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\F3\CSTANFORD, CALIFORNIA 94305
\F4ARTIFICIAL INTELLIGENCE LABORATORY\←L\-R\/'7;\+R\→.\→S   Telephone:
\←S\→.415-497-3330

\F1\COctober 10, 1979

Professor Allan Gottlieb
Dept. of Mathematics
York College
Jamaica, N.Y. 11451

Dear Professor Gottlieb:

\JI am enclosing some results relating to your problem NS16 (Palindromes) in the
Aug./Sept.  Technology Review.  Not having read the previous results on NS16, I
may be repeating what others have done.

It seemed to me that some insight might be gained by a brute-force computer
calculation.  I have considered all numbers up to 99,999, carrying the results
to 100 digits for Intransegent Numbers (those that do not form palindromes).
These results were obtained by simple PASCAL programs.  More efficient programs
should be written to extend the analysis.  Some of your readers may want to do
this.

Significant results are:

1.  Palindromes, when formed, seem to occur with a reasonablely small number
of additions. Numbers forming palindromes fall into classes as defined for
Intransigent numbers below but there are many more classes (all possible classes
not listed for the Intransigents).

2.  Intransigent numbers can be classified in terms of the series of
numbers obtained by adding the first and last digit, the second and next to
last digit, etc., or in terms of the number obtained after the first add.  I
have chosen to report the results in the first of these ways as it seems to
be a bit more useful.

3.  The number of different classes as well as the percentage of intransigency
seems to go up rapidly with the number of digits in the original numbers.

4.  The fact that intransigent cases exist with additions carried to
100 digits, is no proof that some of these cases might not lead to
palindromes if the search were extended.  There are gaps in the tables,
for example, no palindomes were found for 3-digit numbers requiring adds of
12, 13, 16, and from 18 to 21, and there could be a longer gap from 24 to 223.
This does seem unlikely, but we have no proof.  I did carry the analysis for
196 to 2392 additions and 1000 digits, but even this is no proof.  Results
to 169 additions and 75 digits are attached.

Perhaps some your readers might be interested in these results.

\←L\→S\←R\-L\/'2;\+L\→L
Sincerely,



Arthur L. Samuel

\←S\→L
ALS:pdp10