Four Fours/Further Results

Puzzle

Using exactly $4$ instances of the number $4$, it is possible to write an expression for all positive integers from $0$ to $100$, using whatever arithmetical operations are necessary.

It is also possible to populate the table for numbers from $101$ to $200$, although for $197$ we appear to need to use the floor function.

Solution

Four Fours: $101$

$101 = 4! \times 4 + \dfrac {\sqrt 4} {.4}$

Four Fours: $102$

$102 = 4! \times 4 + \dfrac {4!} 4$

Four Fours: $103$

$103 = 4! \times 4 + \map \Gamma 4 + \map \Gamma {\sqrt 4}$

Four Fours: $104$

$104 = \dfrac 4 {.4} \uparrow \sqrt 4 + 4$

Four Fours: $105$

$105 = \dfrac 4 {. \dot 4} \uparrow \sqrt 4 + 4!$

Four Fours: $106$

$106 = 4! \times 4 + \dfrac 4 {.4}$

Four Fours: $107$

$107 = \dfrac {4! + {4!} - {. \dot 4} } {. \dot 4}$

Four Fours: $108$

$108 = \dfrac {4!} {\sqrt 4} \times \dfrac 4 {. \dot 4}$

Four Fours: $109$

$109 = \dfrac {4! + {4!} + {. \dot 4} } {. \dot 4}$

Four Fours: $110$

$110 = \dfrac {4! + 4! - 4} {.4}$

Four Fours: $111$

$111 = \dfrac {4!} { {.4} \times {. \dot 4} } - 4!$

Four Fours: $112$

$112 = \paren {4! + 4} \times \sqrt 4 \times \sqrt 4$

Four Fours: $113$

$113 = \paren {4! + 4} \times 4 + \map \Gamma {\sqrt 4}$

Four Fours: $114$

$114 = \paren {4! + 4} \times 4 + \sqrt 4$

Four Fours: $115$

$115 = \paren {\dfrac {\sqrt 4} {.4} }! - \dfrac {\sqrt 4} {.4}$

Four Fours: $116$

$116 = \paren {4! + 4} \times 4 + 4$

Four Fours: $117$

$117 = \paren {\dfrac {\sqrt 4} {.4} }! - \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $118$

$118 = \paren {\dfrac {\sqrt 4} {.4} }! - \dfrac 4 {\sqrt 4}$

Four Fours: $119$

$119 = \paren {\dfrac {\sqrt 4} {.4} }! - \dfrac 4 4$

Four Fours: $120$

$120 = \paren {\dfrac {4 \times 4 + 4} 4}!$

Four Fours: $121$

$121 = \dfrac {4 + .4} {.4} \uparrow \sqrt 4$

Four Fours: $122$

$122 = \paren {\dfrac {\sqrt 4} {.4} }! + \dfrac 4 {\sqrt 4}$

Four Fours: $123$

$123 = \paren {\dfrac {\sqrt 4} {.4} }! + \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $124$

$124 = \paren {\dfrac {\sqrt 4} {.4} }! + \sqrt 4 + \sqrt 4$

Four Fours: $125$

$125 = \paren {\dfrac {\sqrt 4} {.4} }! + \dfrac {\sqrt 4} {.4}$

Four Fours: $126$

$126 = \paren {\dfrac {\sqrt 4} {.4} }! + 4 + \sqrt 4$

Four Fours: $127$

$127 = \paren {\dfrac {\sqrt 4} {.4} }! + \map \Gamma 4 + \map \Gamma {\sqrt 4}$

Four Fours: $128$

$128 = \paren {\dfrac {\sqrt 4} {.4} }! + 4 + 4$

Four Fours: $129$

$129 = \paren {\dfrac {\sqrt 4} {.4} }! + \dfrac 4 {. \dot 4}$

Four Fours: $130$

$130 = \paren {\dfrac {\sqrt 4} {.4} }! + \dfrac 4 {.4}$

Four Fours: $131$

$131 = \dfrac {4!} {.4 \times {. \dot 4} } - 4$

Four Fours: $132$

$132 = \paren {\dfrac {\sqrt 4} {.4} }! + \dfrac {4!} {\sqrt 4}$

Four Fours: $133$

$133 = \dfrac {4!} {.4 \times {. \dot 4} } - \sqrt 4$

Four Fours: $134$

$134 = \map \Gamma 4 \times 4! - \dfrac 4 {.4}$

Four Fours: $135$

$135 = \map \Gamma 4 \times 4! - \dfrac 4 {. \dot 4}$

Four Fours: $136$

$136 = \map \Gamma 4 \times 4! - 4 - 4$

Four Fours: $137$

$137 = \dfrac {4!} {.4 \times {. \dot 4} } + \sqrt 4$

Four Fours: $138$

$138 = \map \Gamma 4 \times 4! - 4 - \sqrt 4$

Four Fours: $139$

$139 = \map \Gamma 4 \times 4! - \dfrac {\sqrt 4} {.4}$

Four Fours: $140$

$140 = \map \Gamma 4 \times 4! - \sqrt 4 - \sqrt 4$

Four Fours: $141$

$141 = \map \Gamma 4 \times 4! - \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $142$

$142 = \map \Gamma 4 \times 4! - \dfrac 4 {\sqrt 4}$

Four Fours: $143$

$143 = \map \Gamma 4 \times 4! - \dfrac 4 4$

Four Fours: $144$

$144 = \paren {4 + 4 + 4}^{\sqrt 4}$

Four Fours: $145$

$145 = \map \Gamma 4 \times 4! + \dfrac 4 4$

Four Fours: $146$

$146 = \map \Gamma 4 \times 4! + \dfrac 4 {\sqrt 4}$

Four Fours: $147$

$147 = \map \Gamma 4 \times 4! + \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $148$

$148 = \map \Gamma 4 \times 4! + \sqrt 4 + \sqrt 4$

Four Fours: $149$

$149 = \map \Gamma 4 \times 4! + \dfrac {\sqrt 4} {.4}$

Four Fours: $150$

$150 = \map \Gamma 4 \times 4! + 4 + \sqrt 4$

Four Fours: $151$

$151 = \dfrac {4!} {.4 \times .4} + \map \Gamma {\sqrt 4}$

Four Fours: $152$

$152 = \map \Gamma 4 \times 4! + 4 + 4$

Four Fours: $153$

$153 = \map \Gamma 4 \times 4! + \dfrac 4 {. \dot 4}$

Four Fours: $154$

$154 = \map \Gamma 4 \times 4! + \dfrac 4 {.4}$

Four Fours: $155$

$155 = \dfrac {\dfrac {4!} {.4} + \sqrt 4} {.4}$

Four Fours: $156$

$156 = \map \Gamma 4 \times 4! + \dfrac {4!} {\sqrt 4}$

Four Fours: $157$

$157 = \dfrac {\paren {\map \Gamma 4}! \times {. \dot 4} - \map \Gamma 4} {\sqrt 4}$

Four Fours: $158$

$158 = \dfrac {\paren {\map \Gamma 4}! \times {. \dot 4} - 4} {\sqrt 4}$

Four Fours: $159$

$159 = \map \Gamma 4 \times 4! + \dfrac {\map \Gamma 4} {.4}$

Four Fours: $160$

$160 = \dfrac {4 \times 4 \times 4} {.4}$

Four Fours: $161$

$161 = \dfrac {\paren {\map \Gamma 4}! \times {. \dot 4} + \sqrt 4} {\sqrt 4}$

Four Fours: $162$

$162 = \map \Gamma 4 \times 4! + 4! - \map \Gamma 4$

Four Fours: $163$

$163 = \dfrac {\paren {\map \Gamma 4}! \times {. \dot 4} + \map \Gamma 4} {\sqrt 4}$

Four Fours: $164$

$164 = \map \Gamma 4 \times 4! + 4! - 4$

Four Fours: $165$

$165 = \dfrac {\map \Gamma 4 + \dfrac {4!} {.4} } {.4}$

Four Fours: $166$

$166 = \map \Gamma 4 \times 4! + 4! - \sqrt 4$

Four Fours: $167$

$167 = \map \Gamma 4 \times 4! + 4! - \map \Gamma {\sqrt 4}$

Four Fours: $168$

$168 = \map \Gamma 4 \times 4! + \map \Gamma 4 \times 4$

Four Fours: $169$

$169 = \paren {\dfrac 4 {. \dot 4} + 4}^\sqrt 4$

Four Fours: $170$

$170 = \map \Gamma 4 \times 4! + 4! + \sqrt 4$

Four Fours: $171$

$171 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } - \dfrac 4 {. \dot 4}$

Four Fours: $172$

$172 = \map \Gamma 4 \times 4! + 4! + 4$

Four Fours: $173$

$173 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } - \map \Gamma 4 - \map \Gamma {\sqrt 4}$

Four Fours: $174$

$174 = \map \Gamma 4 \times 4! + 4! + \map \Gamma 4$

Four Fours: $175$

$175 = \paren {\map \Gamma 4 + \map \Gamma {\sqrt 4} } \paren {4! + \map \Gamma {\sqrt 4} }$

Four Fours: $176$

$176 = 4 \times \paren {4! + 4! - 4}$

Four Fours: $177$

$177 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } - \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $178$

$178 = \map \Gamma 4 \times \paren {\map \Gamma 4 + 4!} - \sqrt 4$

Four Fours: $179$

$179 = \map \Gamma 4 \times \paren {\map \Gamma 4 + 4!} - \map \Gamma {\sqrt 4}$

Four Fours: $180$

$180 = \dfrac {4!} {.4} \times \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $181$

$181 = \map \Gamma 4 \times \paren {\map \Gamma 4 + 4!} + \map \Gamma {\sqrt 4}$

Four Fours: $182$

$182 = \map \Gamma 4 \times \paren {\map \Gamma 4 + 4!} + \sqrt 4$

Four Fours: $183$

$183 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } + \sqrt {\dfrac 4 {. \dot 4} }$

Four Fours: $184$

$184 = 4 \times \paren {4! + 4! - \sqrt 4}$

Four Fours: $185$

$185 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } + \dfrac {\sqrt 4} {.4}$

Four Fours: $186$

$186 = 4 \times \paren {4! + 4!} - \map \Gamma 4$

Four Fours: $187$

$187 = \dfrac {\map \Gamma {\map \Gamma 4} } {\sqrt {. \dot 4} } + \map \Gamma 4 - \map \Gamma {\sqrt 4}$

Four Fours: $188$

$188 = 4 \times \paren {4! + 4!} - 4$

Four Fours: $189$

$189 = \dfrac {4! + \dfrac {4!} {.4} } {. \dot 4}$

Four Fours: $190$

$190 = 4 \times \paren {4! + 4!} - \sqrt 4$

Four Fours: $191$

$191 = 4 \times \paren {4! + 4!} - \map \Gamma {\sqrt 4}$

Four Fours: $192$

$192 = \sqrt 4 \times \sqrt 4 \times \paren {4! + 4!}$

Four Fours: $193$

$193 = 4 \times \paren {4! + 4!} + \map \Gamma {\sqrt 4}$

Four Fours: $194$

$194 = 4 \times \paren {4! + 4!} + \sqrt 4$

Four Fours: $195$

$195 = \dfrac {4! + \dfrac {4!} {. \dot 4} } {.4}$

Four Fours: $196$

$196 = 4 \times \paren {4! + 4!} + 4$

Four Fours: $197$

$197 = 4! \times \map \Gamma 4 + \floor {\map \Gamma {\map \Gamma 4} \times .\dot 4}$

Four Fours: $198$

$198 = 4 \times \paren {4! + 4!} + \map \Gamma 4$

Four Fours: $199$

$199 = \dfrac {\map \Gamma {\map \Gamma 4} } {.4} \times \sqrt {.\dot 4} - \map \Gamma {\sqrt 4}$

Four Fours: $200$

$200 = 4 \times \paren {4! + 4! + \sqrt 4}$

$\blacksquare$

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 $x$