About

Four fours is a classic mathematical problem where the goal is to find the simplest mathematical expression of every non-negative number using only a combination of exactly four 4s and common mathematical operators.

fourfou.rs was a project I started out of boredom to teach myself web development.

Levels

Each solution in this website has a complexity level that determines how "complex" the solution is. Lower level expressions are considered simpler and thus better than higher level expressions. Each level has a set of allowed operators in addition to allowing the operators from the previous levels.
The levels are described below:

Level 0

Allows addition, subtraction, multiplication, division, square root¹, power, factorial, decimals², and concatenation³ $$+ , - , \times , / , \sqrt{x}, x^{y}, !, .4, 44$$

Level 1

Allows repeating decimal² $$.\overline{4} = 4/9$$

Level 2

Allows the arbitrary root $$\sqrt[y]{x} = x^{1/y}$$

Level 3

Allows the percent sign⁴ $$x\% = x / 100$$

Level 4

Allows the gamma function $$\Gamma(x) = (x - 1)!$$

Level 5

Allows the square function $$sq(x) = x^{2}$$

¹Note: Using an infinite amount of square roots or other unary operators (e.g. $\sqrt{\sqrt{...\sqrt{4}}} = 1$ ) is not allowed.
²Note: Decimalizing an expression such as $.(4 + 4) = .8$ or $.(4!) = .24$ is not allowed.
³Note: Concatenating expressions such as $4||(4 + 4) = 48$ or $(4! - 4)||(4 - 4) = 200$ is not allowed.
⁴Note: Using a percent sign on an expression such as $(4 + 4)\% = 0.08$ and $(.4)\% = 0.004$ is allowed.

All solutions levels 0 to 5 are valid solutions. Solutions that use operators or functions outside of the ones listed above will be considered "semi-valid" and will have a complexity level of 6. For example:

$$69 = \phi(\Gamma(\Gamma({\color{red44}4})) + {\color{green44}4}) + {\color{blue44}4} / .\overline{{\color{purple44}4}}$$

would have a complexity level of 6 because $\phi$ (Euler's totient function) is not an operator listed above.

Similarly, solutions that use the existing operators in a way that is disallowed, such as decimalizing expressions, will also fall under this category. For example:

$$69 = (\Gamma(\Gamma({\color{red44}4})) - \Gamma({\color{green44}4}) + \Gamma(\sqrt{{\color{blue44}4}})) \times .(\Gamma({\color{purple44}4}))$$

would have a complexity level of 6 because an expression is decimalized (Specifically $.(\Gamma(4)) = .6$).

Banned Operators

These following operators are explicitly banned and cannot be used in any solution valid or semi-valid:

Log Functions

Includes $log_{y}(x)$ and $ln(x)$

These are banned because there are many ways to break the challenge using the function. For example:

$$x = log_{\sqrt{4}/4}(log_{4}(\sqrt{\sqrt{...\sqrt{4}}}))$$

can get any number $x$, by adding $x$ amount of nested square roots on the last 4. A more detailed explanation can be found here.

Trig Functions

Includes $sin(x)$, $cos(x)$, $tan(x)$, $sec(x)$, $csc(x)$, $cot(x)$ and their respective inverses

(Credit to Jim Millar) These are banned because all numbers above 4 can be created with a single 4 using this equation:

$$x = sec(atan(...(sec(atan(4)))))$$

To get any number $x$, you can make $x^{2} - 16$ amount of nested secants and inverse tans on the 4. A more detailed explanation can be found here.

Increment and Decrement Operators

Includes $++$ and $--$

These are banned because all numbers can be created with a single 4 using these operators:

$$x = 4++...++$$

$$x = 4--...--$$

Numeric Constants

Includes $\pi$, $e$ and more

These are banned because they are representative of another number that isn't 4.

Contact Me

If you found a mistake or have a solution you want to submit that meets one of these requirements:

The number currently does not have a solution.
Your solution's complexity level for a number is lower than the current solution's.
Your solution's complexity level for a number is the same than the current solution's and it uses less operators.

you can contact me at timsin4444@gmail.com. Also contact me if you have any general inquiries about this website and I hope to answer the best I can!