A Palindrome Question
from Maria Gerardina Bulanon (email) [ May 12, 2001 ]
If I were to write numbers from 1 to k in order (k>1),
I can never make a palindrome out of this.
How do you prove this ?
Carlos Rivera [ May 13, 2001 ] would like to try answering this question.
Here is his argument:
Q: Consider the sequence of k numbers from 1 to k, what is the smallest
number you need at the right extreme in order to form a palindrome?
A: You need the reversible of the number produced by the concatenation of 1 to k
i.e. for 1, 2, 3 you need 321 at the right extreme.
But the previous number to the rightmost one ends necessarily in zero
(320 in the above example) and k+1 can not start with zero
(4 can not start with zero).
So no palindrome can ever be produced writing down the natural numbers.
If at the right extreme you use not the smallest numbers the negative
situation remains (for example if you use at the right extreme not 321 but 4321)
The same reasoning can be done, mutatis mutandis, if we do not start at 1
but at any other number.
Is the above proof partial or complete...
Are there sequences where it is possible to form palindromes ?