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The Nine Digits Page 3with 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).

Topic 3.12 [ *September 15, 2003* ]

“Ninedigital fractions” equations and their distinct integer solutions

( sometimes generalized to “Ninedigital expressions” )

Navigate using these button links

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) = 1Likewise 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.

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.

Are we unlucky ? since the second sum gives 5+8 = 13

What about the third sum 7+9 = 16 ? Subtract its power 3... aargh !

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 :

214358976 = (3 + 6)^{2} + (4 + 7)^{8} + (5 + 9)^{1} |

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 largest palindrome (out of

Samples

Value of the beast ! 12 distinct solutions The beast enclosed |
Some repdigits 1111 = (2–6)^4 + (7–5)^9 + (8–1)^3 7777 = (4–7)^1 + (6–8)^2 + (9–3)^5 |

1 - 3556 values have 2 d.s.2 - 2882 values have 4 d.s.3 - 1124 values have 1 d.s.4 - 1054 values have 8 d.s.5 - 436 values have 6 d.s.6 - 237 values have 12 d.s.7 - 197 values have 16 d.s.8 - 174 values have 10 d.s.9 - 93 values have 3 d.s.10 - 70 values have 20 d.s.11 - 68 values have 24 d.s.12 - 63 values have 5 d.s.13 - 47 values have 14 d.s.14 - 44 values have 18 d.s.15 - 43 values have 32 d.s.16 - 29 values have 28 d.s.17 - 22 values have 9 d.s.18 - 18 values have 36 d.s.19 - 15 values have 7 d.s.20 - 15 values have 22 d.s.21 - 14 values have 11 d.s.22 - 13 values have 40 d.s.23 - 12 values have 44 d.s.24 - 11 values have 26 d.s.25 - 11 values have 38 d.s.26 - 10 values have 30 d.s.27 - 9 values have 13 d.s.28 - 9 values have 34 d.s.29 - 9 values have 48 d.s.30 - 9 values have 64 d.s.31 - 8 values have 15 d.s.32 - 8 values have 21 d.s.33 - 7 values have 46 d.s.34 - 6 values have 35 d.s.35 - 6 values have 42 d.s.36 - 6 values have 60 d.s.37 - 5 values have 17 d.s.38 - 5 values have 52 d.s.39 - 5 values have 58 d.s. |
40 - 4 values have 50 d.s. (–243, –126, 72, 88)41 - 4 values have 56 d.s. (–25, 31, 46658, 262144)42 - 4 values have 68 d.s. (10, 23, 33, 1296)43 - 4 values have 72 d.s. (–27, 264, 6561, 65536)44 - 4 values have 76 d.s. (–4, 29, 129, 273)45 - 3 values have 49 d.s. (28, 84, 144)46 - 3 values have 54 d.s. (–125, 15, 35)47 - 3 values have 92 d.s. (21, 38, 321)48 - 2 values have 29 d.s. (94, 98)49 - 2 values have 78 d.s. (13, 69)50 - 2 values have 79 d.s. (–6, 2)51 - 2 values have 84 d.s. (25, 127)52 - 2 values have 90 d.s. (–5, 261)53 - 2 values have 94 d.s. (–510, 262146)54 - 2 values have 106 d.s. (0, 514)55 - 1 value has 25 d.s. (52)56 - 1 value has 31 d.s. (228)57 - 1 value has 39 d.s. (–12)58 - 1 value has 41 d.s. (26)59 - 1 value has 47 d.s. (12)60 - 1 value has 53 d.s. (218)61 - 1 value has 59 d.s. (114)62 - 1 value has 62 d.s. (–262142)63 - 1 value has 66 d.s. (253)64 - 1 value has 74 d.s. (36)65 - 1 value has 82 d.s. (22)66 - 1 value has 91 d.s. (–2)67 - 1 value has 102 d.s. (8)68 - 1 value has 104 d.s. (27)69 - 1 value has 108 d.s. (11)70 - 1 value has 120 d.s. (4096)71 - 1 value has 128 d.s. (4098)72 - 1 value has 132 d.s. (–3)73 - 1 value has 138 d.s. (81)74 - 1 value has 150 d.s. (9)75 - 1 value has 168 d.s. (5)76 - 1 value has 170 d.s. (16)77 - 1 value has 171 d.s. (6)78 - 1 value has 176 d.s. (64)79 - 1 value has 177 d.s. (256)80 - 1 value has 180 d.s. (4)81 - 1 value has 181 d.s. (66)82 - 1 value has 182 d.s. (7)83 - 1 value has 189 d.s. (258)84 - 1 value has 202 d.s. (18)85 - 1 value has 348 d.s. (3)86 - 1 value has 350 d.s. (–1)87 - 1 value has 544 d.s. (1) |

Variation #12 : the equation A^B/C + D^E/F + G^H/I [Jerry Levy's suggestion]

Values range from

Samples

The largest prime is

Samples

Almost all values (**1364**) have just 1 distinct solution.

Only **64** values show 2 distinct solutions.

Semiprime

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.

The largest prime is

Samples

( Note : d.s. = total distinct solutions ).

1 - 135 values have 1 d.s.2 - 96 values have 2 d.s.3 - 14 values have 4 d.s.4 - 8 values have 6 d.s.5 - 3 values have 8 d.s.6 - 2 values have 3 d.s.7 - 1 value has 7 d.s. |

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.

Samples

The largest prime is

Samples

**152** palindromes can be detected.

The largest one is **327723** = (6+7)*3 + (9–1)/2 + 4^8*5

For reference goals and easy searching I list here all the ninedigitals implicitly displayed in the above fifteen variations (as sample expressions).

Topic 3.12

Variation #0 → 912768534

Variation #1 → 136589724, 163589742, 598163724, 136589724, 589136724, 136589724, 518724936, 524718936, 534689712, 849536712, 538746912, 546738912

Variation #2 → none

Variation #3 → 215637849, 215639847, 217635849, 217639845, 219635847, 219637845, 621845937, 627845931, 426715938

Variation #4 → 124578639, 269378415, 278369415

Variation #5 → 453682791, 462573891

Variation #6 → 435812967, 435867912, 612857934, 634857912, 453821976, 453876921, 621875943, 643875921

Variation #7 → 281493576, 291364578, 291483756, 391472586, 391476582, 576843921, 634758921, 742856931, 746852931, 756821943

Variation #8 → 453672891, 123456789, 123457689, 123458679

Variation #9 → 175346892, 182375694, 195362784, 453691782, 453692781, 164582793, 473582691, 492563781, 214358976, 362478591

Variation #10 → 123456879, 987652143, 189276354, 198275364, 534726819, 634725918, 198276354, 534726918

Variation #11 → 346725819, 436725819, 128476593, 128593746, 146593782, 146593872, 153476892, 153476982, 153746892, 153746982, 218476593, 218593746, 416593782, 416593872, 142396785, 241395678, 241395768, 241395786, 241395876, 285346791, 285436791, 395461782, 95461872, 264759813, 624759813, 694735821, 735821964, 471682935, 471862935, 123456879, 987654231, 138264579, 987653241

Variation #12 → 359618724, 452673891, 271359648, 541673892, 258379461, 471682953

Variation #13 → 197362584, 542781963, 126584793, 139584762, 162584793, 168542793, 186542793, 193584762, 271584693

Variation #14 → 258741639, 276941358, 476291358, 674921358, 258734169, 478531269, 678523941, 372945168, 465312897, 673912485, 165823974, 256381974, 572481396, 672841953, 368241957, 245918673, 245981673, 582941673

Topic 3.11 [ *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 !

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.

Topic 3.10 [ *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 131 divisor 151 divisor 181 divisor 191 divisor 313 divisor 353 divisor 373 |
divisor 383 divisor 727 divisor 757 divisor 787 divisor 797 divisor 919 divisor 929 |

In total we have 61 solutions, a nice prime !

There are some curious palindromes expressable in two ways.

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 :

Topic 3.9 [ *August 20, 2003* ]

Pandigital divisions equal to ratio N/D

Statistics : abcd = N/D * efghi or abcde = N/D * fghij

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)

and finally the statistics, the program output :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)

2 / 5 = 6894 / 17235 4 / 5 = 9876 / 12345 1 unique and beautiful solution 2 / 7 = 3654 / 12789 2 / 9 = 3924 / 17658 |

(numerator ⩾ 1) and (10 < denominator < 20)

2 / 11 = 4716 / 25938 4 / 11 = 6492 / 17853 5 / 11 = 9765 / 21483 1 unique solution 1 / 12 = 3816 / 45792 5 / 12 = 8235 / 19764 1 unique solution 1 / 13 = 5184 / 67392 2 / 13 = 2538 / 16497 5 / 13 = 4765 / 12389 1 / 14 = 1839 / 25746 5 / 14 = 4635 / 12978 1 unique solution 1 / 15 = 1863 / 27945 8 / 15 = 9432 / 17685 1 unique solution 1 / 16 = 2871 / 45936 3 / 16 = 3294 / 17568 5 / 16 = 9765 / 31248 1 unique solution 7 / 16 = 5796 / 13248 11 / 16 = 9867 / 14352 1 unique solution |
1 / 17 = 1579 / 26843 2 / 17 = 8514 / 72369 1 unique solution 3 / 17 = 5184 / 29376 1 unique solution 4 / 17 = 4536 / 19278 5 / 17 = 6435 / 21879 1 / 18 = 1593 / 28674 1 unique solution 7 / 18 = 8379 / 21546 1 unique solution 1 / 19 = 2736 / 51984 2 / 19 = 5136 / 48792 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 7 / 19 = 7938 / 21546 1 unique solution 8 / 19 = 5368 / 12749 |

**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.

1 / 7 = 14076 / 98532 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.

666 / 13 = 65934 / 1287 666 / 18 = 65934 / 1782 666 / 22 = 65934 / 2178 666 / 29 = 65934 / 2871 666 / 72 = 65934 / 7128 666 / 79 = 65934 / 7821 666 / 83 = 65934 / 8217 666 / 88 = 65934 / 8712 |
666 / 122 = 65934 / 12078 666 / 130 = 65934 / 12870 666 / 172 = 65934 / 17028 666 / 180 = 65934 / 17820 666 / 213 = 65934 / 21087 666 / 220 = 65934 / 21780 666 / 283 = 65934 / 28017 666 / 290 = 65934 / 28710 666 / 718 = 65934 / 71082 666 / 720 = 65934 / 71280 666 / 788 = 65934 / 78012 666 / 790 = 65934 / 78210 666 / 829 = 65934 / 82071 666 / 830 = 65934 / 82170 666 / 879 = 65934 / 87021 666 / 880 = 65934 / 87120 |

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.

For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in this topic.

Topic 3.9

Body → 582317469, 987612345

1348526970, 7149620853

Part 1 → 689417235, 869421735, 948623715, 987612345, 365412789, 367412859, 534218697, 741825963, 978634251, 986234517, 392417658, 759634182, 471625938, 943251876, 948652173, 649217853, 783621549, 976521483, 381645792, 612973548, 746189532, 763291584, 823519764, 518467392, 627381549, 728194653, 253816497, 387625194, 391825467, 451829367, 697245318, 789651324, 819653274, 476512389, 834521697, 836521749, 183925746, 195627384, 296741538, 329746158, 367851492, 391254768, 439861572, 471365982, 463512978, 186327945, 618392745, 943217685, 287145936, 458173296, 614798352, 329417568, 367219584, 596731824, 976531248, 579613248, 938721456, 986714352, 157926843, 167928543, 173829546, 217436958, 268945713, 269345781, 321754689, 347859126, 382164957, 384165297, 395267184, 395467218, 451976823, 452376891, 459678132, 461978523, 462378591, 479681532, 491683572, 492183657, 526189437, 526389471, 527389641, 537891426, 546192837, 546392871, 547893126, 851472369, 518429376, 453619278, 742831569, 643521879, 814527693, 159328674, 837921546, 273651984, 429381567, 513648792, 523849761, 725468913, 913286754, 467129583, 349216587, 916534827, 496815732, 869427531, 793821546, 536812749, 586413927, 587213946

Part 2 → 1407698532, 1593028674, 4701652893, 2748650391, 1782930564, 3257160489, 8631092475, 2349075168, 1987256304, 4615298073, 5108472369, 4671052938, 5731260894, 4831279056, 1958420736, 1024897356, 1746082935, 2570481396, 4598167203, 4951867203

Part 3 → 128765934, 659341287, 659341287, 659341782, 659342178, 659342871, 659347128, 659347821, 659348217, 659348712

6593412078, 6593412870, 6593417028, 6593417820, 6593421087, 6593421780, 6593428017, 6593428710, 6593471082, 6593471280, 6593478012, 6593478210, 6593482071, 6593482170, 6593487021, 6593487120

Part 4 → 976531248, 349216587

2349075168, 1746082935

Topic 3.8 [ *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 Recreations

investigated the ninedigital versions and found that

Topic 3.7 [ *February 10, 2001* ]

Pandigital pointing to Ninedigital

The 1234567890^{th} palindrome is 234567891198765432

Eric Schmidt developed a nice Java Applet where you can enter any

positive integer N and after pressing ENTER calculates the N

Topic 3.6 [ *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."

Preon wonders if someone has a proof or explanation

cause he hasn't a clue why this amazing property should be true...

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 !

(3÷04) + (6÷78) + (9÷52) = 1(3÷48) + (7÷16) + (9÷02) = 5 (4÷28) + (5÷63) + (7÷09) = 1 (4÷07) + (5÷28) + (9÷36) = 1 (4÷28) + (5÷07) + (9÷63) = 1 (4÷06) + (7÷39) + (8÷52) = 1 (4÷36) + (7÷18) + (9÷02) = 5 (5÷02) + (6÷14) + (7÷98) = 3 (5÷38) + (6÷04) + (7÷19) = 2 (5÷13) + (6÷04) + (9÷78) = 2 (5÷09) + (7÷63) + (8÷24) = 1 (5÷36) + (7÷04) + (9÷81) = 2 |

Topic 3.5

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 exists

in 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

Topic 3.4 [ *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

Topic 3.3

Amazing Number Facts

Amazing Number Facts : Nine Digits 17469 divided by 5823 is 3. Every digit except 0 appears once and only once. [Kordemsky] |

Topic 3.2

Difficult Digits Puzzle

Here is a pandigital puzzle I found at address http://pegasus.cc.ucf.edu/~mathed/digits.html :

Arrange the digits 0 1 2 3 4 5 6 7 8 9 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!

Topic 3.1

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 !

9 | 3 | 1 | 13 | |
---|---|---|---|---|

8 | ||||

13 | 4 | 7 | 2 | |

5 | ||||

13 | 6 |

For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.

Topic 3.6 → 912768534

304678952, 348716902, 428563709, 407528936, 428507963, 406739852, 436718902, 502614798, 538604719, 513604978, 509763824, 536704981

Topic 3.5 → 598624173

2865174093

Topic 3.4 → 862759134, 897625134

Topic 3.3 → 174695823, 134852697, 318725496, 123456789

Topic 3.2 → 1234567890

Topic 3.1 → 931847256

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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