Four Fours
Problem
Using exactly four fours and the simple rules of arithmetic, express all of the integers from 1 to 25.
For example,  1 =  4 4  + 4 – 4,  2 =  4 4  +  4 4  and 3 =  4 + 4 + 4 4 
Solution
This problem is ambiguous in its wording, "simple rules of arithmetic"; that is, are we to interpret it as only using +, –, , and ÷, or will we need to employ a few tricks? Unfortunately it is not possible to obtain all of the integers from 1 to 25 otherwise.
Useful building blocks are square roots and factorial, however, as 4 = 4^{1/2} and 4! = 4 3 2 1, they both use numbers that we do not have. What about the use of place positions, as in 44 or .4? I personally object to this type.
Wherever possible, I have tried to use methods that are as close to the basic rules of arithmetic as possible: add, subtract, multiply and divide, submitting to the use of square root and factorial only where necessary.

Can you produce all of the integers from 1 to 100?
The integer part function, [x], which has the effect of stripping away the decimal fraction of a number and leaving the integer part, can be used to derive some of the more stubborn numbers. Hence, [(4)] = [2] = [1.41...] = 1. Similarly, [4cos4] = [3.99...] = 3 (working in degrees) and [log(4)] = [0.602...] = 0 (working in base 10), and this could be used to eliminate a surplus 4.
Once you"ve given the integers 1 through 100 a good shot, you may like to check out David Wheeler's rather outstanding, Definitive Four Fours Answer Key; found at: http://www.dwheeler.com/fourfours/. Wherever he employs contentious functions: for example, square(n)=n², which makes use of a 2 in the exponent, he has gone to great lengths to provide an "impurity" index admitting the use of functions which some people may object to using.
Extensions
 What is the first natural number that cannot be derived?
 Which is the first number that cannot be obtained if you are only permitted to use the basic rules of arithmetic (+, –, , and ÷)?
 The maximum integer is not a sensible question, as we could apply factorial any finite number of times. But, what is the largest known prime you can produce?
 Using three fours (or threes), which integers can you make?
 Using any number of fours and only addition, subtraction, multiplication and division, produce all of the integers from 1 to 25 in the most efficient way possible.
Notes
Surprisingly it is possible to produce any finite integer using logarithms in a rather ingenious way.
We can see that,
4 = (4^{1/2})^{1/2} = 4^{1/4}
4 = ((4^{1/2})^{1/2})^{1/2} = 4^{1/8} and so on.
By definition,
log_{4}(4^{2}) = 2
log_{4}(4^{3}) = 3, leading to log_{4}(4^{x}) = x.
Therefore,
log_{4}(4) = 1/4
log_{4}(4) = 1/8 and so on.
In the same way,
log_{1/2}(1/4) = 2
log_{1/2}(1/8) = 3, ...
By writing log_{1/2}(x) as log_{(4/4)}(x) we can now produce any finite integer using four fours.
log_{(4/4)}(log_{4}(4)) = 1
log_{(4/4)}(log_{4}(4)) = 2
log_{(4/4)}(log_{4}(4)) = 3, ...