Calculator without Digit Buttons

Last modified: Wed Aug 29 22:25:12 2001 JST [ English | Japanese ]

[ source ]

Please generate numbers by pushing function keys.
This is my original puzzle. Trigonometric functions are calculated in degrees.

Number	Sequence	Steps
1	`cos`	1
2	`cos 10^x x^2 log`	4
3	`cos 10^x sqrt 1/x acos tan`	6
4	`cos 10^x x^2 log x^2`	5
5	`cos 10^x 10^x sqrt log`	5
6	`cos 10^x sqrt 1/x acos tan 10^x x^2 log`	9
7	`cos atan cos 10^x sqrt log acos tan x^2`	9
8	`cos 10^x x^2 x^2 x^2 log`	6
9	`cos 10^x sqrt 1/x acos tan x^2`	7
10	`cos 10^x`	2
11	`cos 10^x sqrt atan cos 1/x x^2`	7
12	`cos 10^x sqrt atan cos atan sin 1/x x^2`	9
13	`cos 10^x x^2 x^2 sin acos sqrt 1/x acos tan`	10
14	`cos 10^x sqrt log x^2 asin tan acos tan x^2`	10
15	`cos 10^x sqrt log asin 10^x sqrt log`	8
16	`cos 10^x x^2 log x^2 x^2`	6
17	`cos 10^x x^2 log x^2 atan cos 1/x x^2`	9
18	`cos 10^x sqrt 1/x acos tan x^2 10^x x^2 log`	10
19	`acos sqrt 10^x x^2 log atan cos 1/x`	8
20	`cos 10^x 10^x x^2 log`	5

Please prove that any integer can be generated by only using these function keys.
[ Short Answer ]
Please prove that any rational number can be generated by only using these function keys.
[ Short Answer ]
Please prove that any computable function f: N --> N can be defined by these functions and the following instruction.
```
    if integer(x) goto L
```
[ Short Answer ]