Symmetries of the Sum of Square of Digits of Natural Numbers up to 100

In this paper, we have identified cycles by consecutively adding the cross digit sum of squares (or Beejank) of the number and then finding the cross digit sum of the squares of the resulting number up to 100. It was found that these cycles either end at 89 or 1. We calculated the sum of the squares of the digits for all positive integers up to 100 and discovered some useful relationships among the consecutive values of these sums. Additionally, we observed relationships between the cross digit sums of these numbers. Finally, we calculated the cross digit sum and the sum of the squares of the digits for the multiplication tables of numbers from 2 to 19.