A follow-up to WONplate 36
and a new ninedigital puzzle...

Every integer X can be multiplied with another
integer Y to produce a palindrome P.
Which one are the hardest cases ?

In WONplate 36 I solved Fred W. Helenius's (email) puzzle
"Find the smallest palindrome divisible by 8181"
which turned out to be 9599999998999999959

What I wanted to know is how he came about that 8181
would be a challenge in the first place ?

Here is Fred's answer from March 12, 2001 On 27 April 1997, I sent you an email message in response to your post on sci.math in which I mentioned a C program that I used to verify that 999999999 was the smallest palindrome divisible by 81. A few days later, a much-improved version of that program was able to find the smallest palindrome divisible by each integer (other than multiples of 10) up to 10000, with three exceptions: 8181, 8991 and 9801. The program checked palindromes of up to 15 digits, and took only a few minutes to run. So the short answer is that I knew 8181 was hard because I knew it wasn't easy. Also, a special mention should be given to 8891, which is the only number up to 10000 other than the multiples of 81 which is at all difficult; its smallest palindromic multiple has 14 digits. And lastly, was the above solution for 8181 already known ? Probably not. Although I checked yesterday that my program could handle 8181 in about 20 minutes, I don't think I bothered to do so back in 1997. 9801 is of similar difficulty (20 digits); 8991 is worse (more than 20 digits; once again, I didn't take the time to look further).

Things are recapitulated in the following table.
Let us try now to (re)discover Fred Helenius results

Multiplicand	Smallest Possible Multiplier	Palindrome
81	12345679 - (easy)	999999999
8181	1173450678278939 - (March 10, 2001)	9599999998999999959
8891	8546948927	75990922909957
8991	unknown	?
9801	2030405060708091 - (May 16, 2003)	19899999999999999891

Some things for the margin
8991 can be expressed palindromically.
Is is namely equal to 9990 – 0999

I'd like to introduce now the next ninedigital variation to the puzzle.

Which ninedigital multiplicand needs the smallest/largest
multiplier to become palindromic ?

There are 9! or 362880 ninedigital numbers to search through.
I started by checking out a few limit values to get the puzzle going !
Can you come up with better results than these ?

Ninedigital	Multiplier	Smallest (?) Palindrome
987654321	99	97777777779
?	?	?

Ninedigital	Multiplier	Largest (?) Palindrome
123456789	4340315	535841353148535
123456798	343340038	42387661716678324

An extra challenge !
Exist there other 'palindromic' multipliers for some ninedigital multiplicands ?
World!Of Ninedigitals