Four Fours/Lemmata

Puzzle

It is a useful exercise to determine which numbers can be made from $1$, $2$ and $3$ instances of $4$, and how this can be done.

One Four

$4$ 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: $64$

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

$\Box$

Two Fours

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: $120$

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

$\Box$

Three Fours

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

Three Fours: $0$

$0 = 4 - \paren {\sqrt 4 + \sqrt 4}$

Three Fours: $1$

$1 = \dfrac {4 - \sqrt 4} {\sqrt 4}$

Three Fours: $2$

$2 = \dfrac {4 + 4} 4$

Three Fours: $3$

$3 = \dfrac {4 + \sqrt 4} {\sqrt 4}$

Three Fours: $4$

$4 = 4 + 4 - 4$

Three Fours: $5$

$5 = 4 + \dfrac 4 4$

Three Fours: $6$

$6 = 4 + 4 - \sqrt 4$

Three Fours: $7$

$7 = \dfrac {4! + 4} 4$

Three Fours: $8$

$8 = \dfrac {4 \times 4} {\sqrt 4}$

Three Fours: $9$

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

Three Fours: $10$

$10 = \dfrac {4! - 4} {\sqrt 4}$

Three Fours: $11$

$11 = \dfrac {4! - \sqrt 4} {\sqrt 4}$

Three Fours: $12$

$12 = 4 + 4 + 4$

Three Fours: $13$

$13 = \dfrac {4! + \sqrt 4} {\sqrt 4}$

Three Fours: $14$

$14 = \dfrac {4! + 4} {\sqrt 4}$

Three Fours: $15$

$15 = \dfrac {4 + \sqrt 4} {.4}$

Three Fours: $16$

$16 = \paren {4 + 4} \times \sqrt 4$

Three Fours: $17$

$17 = \dfrac {\map \Gamma 4} {.4} + \sqrt 4$

Three Fours: $18$

$18 = 4! - 4 - \sqrt 4$

Three Fours: $19$

$19 = 4! - \dfrac {\sqrt 4} {.4}$

Three Fours: $20$

$20 = 4! - \sqrt 4 - \sqrt 4$

Three Fours: $21$

$21 = 4! - \sqrt {\dfrac 4 {. \dot 4} }$

Three Fours: $22$

$22 = 4! - 4 + \sqrt 4$

Three Fours: $23$

$23 = 4! - \dfrac 4 4$

Three Fours: $24$

$24 = 4! + 4! - 4!$

Three Fours: $25$

$25 = 4! + \dfrac 4 4$

Three Fours: $26$

$26 = 4! + 4 - \sqrt 4$

Three Fours: $27$

$27 = 4! + \sqrt {\dfrac 4 {. \dot 4} }$

Three Fours: $28$

$28 = 4! + \sqrt 4 + \sqrt 4$

Three Fours: $29$

$29 = 4! + \dfrac {\sqrt 4} {.4}$

Three Fours: $30$

$30 = 4! + 4 + \sqrt 4$

Three Fours: $32$

$32 = 4! + 4 + 4$

Three Fours: $33$

$33 = 4! + \dfrac 4 {. \dot 4}$

Three Fours: $34$

$34 = 4! + \dfrac 4 {.4}$

Three Fours: $36$

$36 = \dfrac {4 \times 4} {. \dot 4}$

Three Fours: $40$

$40 = \dfrac {4 \times 4} {.4}$

Three Fours: $42$

$42 = \dfrac {4! + \sqrt 4} {\sqrt {. \dot 4} }$

Three Fours: $44$

$44 = 4! + 4! - 4$

Three Fours: $45$

$45 = \dfrac {4! - 4} {. \dot 4}$

Three Fours: $46$

$46 = 4! + 4! - \sqrt 4$

Three Fours: $48$

$48 = 4! \times \paren {4 - \sqrt 4}$

Three Fours: $50$

$50 = \dfrac {4!} {. \dot 4} - 4$

Three Fours: $52$

$52 = 4! + 4! + 4$

Three Fours: $53$

$53 = \dfrac {4! - {. \dot 4} } {. \dot 4}$

Three Fours: $54$

$54 = \dfrac {\paren {\sqrt 4 + \sqrt 4 }!} {. \dot 4}$

Three Fours: $55$

$55 = \dfrac {4! - \sqrt 4} {.4}$

Three Fours: $56$

$56 = \dfrac {4!} {.4} - 4$

Three Fours: $58$

$58 = \dfrac {4!} {. \dot 4} + 4$

Three Fours: $60$

$60 = \dfrac {\paren {\sqrt 4 + \sqrt 4 }!} {.4}$

Three Fours: $61$

$61 = \dfrac {4! + .4} {.4}$

Three Fours: $62$

$62 = \dfrac {4!} {.4} + \sqrt 4$

Three Fours: $63$

$63 = \dfrac {4! + 4} {. \dot 4}$

Three Fours: $64$

$64 = \dfrac {4!} {.4} + 4$

Three Fours: $65$

$65 = \dfrac {4! + \sqrt 4} {.4}$

Three Fours: $70$

$70 = \dfrac {4! + 4} {.4}$

Three Fours: $72$

$72 = \dfrac {4! + 4!} {\sqrt {. \dot 4} }$

Three Fours: $78$

$78 = \dfrac {4!} {. \dot 4} + 4!$

Three Fours: $81$

$81 = \paren {\dfrac 4 {. \dot 4} }^{\sqrt 4}$

Three Fours: $84$

$84 = \dfrac {4!} {.4} + 4!$

Three Fours: $90$

$90 = \dfrac {4!} {.4 \times \sqrt {. \dot 4} }$

Three Fours: $96$

$96 = 4! \times \paren {\sqrt 4 + \sqrt 4}$

$\Box$

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 $x$