Broken Calculator - simplified but not simple

Here is an example of how to use this applet - - set a goal number to reach - say 2345 - disable the 2,3,4,5,6,7,8, and 9 keys - use only the remaining 0 and 1 keys along and the addition & subtraction keys to get a result of 2345. How many steps do you need? Can you do it with fewer steps? What about a goal of 9999? How many steps do you need? Can you do it in fewer steps? This applet is a revised version of a program I wrote in the early 1980's entitled "What do you do with a Broken Calculator". The program was widely imitated - mostly by people with less flexible pedagogical views than I am comfortable with. For some problem types as well as the underlying theory that drove the program's development read the essay on the Broken Calculator on the MathMindHabits website. What problems could / would you put to your students using this applet?
Are all calculators broken? any calculator (or digital computer, for that matter) is a rational number machine - it cannot represent real but irrational numbers numerically. Thus, there is a large collection of problems that calculators must produce INCORRECT answers to - for example, 1 divided by 3. In this case one might suppose that the calculator might distinguish between .333...3 as a truncated decimal which is incorrect and .3 with a line over (or under) the 3 to indicate the repeating pattern. That would be a correct answer, but one has to consider how it is arrived at. [N.B. repeating decimals indicated with an underline, thus 1/6 = .16, 1/3 = .3, 1/7 = .142857 etc.] Does the calculator ‘know’ that a pattern repeats - -by calculating until the pattern repeats and then assumes it will continue to do so? Or, -does the calculator ‘know’ the way we humans ‘know’? How do we humans know that 1/7 = .142857 ?