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

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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

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