Additional theorems ,By analogy, further theorems can be developed from the chart by the same sequence of activities- guessed at first then verified on the calculator, and finally proved by exhausting all possible cases the help of the calculator), (with or, more elegantly, proved by algebraic methods, as in the case of h u 1, which was just illus trated. After further investigations we can determine that we arrive at zero with(1) after three steps, if and only if either h(2) after four steps, if and only if either(3) after five steps, if and only if either(4) after six steps, if an only if either h u 2 or h u-9 These theorems are at first obtained and proved only in the"if then" form and can be formulated in the"if and only if form only after the complete exhaustion of all possibilities