On Friday, August 14, 2020 at 4:38:31 AM UTC-7, Aaron Liang wrote:
> On Sunday, October 13, 1996 at 8:00:00 AM UTC+1, Peter Kirby wrote:
> > I have been puzzling over this for a while. Please help!
> > There is a certain math teacher of mine who holds that all positive
> > integers can be written as an expression with only mathematical symbols
> > and four 4's. For example,
> > 1: (4 + 4) / (4 + 4)
> > 2: 4 / 4 + 4 / 4
> > 3: (4 + 4 + 4) / 4
> > Et cetera. Just to make sure your following along, find a Four Fours
> > for 31.
> > (Hint: You can use +, -, *, /, !, and sqrt. 4! = 4*3*2*1 = 24.)
> > Before we get to my questions, we must first consider which
> > operators/functions we can use. I suggest that we can use anything but
> > a rounding function. That means +, -, *, /, sqrt, !, ^, (, ), ., log,
> > sin, cos, tan, and anything else you can think of (that won't let you
> > build a rounding function). You can use some fours to make a decimal
> > number (i.e., use two fours to make 44, 4.4, or .44). Of course, you
> > can only use the digit four, it can only appear four times, and you must
> > use base decimal.
> > It isn't difficult up to 32, but 33 is a stumbling block. Can you
> > A) Find a Four Fours expression for 33?
> > B) Explain why you can't?
> > In general, can you demonstrate that
> > A1) Every whole number has a Four Fours expression?
> > A2) Given a rounding function, every whole number has a Four Fours
> > expression?
> > B1) There is a whole number that doesn't have a Four Fours expression?
> > B2) Even with a rounding function, there isn't a Four Fours expression
> > for every whole number?
> > Any insights appreciated,
> > Peter Kirby
> > PS-
> > Perhaps this will save you a little grief. To try to answer my
> > questions, I wrote a program to generate a subset of Four Fours
> > expressions. It found the combinations of the operators +, -, *, /, and
> > ^ on four numbers 2, 4, and 24. (2 is sqrt(4) and 24 is 4!)
> > That is, it would find:
> > (4 + (24 / 24)) ^ 2 = 25
> > but it wouldn't find
> > (sqrt(24) ^ 2) + (4 - 4) = 24
> > because sqrt isn't one of the operators used by the program.
> > Only a finite number of expressions can be found this way, but that was
> > what I wanted. After all these expressions were found, the program
> > printed out an expression (if any) that it found for each number 1 to
> > 1000. The first number that was not discovered was 33, and most of the
> > larger numbers were also not found. This doesn't really show anything
> > except that anyone who wants to find 33 with Four Fours will have to be
> > pretty creative.
> > (4 - .4) / .4 + 4! = 33
33 = arccsc( sqrt( 4 ) ) + 4! / ( 4 + 4 ) This uses the inverse cosecant function to get 33.
= 30 + 24 / 8 = 30 + 3