Four 4's puzzle

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Many moons ago, a maths teacher gave me the following problem to shut me up:

Using four digit 4's, and any other mathematical symbols you like, what's the largest consecutive number you can count to, starting from 1?

For example:

1 = 4/4
2 = 4/4 + 4/4
3 = 4 - 4/4
4 = 4 (easy!)
5 = 4 + 4/4
6 = 4 + sqrt(4) (imagine a square root symbol here)
7 = 4 + 4 - (4/4)
8 = 4 + 4
9 = 4 + 4 + 4/4
10 = 4 + 4 + sqrt(4)
...etc...

There are a few rules:

- No need to use all four 4's each time (though it can make it more challenging)
- No digits other than 4 (including zero)
- You're allowed to group 4's together, e.g. 11 = 44/4
- Use of a single dot over a 4 to indicate a recurring decimal is allowed. E.g: .4' is equivalent to .444444444..., so (4 / .4) = 10 while (4 / .4') = 9, and sqrt(4/.4') = 3.

It's surprising how far you can get: I seem to remember getting over 70, though it's a while since I've tried it.

Anyway, might give some relief from the countdown thread if anyone wants to give it a try? I suggest posting additional entries to the list in bunches of five or so until they start getting hard, otherwise there'll be a ton of messages. If a number proves particularly difficult, skip over it and we can come back to it later, but don't give up too easily.

Syth

As someone with a strong computer background, my first thought was to programme a computer to to a brute force attack, which might work.

But if you want to be mathematical I might have spotted a slight flaw in your argument that would allow someone to be able to write every single number in this way. Here it is:"and any other mathematical symbols you like". In your example you used +, - etc. All are functions, so one is allowed to use any function one likes. So you could make them up as you go along. Eg one could define a function for every number you want that will return the number you want independ of the arugement passed in. Eg to express 71 in this way just define f:R->R st f(x) = 71 for all x in R. Then just write 71 = f(4). So there must be a limit on the functions one can use.

I'd suggest only allowing +, -, *, /, sqrt(). You could make it harder by restricking the functions allowed. Interesting problem though.

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User-defined functions of any kind are disallowed (though it's a nice loophole). Not least because you'd need to define the function itself as part of the expression, and your example would thus use digits other than 4...

Restricting the possible symbols too much makes it too easy to stumble. However, for clarity, things like sin, cosine, and other functions are also disallowed; only "pure" mathematical symbols are allowed. (Makes it a bit messy for expressing in HTML, of course, so equivalent words are fine.)

After posting last night, I sat down to have a go. Somewhat to my surprise, I got to 130 before running out of steam.

Syth

only "pure" mathematical symbols are allowed.

What are pure mathematical symbols? What functions did you use for getting up to 130?

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By "pure", I just meant well understood symbols.

Here are the significant symbols I used to get to 130:

+, -, *, /, () - all as you would expect

44 = 44 (concatenation)
.4 = 2/5 (decimal)
.4' = 4/9 (recurring decimal)
4! = 24 (factorial)
sqrt(4) = 2 (square root)
4^4 = 256 (raising to a power)

There is also one slightly cheeky one:

sqrt'(4) = 1 (infinite square root)

I can't remember the exact mathematical symbol, but it's essentially a square root with a slight modifier, to indicate infinite recursion. I.e. if you take an infinite number of square roots of anything, you end up with 1. Without using this symbol, the maximum target I can reach is a lot lower - only 36.

(The full set of answers I found using sqrt'() is: 37, 63, 65, 69, 71, 73, 75, 76, 82, 87, 89, 95, 97, 98, 103, 109, 113, 119, 121, 123, 126, 127, 129 -- it's a very useful trick.)