Four Fours/Lemmata/One Four

Puzzle

$4$ can be used on its own to make the following:

One Four: $\dfrac 4 {10}$

$\dfrac 4 {10} = .4$

One Four: $\dfrac 4 9$

$\dfrac 4 9 = {. \dot 4}$

One Four: $\dfrac 2 3$

$\dfrac 2 3 = \sqrt {. \dot 4}$

One Four: $1$

$1 = \map \Gamma {\sqrt 4}$

One Four: $2$

$2 = \sqrt 4$

One Four: $4$

$4 = 4$

One Four: $6$

$6 = \map \Gamma 4$

One Four: $24$

$24 = 4!$

One Four: $64$

$64 = \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \surd \surd \surd \surd \surd \surd \surd \floor { \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \surd \floor {\surd \floor {\surd \floor {\surd \surd \surd \surd \surd \paren {4!} !} !} !} !} !} !} !} !}$

$\blacksquare$

Glossary

Symbols used in the Four Fours are defined as follows:

\(\ds . \dot 4\)

\(:=\)

\(\ds 0.44444 \ldots\)

$.4$ recurring, equal to $\dfrac 4 9$

\(\ds \sqrt 4\)

\(:=\)

\(\ds 2\)

square root of $4$

\(\ds 4!\)

\(:=\)

\(\ds 1 \times 2 \times 3 \times 4\)

$4$ factorial

\(\ds \map \Gamma 4\)

\(:=\)

\(\ds 1 \times 2 \times 3\)

gamma function of $4$

\(\ds a \uparrow b\)

\(:=\)

\(\ds a^b\)

Knuth uparrow notation

\(\ds \floor x\)

\(:=\)

\(\ds \text {largest integer not greater than $x$}\)

floor function of $x$

\(\ds \map \pi x\)

\(:=\)

\(\ds \text {number of primes less than $x$}\)

prime-counting function of $x$