Perhaps you should group operations according to their desirability, and have your script come up with a purity factor based on how low it has to stoop to come up with a solution.
I sort of tried that by increasing the number of unary functions after each iteration. The first version gave extremely complicated solutions even for one-digits, so I made it this way to reduce complex stuff. I also ordered the functions by desirablity, but as long as possible combinations are tested in nested foreach loops a few nasty solutions will always creep in.
I'd be more interested if someone can define a minimal set of functions with which the four4 problem is solvable for all integers.
The trivial set of three functions is:
unary:
- 0(a) - returns zero
- +(a) - returns a + 1
binary:
- f(a,b) - returns a
With this every number n can be represented as
n = f( f( f(+^n(0(4)) , 4) , 4) , 4)
where +^n(a) is the function "+" n times applied to a.
Can someone do it in less functions?
So I guess the scope of this topic has moved from figuring out the expressions by hand to writing a program that can produce them for you?
It seems like that.. and you know, it's sad really.. I was actually begining to enjoy this and was looking forward to more being solved and constantly going on... but now :( Sigh...
Mr. Kelly R. Flewin
Mr. Kelly R. Flewin
Just another random gamer
----
<OmnipotentEntity> How do you people get bored in the span of 10 seconds? Worst ADD ever.
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Yeah, here I calculated with pen and paper a lot of these equations, and here they go coding programs that do that stuff for them! Mathematic judges you! :)
Just kidding, but seriously, I agree with you, Mr. Kelly R. Flewin, that this is quite sad... I was prepared to go to at least 500 with this, but hey, making programs that does this for you is also kind of cool, although the topic should change name to perhaps The Four Fours challenge (programming) or something.
Why don't you guys have a race. You hand guys try to get to 500 the hard way, while the other bunch try to code a program to get to 500. Or have they done that already?
You ever read that Asimov story where no one can do math anymore because machines do it all? And they find this one janitor somewhere who can multiply in his head and it's like the most stunning thing ever? I think it's relevent here somehow. Even though I am sure it takes an extreme knowledge of math to be able to program these programs.
As you could read, 12motion, Gorash has written a program, and he went without any major problems up to 3947... so 500 is really a very low challenge for programmers...
And doing it the hard way was fun at the beginning but I'm (and maybe randil is as well) a little bored of 4 4s... Though this could have been a good idea... The problem is that when you write a script, most of the time, even if it takes you time, it goes rapidly beyond the speed of human mind (logically, because that's the aim of program scprits i guess) so, in my opinion, a race would not be a fair thing...
A last thing is that to program that kind of script, languages such as PERL or LISP are extremely effective and those script can be written very "easily"... It's even more unfair...
Human mind has been, in a way, outpaced by computers :) (I'm good in philosophy... Nah just kidding)
Oh c'mon guys, it's not like I forced you to use programs.
I was interested how far this would go, and because I had free time to spend, and because it was a nice excuse to refresh my perl references knowledge (I tend to forget the more weird features of perl everytime).
I didn't even intend to post it, but the question for a solution for 1337 was just too fitting. It is fun to do this by hand, but at some point it also becomes rather pointless not to let a stupid machine do the number crunching.
Anyway, when it comes to solutions using periodic fractions or other stuff I firmly believe you won't solve it with pure brute force. A race up to 500 would probably have been lost by the humans. 200 would have been hard for an average programmer though.
Oh 12motion... the math in this script is 9th grade at most. The most complicated thing maybe is knowing that n? can be computed as (n+1)*n/2...
I just see Xaphan has posted almost the same, but I disagree on one thing:
Human mind ahs been outpaced by itself with the invention of computers. :)
(And the script didn't went to 3947 without problems, it was just that is didn't find anything for that value... it computed solutions up to 10000000 without problems...)
Joined: 4/20/2005
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Here's an idea: Why not see who can find out the highest integer that be be represented by 4 fours, using programs. Although that will turn all this to who can code the best program, I, for one, would find it quite interesting. What do you think?
What functions would you be allowed to use for the programs? Using the functions we have used so far, I don't think it would be possible to find a "highest" number. You could always add a ! or ? to the end to make it even higher, so there would need to be a limit on the unary operators that you could use.
Joined: 4/20/2005
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Let's say that you are allowed to use 12 operators at most, and you still have to use your four fours.
An operator is something that changes the value, such as +, - or !. You may still use () and [] freely.
If it's okay with you, you can use the operators that xebra talked about in his post, such as hyperpower. See his post for more info on this.
This is just a suggestion for the rules for this, if you don't like something about them it's okay if you change them.
Well lets just say regardless of which grade of math you used, I would never be able to write mathimatic forumlas that make sense, or apply math in any prectical purpose whatsoever. So my hat is off to you guys. Sorry I missed how high you got. I was under the impression the program was still in beta testing. I don't know anything about math so reading ever post here doesn't interest me, just the overall progress.
Let's say that you are allowed to use 12 operators at most, and you still have to use your four fours.
An operator is something that changes the value, such as +, - or !. You may still use () and [] freely.
Even 4!!! is greater than everything imaginable... Using the Stirling approximation I figure that the count of digits (!) of that number is around 10^26
(guessed number of particles in the universe has 80 digits...)
But the greatest value will probably be:
4 ^ ( 4 ^ ( 4 ^ ( 4!!!!!!!!! )))
That number is... huge.
Joined: 4/20/2005
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Gorash wrote:
But the greatest value will probably be:
4 ^ ( 4 ^ ( 4 ^ ( 4!!!!!!!!! )))
That number is... huge.
Yup, that number is really, really huge. I thought about the exact same thing when I wrote my previous post, that the highest value would probably be when you use all of your ! in your highest exponent. But perhaps there is a number that is higher than 4 ^ ( 4 ^ ( 4 ^ ( 4!!!!!!!!! ))), but since that number i amazingly huge, I'm guessing you need tremendous computer power to compare number of that size.
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Here's a bigger one:
4^(4^(4^[4↑]!!!!!!!))
Or, if division with 0 is allowed: (4+4)/(4-4)
That number goes towards infinity, IF division with 0 is allowed here. If it is, than there is no number that could possibly be bigger than this.
Randil:
↑ is a binary operator, your statement makes no sense. Besides which, if you are going to use hyperpowers, 4↑↑↑↑↑↑↑↑↑↑↑↑444 is probably the biggest.
Furthermore, (4+4)/(4-4) is not a number, so it's true there can be no number bigger than it, but then again, there can be no number smaller than it, either. (Because no comparison can be made.)
Joined: 4/20/2005
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xebra: Okay, sorry for misunderstanding the hyperpower function. And I am aware of the fact that a number divided by zero is not actually a defined number, I mostly took that example up to make sure i'd win if you accepted that as a number :)
Maybe 4*(4↑↑↑↑↑↑↑↑↑↑↑44) is higher than 4 ^ ( 4 ^ ( 4 ^ ( 4!!!!!!!!! ))), I don't know.
As incomprehensibly huge 4^4^4^4!!!!!!!!! is, 4*4↑↑↑↑↑↑↑↑↑↑↑44 is so much bigger that the ratio of the first to the second is smaller than the ratio of the size of an electron to the size of the entire universe. Furthermore, the number that I mentioned in my previous post, 4↑↑↑↑↑↑↑↑↑↑↑↑444, is so much bigger than your number, 4*4↑↑↑↑↑↑↑↑↑↑↑44, that if every particle in the unverse were counting by 4*4↑↑↑↑↑↑↑↑↑↑↑44's every 1/4*4↑↑↑↑↑↑↑↑↑↑↑44th of a second, the universe would be a uniform temperature a fraction above absolute zero before you ever reached 4↑↑↑↑↑↑↑↑↑↑↑↑444.
Remember, 4↑↑↑↑↑↑↑↑↑↑↑↑444=
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4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑...
4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4↑↑↑↑↑↑↑↑↑↑↑4
As incomprehensibly huge 4^4^4^4!!!!!!!!! is, 4*4↑↑↑↑↑↑↑↑↑↑↑44 is so much bigger that the ratio of the first to the second is smaller than the ratio of the size of an electron to the size of the entire universe. Furthermore, the number that I mentioned in my previous post, 4↑↑↑↑↑↑↑↑↑↑↑↑444, is so much bigger than your number, 4*4↑↑↑↑↑↑↑↑↑↑↑44, that if every particle in the unverse were counting by 4*4↑↑↑↑↑↑↑↑↑↑↑44's every 1/4*4↑↑↑↑↑↑↑↑↑↑↑44th of a second, the universe would be a uniform temperature a fraction above absolute zero before you ever reached 4↑↑↑↑↑↑↑↑↑↑↑↑444.
My number is so much bigger than yours! No mine is, I'm going to get my brother, he's going to tell you!!
Sorry, couldn't resist.
Could you please stop arguing about numbers beyond any reasonable size? If you want huge numbers with a sense, go solve Go.
Joined: 4/20/2005
Posts: 2161
Location: Norrköping, Sweden
Hehe, so if 4*(4↑↑↑↑↑↑↑↑↑↑↑44) is that huge, I'm guessing that 4^(4↑↑↑↑↑↑↑↑↑↑↑44) is so huge that our brains would explode if we found out the magnitude of that number. :)
The recent discussion reminds me of another interesting problem I saw a few years ago.
Write the largest natural number you can. Any notation is acceptable as long as:
1) It is easily understood (this is subjective and depends on your readership, but I think posters in this thread will be reasonable)
2) The representation of your number (and any function definitions, explanations, etc.) must not exceed a certain number of characters. I don't remember the character limit on the original problem, but let's make it 500 characters.
As you have seen we can exceed the means of reasonable numbers even with only 4 characters.
This reminds me more of the child game to count higher than others. At some point (possibly in the hundreds or in the thousands) the child realises that there will be no end.
Joined: 11/27/2004
Posts: 688
Location: WA State, USA
Randil wrote:
Hehe, so if 4*(4↑↑↑↑↑↑↑↑↑↑↑44) is that huge, I'm guessing that 4^(4↑↑↑↑↑↑↑↑↑↑↑44) is so huge that our brains would explode if we found out the magnitude of that number. :)
*Hooks Randil up to the Total Perspective Vortex*
*Whistles*
Nach wrote:
I also used to wake up every morning, open my curtains, and see the twin towers. And then one day, wasn't able to anymore, I'll never forget that.
