Four Fours

This is an old puzzle from the days when calculators were new enough that they were still thought of as "neat." (I'm not sure which part of that statement dates me more....:-)

Using only the number "4" on your calculator keypad exactly four times, along with any of the standard mathematic function keys on your calculator, get a result of any positive integer, starting with 1. How high can you go?

I've actually gotten every number from 1 to 306. (A few challenging numbers on the way to 306 were 183 and 185.)

For clarity's sake, and since calculators these days are pretty complex, here is the list of allowable function keys:

+, -, X, /
1/x
x^2
SQRT(x)
y^x
! (factorial)
. (decimal point)

If no one can get 307 using just these, we can open it up to trigonometric functions, etc., but I know that all of the numbers up to 306 can be solved with just these keys.

I'm impressed that you got that far. Using an old calculator without brackets really limits what you can do. How does one make 3, for example?

Is it allowed to keep pushing the SQRT button until it rounds it to 1?

Originally posted by Acolyte
I'm impressed that you got that far. Using an old calculator without brackets really limits what you can do. How does one make 3, for example?

1 = 4 - 4 + 4/4
2 = 4/4 + 4/4
3 = 4 - sqrt(4) + 4/4
4 = 4 + 4 - sqrt(4) - sqrt(4)
5 = 4 + sqrt(4) - 4/4

etc

Originally posted by piderman
3 = 4 - sqrt(4) + 4/4

Ah, but that's not how an old calculator works. For example, pressing 4,-,4,SQRT,+,4,/,4 in order will give you (sqrt(4-4) + 4)/4, which is 1.

One interesting trick is that you can dispose of excess 4s by starting with '* 4 4 +' or whatever, something that you wouldn't be allowed to do in the pen-and-paper version.

Originally posted by Acolyte
Ah, but that's not how an old calculator works. For example, pressing 4,-,4,SQRT,+,4,/,4 in order will give you (sqrt(4-4) + 4)/4, which is 1.

Well, then you must have a very old calculator. I have a normal one (not the graphical kind - for those who want to check, it's a Casio fx-82SOLAR 🙂) and if I push the buttons in that order I get 3. What you are describing here is a calculator which doesn't know the prority rules of calculation, and I may expect that those apply, don't I?

Originally posted by piderman
Well, then you must have a very old calculator. I have a normal one (not the graphical kind - for those who want to check, it's a Casio fx-82SOLAR 🙂) and if I push the buttons in that order I get 3. What you are describing here is a calculator which doesn't know the prority rules of calculation, and I may expect that those apply, don't I?

Originally posted by The Plumber
This is an old puzzle from the days when calculators were new enough that they were still thought of as "neat."

I was going by this.

What about Reverse Polish notation? I'm not sure about this, but I think the oldest pocket calculators worked like:

4 ENTER 4 + ENTER 4 * ENTER 4 - ENTER 4 - ENTER 4 / ENTER 4 sqrt / -----> (((4+4)*4 - 4 - 4)/4)/sqrt(4) = 3.

Oh, I just saw that we only get four fours.

4 ENTER 4 + ENTER 4 + ENTER 4 / -----> (4 + 4 + 4)/4

I haven't used an RPN calculator in several years, so I'm not sure exactly where the ENTERs are necessary, but in both cases I've used at least enough. For example, I think 4 ENTER 4 + 4 + will give 12.

Originally posted by Acolyte
Originally posted by The Plumber
[b]This is an old puzzle from the days when calculators were new enough that they were still thought of as "neat."

I was going by this.[/b]

Maybe I'm just not old enough for this poser 😉

Anyway, I doubt you would get to 306 using those old calculators, though it would be neat to be proven wrong.

BTW,
3 = 4, /, 4, +, 4 + 4, SQRT 😛
or, in RPN: 4 ENTER 4 / 4 + 4 + SQRT

Multiple pushes of the SQRT button to get to 1 are not what was in mind with the original question.

Regarding the priotity of calculation issue, the original question comes from "The Great International Math on Keys" book which came with the old TI-30 calculators which had parentheses. You can use parentheses, etc. to solve the question at hand. Please note, it was not my intention that the use of a calculator be taken too literally.

To get 306, I did the following: 4*SQRT(4)/(.4)^2 + (4^2)^2 {the actual key punches would look a little different, but still quite doable}

By the way, my current calculator is RPN (reverse polish notation). I always liked RPN better. Of course, my "current" calculator is 20+ years old, and I don't know if they even make RPN calculators any more. (I'm starting to feel really old....😵)