Answers

1. The sequence of key-presses
   
   increases the number in the display window by one. Thus, the sequence is equivalent to the increment key (operator) . This solution is based on the following formula.

   
   
   

   Generally speaking, if we have the formula
   
   we can always obtain
    
   by reducing x with f(x)=n-1, in which is the inverse function of f. For example, the above solution is using the equation
    
   There are such equations
    
   and
    
   Based on the equations (6) and (7), we obtain another solutions
   
   and
    
   The last sequence (9) is derived by (Kaneko, 1989).

2. The following sequence of key-presses
   
   provide us -operator (Kaneko, 1989), where k is the number of . Using (5) - (7), several decrement key-sequences are given by
   
   
   
   and
   
   which are based on
   
   in equation (4), where is the inverse function of g.
3. If your function calculator has [e] key which provide e=2.718281828..., is obtained by the number of a set of key-presses that is