  This paper focuses on finding the smallest happy number for each height in
any numerical base. Using the properties of the height, we deduce a recursive
relationship between the smallest happy number and the height where the initial
height is function of the numerical base. With the usage of the recursive
relationship, we build an algorithm that exploits the properties of the height
in order to find all of those smallest happy numbers with unknown height.
However, with the modular arithmetic, we conclude on an equation that
calculates the smallest happy numbers based on known heights for binary and
ternary bases.
