The numbers that give a perfect square on adding as well as subtracting their reverse are rare, like 65, 621770, etc. and hence have been termed as rare numbers by the author. This chapter is extensively devoted to computing and investigating rare numbers up to 1024. Computations of rare numbers need to be extended further to find the answers to the conjecture and problems raised. An interesting analysis has been made based on the results of the computation of palindromic and non-palindromic rare numbers. The methodology discussed in the chapter for fast computation of rare numbers, along with the properties, can be helpful to enable the readers to further explore the subject. Some interesting conjectures and problems raised such as ‘finding an example of an odd abundant rare’, ‘finding an example of a rare number of the 4K+1 form’, ‘to characterize the infinite family of non-palindromic rare numbers’, etc. can be challenging to explore.