
Preon investigated also the following pandigital variant but found only solutions with the zero as a nonsignificant leading zero !
Extra pan- & ninedigital equations some smallest solutions for each type by G.L. Honaker, Jr. and Patrick De Geest
the digits of n and the prime factorization of n are from the set 1 to 9 with each digit occurring exactly one time.
the digits of n and the prime factorization of n are from the pandigital set 0 to 9 with each digit occurring exactly one time.
the digits of n and the prime factorization of n are from the set 1 to 9 with each digit occurring exactly two times.
the digits of n and the prime factorization of n are from the pandigital set 0 to 9 with each digit occurring exactly two times.
It has me wondering if such a composite number existsin which each digit appears thrice and only thrice !
Special thanks goes to G. L. Honaker, Jr. for providing the original idea ! Prime Curios! sources : 5986 28651 14368485 40578660
[ August 13, 2000 ] Friedman numbers
Problem of the month (August 2000) from Erich Friedman is very interesting It investigates positive integers which can be written in some non-trivial way using its own digits, together with the symbols + - x / ^ ( ) and concatenation. The following subquestion is posed :
Solutions from Mike Reid and Philippe Fondanaiche
Amazing Number Facts
17469 divided by 5823 is 3. Every digit except 0 appears once and only once. [Kordemsky] 13485 divided by 2697 is 5. Idem. The prime 3187 divides 25496. Note all nine digits are used. 394521678 The sum of the prime numbers from 2 upto 92857 is a nine digits number. [Honaker] PI(2543568463) = 123456789. 5897230146 The pandigital sum of the first 32423 prime numbers. [Honaker] 12 * 3456^{7} + 89 is prime. [Honaker]
Difficult Digits Puzzle
Here is a pandigital puzzle I found at address http://pegasus.cc.ucf.edu/~mathed/digits.html : Arrange the digits 1 2 3 4 5 6 7 8 9 0 using only one operation sign so that they will equal 100. Good Luck! As I don't know the answer(s) myself yet I'd appreciate if someone could send them to me!
On [ Januari 20, 2001 ] I received the following solution from Eric Poindessault (email) He used as asked only operation sign namely the minus sign ¬ 1 – – 2 – – 3 – – 4 – – 5 – 6 – 7 – – 8 – – 90 = 100 Thanks Eric, well done!
The Nine Digits staircase
Gérard Villemin maintains an exciting maths website (in the French language) and some of the pages deals also with Nine Digits and Pandigital numbers :
Arrange the nine digits in steps and rises (paliers et marches [Fr.]) of length 3 such that the sums are always... 13 !
Jean Claude Rosa (email) - go to topic
Terry Trotter (email) - go to topic
Shyam Sunder Gupta (email) - go to topic
Eric Pointdessault (email) - go to topic
Paul E. Triulzi (email) - go to topic
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