### 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.

it is easy to prove by induction that

ReplyDeleteif 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.

0 -> cos -> 1

ReplyDelete1 -> 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

I think this can be solved like this.

ReplyDeleteStep 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.

pow(sin(arctan(cos(0)), -2) = 2

ReplyDeletelog(pow(2,3),2) = 3

Is it okay?

sec(arctan(cos(0))) = 2

ReplyDeletesec(arctan(sec(arctan(cos(0))))) = 3

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

ReplyDelete@Siddhartha.. Very interesting generalisation. Thanks.

2= [exp]

ReplyDelete3={exp}

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

e^ln2=2

ReplyDeletee^ln3=3