Four Fours/Lemmata/Two Fours

Puzzle

Two instances of $4$ can be used to make the following:

Two Fours: $0$

$0 = 4 - 4$

Two Fours: $1$

$1 = \dfrac 4 4$

Two Fours: $2$

$2 = 4 - \sqrt 4$

Two Fours: $3$

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

Two Fours: $4$

$4 = \sqrt 4 + \sqrt 4$

Two Fours: $5$

$5 = \dfrac {\sqrt 4} {.4}$

Two Fours: $6$

$6 = 4 + \sqrt 4$

Two Fours: $7$

$7 = \map \Gamma 4 + \map \Gamma {\sqrt 4}$

Two Fours: $8$

$8 = 4 \times \sqrt 4$

Two Fours: $9$

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

Two Fours: $10$

$10 = \dfrac 4 {.4}$

Two Fours: $12$

$12 = \dfrac {4!} {\sqrt 4}$

Two Fours: $15$

$15 = \dfrac {\map \Gamma 4} {.4}$

Two Fours: $16$

$16 = 4 \times 4$

Two Fours: $18$

$18 = 4! - \map \Gamma 4$

Two Fours: $20$

$20 = 4! - 4$

Two Fours: $22$

$22 = 4! - \sqrt 4$

Two Fours: $23$

$23 = 4! - \map \Gamma {\sqrt 4}$

Two Fours: $24$

$24 = \paren {\sqrt 4 + \sqrt 4}!$

Two Fours: $25$

$25 = 4! + \map \Gamma {\sqrt 4}$

Two Fours: $26$

$26 = 4! + \sqrt 4$

Two Fours: $28$

$28 = 4! + 4$

Two Fours: $30$

$30 = 4! + \map \Gamma 4$

Two Fours: $36$

$36 = \dfrac {4!} {\sqrt{. \dot 4} }$

Two Fours: $44$

$44 = 44$

Two Fours: $48$

$48 = 4! + 4!$

Two Fours: $54$

$54 = \dfrac {4!} {. \dot 4}$

Two Fours: $60$

$60 = \dfrac {4!} {.4}$

Two Fours: $64$

$64 = \paren {\sqrt 4}^{\map \Gamma 4}$

Two Fours: $96$

$96 = 4 \times 4!$

Two Fours: $120$

$120 = \paren {\dfrac {\sqrt 4} {.4} }!$

$\blacksquare$

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 $x$