Four Fours/Lemmata/Three Fours
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Puzzle
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: $96$
- $96 = 4! \times \paren {\sqrt 4 + \sqrt 4}$
$\blacksquare$
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 $x$ |