Task 1: "Rare Numbers

You are given a positive integer $N.

Write a script to generate all Rare numbers of size $N if exists. Please
read http://www.shyamsundergupta.com/rare.htm for more information about it.

Examples

(a) 2 digits: 65
(b) 6 digits: 621770
(c) 9 digits: 281089082
"

My notes: ok, interesting question.  In summary: R (a +ve no) is a Rare No
iff R + reverse(R) and R - reverse(R) are both perfect squares.
So: generate and test time.

SEE ALSO OptimizingTask1 for an account of a series of profiling-driven
optimizations that I made to speed up finding rare numbers.  Overall, my
fastest version ran approx 18 times faster than the original.



Task 2: "Hash-counting String

You are given a positive integer $N.

Write a script to produce Hash-counting string of that length.

The definition of a hash-counting string is as follows:

- the string consists only of digits 0-9 and hashes, '#'

- there are no two consecutive hashes: '##' does not appear in your string
- the last character is a hash
- the number immediately preceding each hash (if it exists) is the position
  of that hash in the string, with the position being counted up from 1

It can be shown that for every positive integer N there is exactly one
such length-N string.

Examples:

	(a) "#" is the counting string of length 1
	(b) "2#" is the counting string of length 2
	(c) "#3#" is the string of length 3
	(d) "#3#5#7#10#" is the string of length 10
	(e) "2#4#6#8#11#14#" is the string of length 14
"

My notes: ok, weird.  Can we directly generate the single
hash-counting-string of length N?  I think we can..  Turns out to be easy.