Arithmetic Puzzle: Broken Calculator

Source: Quantnet Forum



Problem:
There is a calculator in which all digits(0-9) and the basic arithmetic operators(+,-,*,/) are disabled. However other scientific functions are operational like exp, log, sin, cos, arctan, etc. The calculator currently displays a 0. Convert this first to 2 and then to 3.

Update (8/7/2012):
Solution posted in comments by Siddhartha, Sumit, Salman Khan and Kapil Dubey. Thanks. Interesting generalisation proposed by Siddhartha.

Comments

  1. it is easy to prove by induction that
    if f(x)= sec(tan^-1(x)),
    then
    fofofo....f (0) {n times} = sqrt(n),or that
    fofofo....f (0) {n^2 times} = n.

    In particular,
    fofofof(0)=2, and
    fofofofofofofofof(0)=3.

    ReplyDelete
  2. 0 -> cos -> 1
    1 -> arctan -> 45
    45 -> sin -> 1/root(2)
    1/root(2) -> inverse -> root(2)
    root(2) -> square -> 2

    2 -> inverse -> 1/2
    1/2 -> arcsin -> 30
    30 -> tan -> 1/root(3)
    1/root(3) -> inverse -> root(3)
    root(3) -> square -> 3

    ReplyDelete
  3. I think this can be solved like this.

    Step 1 :
    There is 0 on the screen. So, apply cos(0). It is 1 now. Apply e^x on 1 (if available). We get e. Now square it using x^2. We get e^2. Apply ln. We get 2.

    Step 2 :
    Now we have 2. Apply 1/x. We get 1/2. Apply arcsin. We get pi/6. Apply tan. We get 1/sqrt(3). Square it. We get 1/3. Apply 1/x. We get 3.

    Alternative for step 2:
    From 2, go back to 1 (reversing step1). Now, apply e^x, apply x^3, and apply ln. We get 3.

    ReplyDelete
  4. pow(sin(arctan(cos(0)), -2) = 2
    log(pow(2,3),2) = 3

    Is it okay?

    ReplyDelete
  5. sec(arctan(cos(0))) = 2
    sec(arctan(sec(arctan(cos(0))))) = 3

    ReplyDelete
  6. @Sumit, @Salman, @Kapil. Nice intuitive solutions. Thanks

    @Siddhartha.. Very interesting generalisation. Thanks.

    ReplyDelete
  7. 2= [exp]

    3={exp}

    where [ ] is greatest integer function and { } is shortest integer function.

    ReplyDelete

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