Using exactly four fours and the simple rules of arithmetic, express all of the integers from 1 to 25.
|For example,||1 =|
|+ 4 4,||2 =|
|and 3 =|
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 = 41/2 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)] =  = [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.
- 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.
Surprisingly it is possible to produce any finite integer using logarithms in a rather ingenious way.
We can see that,
4 = (41/2)1/2 = 41/4
4 = ((41/2)1/2)1/2 = 41/8 and so on.
log4(42) = 2
log4(43) = 3, leading to log4(4x) = x.
log4(4) = 1/4
log4(4) = 1/8 and so on.
In the same way,
log1/2(1/4) = 2
log1/2(1/8) = 3, ...
By writing log1/2(x) as log(4/4)(x) we can now produce any finite integer using four fours.
log(4/4)(log4(4)) = 1
log(4/4)(log4(4)) = 2
log(4/4)(log4(4)) = 3, ...