Problems & Puzzles: Puzzles

Puzzle 300. UFO message?

George Peite, sent the following amusing puzzle:

it happened that i have a book with three bibliographical numbers printed inside its cover :        23 - 114 - 6236 when i arrange every pair of numbers side by side i have the following amusing phenomena:

23114=7*13*254
41132=7*13*452

the reversal of the above number, also note that 452 is the reversal of 254

1146236=7*13*12596
6326411=7*13*69521

the reversal of the above number, also note that 69521 is the reversal of 12596

236236=7*13*2596
632632=7*13*6952

also note that 6952 is the reversal of 2596

I think I have a UFO message, else the puzzle is how rare this phenomena will reoccur in the triple numbers taken pair by pair and their reversals?

And the question now is:

1. Can you answer the George's question?

Contributions came from Faride Firoozbakht, Anurag Sahay & W. Edwin Clark. It seems that if UFOs exist they are not necessarily responsible for such messages.

Faride wrote:

I think there exists many such triples because there exists infinitely many numbers n such that

91*reversal(n) = reversal(91*n)

the length of the set A of all numbers n where n<16000000 and 91*reversal(n) = reversal(91*n) is 1120288.

A={11,22,33,44,55,66,77,88,99,110,111,121,122,132,133,143
,144,154,155,165,...,15999988,15999990,15999991,15999992
,15999993,15999994,15999995,15999996,15999997,15999998,
15999999}

Some triples like 23-114-6236 :

10-101-1010, 10-101-1101, 10-101-2011, 10-101-2102, ... , 23-114-5053, 23-114-5144, 23-114-5235, 23-114-5326, 23-114-5417, 23-114-5508, 23-114-6054, 23-114-6145.

Anurag wrote:

Some interesting solutions to puzzle 300:

Here is a set of 4 numbers(quadruplet):
51-233-324-142
51233 = 563*7*13
and other equations...

I found many pairs which retain their properties when they are arranged in any order .
Following are few of the closed pairs:
254-163-527(527-163-254)

171-535-353-717
We get 12 different numbers when every pair of 2 numbers is joined.

And other examples follow..

718-263-172
361-452-816
814-723-632-541
436-618-254
958-276-137
716-352-534

A closed pair of 6 numbers:(4 of them are prime)
727-545-181-272-818-454
Arrange them in any of the 720 ways , they retain the properties.

...

According to me,
with numbers of atleast m-digits the largest set has atleast 10^(m-f) numbers.
where f is the length of the common factor.
For example, the largest set for m=3 and f=2 is
3275-181-272-363-454-545-636-727-818-909
3275181 = 35991*91
1815723 = 19953*91
...
How about concatenating all the numbers in a set - It also works:
For the following closed set
43531-23237474-23235959-....
After joining all the above numbers we get
435312323747423235959 = 4310023007400230059 and there is no limit to the number of zeroes within the numbers: 2323000000..7474

W. Edwin Clark wrote did not solve the question, instead he switched it to a new puzzle interesting enough to include it here extensively:

I don't know how rare the phenomenon mentioned in the 300 is. There only one factor is reversed and the product gives the reverse of the original product. I think it is not rare. To make it harder I decided to search for numbers a,b,c such that if rev(x) denotes the reverse of the number x, then

n = x*y*z
rev(n) = rev(x)*rev(y)*rev(z)

and to make it more difficult I insist that all the numbers
x,y,z,rev(x),rev(y),rev(z) be different. It is easier to find such x,y,z if you allow divisibility some of x,y,z,rev(x),rev(y),rev(z) to the the same.

Here are some examples (I expect that there are lots more)

4034030=65*91*682
304304=56*19*286

23580150=93*275*922
5108532=39*572*229

10010530=94*95*1121
3501001=49*59*1211

2172170=65*98*341
712712=56*89*143

2896812=12*201*1201
2186982=21*102*1021

1264830=14*95*951
384621=41*59*159

1238688=12*102*1012
8868321=21*201*2101

1348848=12*102*1102
8488431=21*201*2011

6025448=92*143*458
8445206=29*341*854

4435872=46*98*984
2785344=64*89*489

2785344=64*89*489
4435872=46*98*984

6998560=83*85*992
658996=38*58*299

1000160=94*95*112
610001=49*59*211

2352350=91*94*275
532532=19*49*572

23453030=95*274*901
3035432=59*472*109

63331884=268*341*693
48813336=862*143*396

2186982=21*102*1021
2896812=12*201*1201

21965870=85*314*823
7856912=58*413*328

48813336=286*396*431
63331884=682*693*134

48813336=143*396*862
63331884=341*693*268

1237464=12*102*1011
4647321=21*201*1101

1347624=12*102*1101
4267431=21*201*1011

2530280=61*85*488
820352=16*58*884

11001760=94*95*1232
6710011=49*59*2321

4231872=48*93*948
2781324=84*39*849

140298620=385*391*932
26892041=583*193*239

1226448=12*102*1002
8446221=21*201*2001

8445206=29*341*854
6025448=92*143*458

1419552=93*96*159
2559141=39*69*951

11010690=94*95*1233
9601011=49*59*3321

2998840=65*73*632
488992=56*37*236

11932160=32*395*944
6123911=23*593*449

2781324=39*84*849
4231872=93*48*948

***

Later Anurag Sahay sent the next follow up to the Clark's contribution:

Clark's numbers have 3 factors. Here are some examples with 4 factors.

4490850 = 94*21*25*91
44347792 = 883*43*73*16
25590560 = 235*83*32*41
268866720=285*951*31*32
409108104 = 723*813*29*24
6998560 = 83*85*31*32

***

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