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 this? Jim 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:

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:

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

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

(*) 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:

(*) 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

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:

* 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

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

...

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