Problems & Puzzles: Puzzles

Puzzle 41.- Palindromic Carnival (dedicated to Patrick De Geest)

Two weeks ago I started trying to create a puzzle thinking in the beauty and inspiring pages of my friend Patrick De Geest. My very first intention was to make a simple puzzle dealing only with Palprimes, but the Anagram of the Nine digits (A9D) soon invaded the searching space. I beg your pardon if - at the end – the “single puzzle” resulted in a bizarre medusa. Try to enjoy this.

A.- (A9D)/SOD(A9D)=Palindrome has 13 solutions.

164729835 /(1+2+3+4+5+6+7+8+9)= 3660663
231498675 /(1+2+3+4+5+6+7+8+9)= 5144415
248136975 /(1+2+3+4+5+6+7+8+9)= 5514155
248316975 /(1+2+3+4+5+6+7+8+9)= 5518155
248361975 /(1+2+3+4+5+6+7+8+9)= 5519155
321498765 /(1+2+3+4+5+6+7+8+9)= 7144417
326498715 /(1+2+3+4+5+6+7+8+9)= 7255527
328496715 /(1+2+3+4+5+6+7+8+9)= 7299927
347182965 /(1+2+3+4+5+6+7+8+9)= 7715177 (prime!)
348271965 /(1+2+3+4+5+6+7+8+9)= 7739377
632971845 /(1+2+3+4+5+6+7+8+9)= 14066041
736981245 /(1+2+3+4+5+6+7+8+9)= 16377361
812973645 /(1+2+3+4+5+6+7+8+9)= 18066081

Or maybe you’ll like them the same numbers this way better:

164729835 /(1+6+4+7+2+9+8+3+5)= 3660663
231498675 /(2+3+1+4+9+8+6+7+5)= 5144415
248136975 /(2+4+8+1+3+6+9+7+5)= 5514155
248316975 /(2+4+8+3+1+6+9+7+5)= 5518155
248361975 /(2+4+8+3+6+1+9+7+5)= 5519155
321498765 /(3+2+1+4+9+8+7+6+5)= 7144417
326498715 /(3+2+6+4+9+8+7+1+5)= 7255527
328496715 /(3+2+8+4+9+6+7+1+5)= 7299927
347182965 /(3+4+7+1+8+2+9+6+5)= 7715177 (prime!)
348271965 /(3+4+8+2+7+1+9+6+5)= 7739377
632971845 /(6+3+2+9+7+1+8+4+5)= 14066041
736981245 /(7+3+6+9+8+1+2+4+5)= 16377361
812973645 /(8+1+2+9+7+3+6+4+5)= 18066081

Questions:

A1.- None of the Palindromes resulted in a prime number? Is there any special reason for thisJim Howell pointed out that "actually 7715177 is prime..." (23/2/99), so the entire question has been settled down.

B.- Pal1/SOD(Pal1) = Pal2 has the following first solutions:

 Pal1 SOD(Pal1) Pal2 279972 36 7777 2774772 36 77077 5499945 36 122221 25477452 36 707707 27722772 36 770077 254545452 36 7070707 277202772 36 7700077 279999972 36 4444444

Questions:

B1.- Why Pal2 is not a prime number?

B2.- Why SOD(Pal1) is always 36??!!

B3.- Why the digital roots of Pal1= 9, or why the digital root of Pal2 =1?

Nota bene: For the sections B, C, D & E the search was extended up to 10^9

C.- PalPrime1*SOD(PalPrime1) = Pal2 has the following first solutions:

 PalPrime1 SOD(Palprime1) Pal2 11 2 22 101 2 202 353 11 3883 13331 11 146641 1123211 11 12355321 1221221 11 13433431 1303031 11 14333341 1311131 11 14422441 3103013 11 34133143 100111001 5 500555005 110111011 7 770777077 110232011 11 1212552121 111010111 7 777070777 111050111 11 1221551221 112030211 11 1232332321 112111211 11 1233223321 121111121 11 1332222331 130030031 11 1430330341 301111103 11 3312222133

Question C1.- Any special reason why SOD(Palprime1) is a prime number?

D.- Pal1/POD(Pal1) = Pal2 has the following first and few solutions:

 Pal1 (*) POD(pal1) Pal2 4224 64 66 42624 384 111

(*) We are not showing the trivial cases where Pal1 = R(k) for k=>1

Questions:

D1.- Is any hidden rule behind this relation?

D2.- Can exists a case with Pal2 being a prime number?

D3.- Can you extend this sequence?

E.- Pal1*POD(Pal1)=Pal2 has the following solutions:

 Pal1 POD(Pal1) (*) Pal2 77 49 3773 464 96 44544 22622 96 2171712 22622 96 2171712 2223222 192 426858624 4213124 192 808919808 122232221 192 23468586432 142131241 192 27289198272 212232212 192 40748584704 221232122 192 42476567424 222131222 192 42649194624 241131142 192 46297179264 421131124 192 80857175808

(*) We are not showing the trivial cases where POD(Pal1)=1, 2 , 3 or 4.

Questions:

E1.- Is there any special reason why POD(Pal1) takes specifically the values 49, 96 or 192 (49 =7^2, 96 =3*2^5; 192 =3*2^6)?

F.- ABS(A9D - Reverse(A9D)) = Palindrome has several ( a lot of ) solutions. But all of them can be divided in two cases: or Palindrome is 544505445 or Palindrome is 90900909.

Examples:

- 128954376 + 673459821 = 544505445
- 145897632 + 236798541 = 90900909

Question

F1.- Can you explain why this occurs?

G.- A9D\the Beast =Palindrome has only two solutions

913572846\666 = 1371731(Palprime!)
264197538\666 =   396693 (Palindrome)

No questions. Take this only as a curio.

H.- Prime factors of the A9D’s

961327458 = 2*3^2*7*13*17*19*23*79 (8 distinct prime factors)
943128576 =2^16*3^3*13*41 (21 prime factors)

Questions (in this case I have not had time to make an exhaustive search):

Find another A9D:

H1.- with more than 8 distinct prime factors

H2.- with more than 21 prime factors

I.- A9D/9 =   Palindrome has 34 solutions. The entire palindromes resulted to be composite numbers. Any special reason for this?

Example : 867594321/9 = 96399369

Note: Remember that A9D mod 9 = 0 always.

J.- A9D=3^2*Prime = Expressible with all the numbers from 0 to 9.

Example: 149256873= 3^2*16584097

Question: Can you find all the other A9D’s?

K.- AD9’s whose prime factors are all palindromes

Here are some conspicuous of them:

127935864 = 2*2*2*3*3*7*7*36263 (the least A9D)

987643125 = 3*3*3*3*3*5*5*5*5*7*929 (the largest A9D)

258164793 = 3*3*3*9561659 (the least quantity – 4 -of factors and the largest palindromic factor also)

536481792   = 2*2*2*2*2*2*2*2*2*2*2*2*3*3*3*3*3*7*7*11 (the largest quantity – 20 – of factors)

No questions.

L.- Sum of k Consecutive A9D’s = A9D

 k The least example 2 123456789 + 123456798 = 246913587 3 123954786 + 123954867 + 123954876 = 371864529 4 123465978 + 123465987 + 123467589 + 123467598 =   493867152 5 123456798 + 123456879 + 123456897 + 123456978 +   123456987 =  617284539 6 123768549 + 123768594 + 123768945 + 123768954 +   123769458 + 123769485 = 742613985 7 123675894 + 123675948 + 123675984 + 123678459 + 123678495 + 123678549 + 123678594 =   865741923 8 Can you find an example of eight A9D’s – not necessarily consecutive - such that added they result in another A9D?Jim Howell has found (23/2/99) two solutions to this question: "a) 123456789 + 123456789 + ... (8 times) = 987654312 b) Replace one of the 123456789 with 123456798, and the sum is 987654321 The examples above are the smallest sums of 8 anagrams, so that any other sum of 8 (or 9), including those with distinct anagrams, will be greater than 987654321.  So there are no solutions possible with 8 distinct anagrams."

Solutions

J. C. Colin sent the following solutions (October 29, 2002) to question J:

365928471  = 3^2* 40658719
438216759  = 3^2* 48690751
459871623  = 3^2* 51096847
491562873  = 3^2* 54618097
526149873  = 3^2* 58461097
639527841  = 3^2* 71058649
671382549  = 3^2* 74598061
716854329  = 3^2* 79650481
871654239  = 3^2* 96850471
871659423  = 3^2* 96851047
873526149  = 3^2* 97058461

***

Later (on August, 2005) J. C. Coling found the following new results:

In the table of the section B, there are two mistakes: SOD of 5499945 & 279999972 are 45 & 63, not 36 as wrongly stated.

For section D, he found four more results:

 digit(pal1) pal1 pod(pal1) pal2 = pal1/pod(pal1) 10 2114114112 64 33033033 11 21141314112 192 110111011* 12 211221122112 64 3300330033 13 2112213122112 192 11001110011

* Palprime!...I calculate for all pal1 as far as 20 digits and there is no other solution than this six solutions.

During calculation, I capture some other strange regularities I propose to your wonder :

11111133111111       /9 = 1234570345679 = 11 *333367*336667
11111311311111       /9 = 1234590145679 = 11 *
112235467789
11113111131111       /9 = 1234790125679 = 11 *29*4357*888413
11131111113111       /9 = 1236790123679 = 11 *337*367*909091
11311111111311       /9 = 1256790123479 = 11
*114253647589
13111111111131       /9 = 1456790123459 = 11 *31*53*673*119771
31111111111113       /9 = 3456790123457 = 11 *11*23*47*26427857

 212 2112 21112 211112 2111112 21111112 211111112 2111111112 .... n>1,   2(1)n2 /4 = /4 = /4 = /4 = /4 = /4 = /4 = /4 = ... /4 = 53 528 5278 52778 527778 5277778 52777778 527777778      ... 52(7)n-28

2111111133311111112 / 108  =  19547325308436214
2111111313131111112 / 108  =  19547326973436214
2111113113113111112 / 108  =  19547343639936214
2111131113111311112 / 108  =  19547510306586214
2111311113111131112 / 108  =  19549176973251214
2113111113111113112 / 108  =  19565843639917714
2131111113111111312 / 108  =  19732510306584364
2311111113111111132 / 108  =  21399176973251029

...

111111111333111111111 / 27  =  4115226345670781893  0

111111113131311111111 / 27  =  4115226412270781893  0

111111131131131111111 / 27  =  4115227078930781893  0

111111311131113111111 / 27  =  4115233745596781893  0

111113111131111311111 / 27  =  4115300412263381893  0

111131111131111131111 / 27  =  4115967078930041893  0

111311111131111113111 / 27  =  4122633745596707893  0

113111111131111111311 / 27  =  4189300412263374493  0

131111111131111111131 / 27  =  4855967078930041153  0

311111111131111111113 / 27  =  11522633745596707819  0

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