4
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The least number that cannot be written using the numbers 0, 1, 2, and 3, each exactly once, and any combination of standard arithmetic operations (including factorials) is 41. What is the least such number if the numbers 0, 1, 2, 3, and 4 are allowed?

Allowed operations are addition, subtraction, multiplication, division, factorials, exponents, square roots, decimal points, intermediate non-integer results, concatenation, recurring decimals, and any amount of parenthesis and brackets. No other digits besides one of each of 0, 1, 2, 3, and 4.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Aza Aug 10 '16 at 19:10
  • $\begingroup$ What's an "intermediate non-integer result"? $\endgroup$ – theonlygusti Mar 21 '17 at 0:39
  • $\begingroup$ Is unary negation allowed? $\endgroup$ – theonlygusti Mar 21 '17 at 0:42
7
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For a start, here is a list of composing all the numbers up to

100

(I avoided using concatenation and decimal points, as based on the conversation in the comments there is some ambiguity if (or when) they are allowed or not.)

$0=0\times(1+2+3+4)$
$1=1^{0+2+3+4}$
$2=0\times(1+3+4)+2$
$3=0\times(1+2+4)+3$
$4=0\times(1+2+3)+4$
$5=0\times(2+3)+4+1$
$6=0\times(1+3)+4+2$
$7=0\times(1+2)+4+3$
$8=0\times2+4+3+1$
$9=0\times1+4+3+2$
$10=0+1+2+3+4$
$11=0\times2+4+3!+1$
$12=0\times1+4+3!+2$
$13=0+2+4+3!+1$
$14=2\times(0!+1)+4+3!$
$15=0+1+2+3\times4$
$16=0\times2+4\times(3+1)$
$17=4^2+0\times3+1$
$18=4^2+0+3-1$
$19=4^2+0+3\times1$
$20=4^2+0+3+1$
$21=4\times(3+2)+0+1$
$22=4\times(3+2)+0+1$
$23=4\times3\times2+0-1$
$24=4\times3\times2+0\times1$
$25=4\times3\times2+0+1$
$26=4!+3+0\times2-1$
$27=4!+3+0\times2\times1$
$28=4!+3+0\times2+1$
$29=4!+3+0\times1+2$
$30=4!+3+0+1+2$
$31=4!+3!+2\times0+1$
$32=4!+3!+1\times0+2$
$33=4!+3!+0+1+2$
$34=4!+3!+0!+1+2$
$35=3!\times(4+2)+0-1$
$36=3!\times(4+2)+0\times1$
$37=3!\times(4+2)+0+1$
$38=3!\times(4+2)+0!+1$
$39=(3!-0!)\times4\times2-1$
$40=(3!-0!)\times4\times2\times1$
$41=(3!-0!)\times4\times2+1$
$42=(3!+0!)\times(4+2)\times1$
$43=(3!+0!)\times(4+2)+1$
$44=4\times(3!+(2+1)!-0!)$
$45=3\times(4^2-1)+0$
$46=3\times(4^2-1)+0!$
$47=3\times4^2-1+0$
$48=3\times4^2+1\times0$
$49=3\times4^2+1+0$
$50=3\times4^2+1+0!$
$51=3\times(4^2+1)+0$
$52=3\times(4^2+1)+0!$
$53=2\times(4!+3)+0-1$
$54=2\times(4!+3)+0\times1$
$55=2\times(4!+3)+0+1$
$56=2\times(4!+3)+0!+1$
$57=2\times(4!+3+0!)+1$
$58=2\times(4!+3+0!+1)$
$59=2\times(4!+3!)+0-1$
$60=2\times(4!+3!)+0\times1$
$61=2\times(4!+3!)+0+1$
$62=2\times(4!+3!)+0!+1$
$63=2\times(4!+3!+0!)+1$
$64=2\times(4!+3!+0!+1)$
$65=4^3+1^2+0$
$66=4^3+2^1+0$
$67=4^3+2+1+0$
$68=4^3+2+1+0!$
$69=4!\times3-2-1+0$
$70=4!\times3-2+1\times0$
$71=4!\times3-2+1+0$
$72=4!\times3+2\times1\times0$
$73=4!\times3+2\times0+1$
$74=4!\times3+2+1\times0$
$75=4!\times3+2+1+0$
$76=(4!+1)\times3+2-0!$
$77=(4!+1)\times3+2+0$
$78=(4!+1)\times3+2+0!$
$79=(4+1)!\times\frac23-0!$
$80=(4+1)!\times\frac23+0$
$81=(4+1)!\times\frac23+0!$
$82=3^4+2-1+0$
$83=3^4+2+1\times0$
$84=3^4+2+1+0$
$85=3^4+2+1+0!$
$86=3^4+(2+1)!-0!$
$87=3^4+(2+1)!+0$
$88=3^4+(2+1)!+0!$
$89=(4!-2)\times(3+1)+0!$
$90=(3+2)!\times(0!-\frac14)$
$91=(4!+(1+2)!)\times3+0!$
$92=(4!-1)\times(3+2-0!)$
$93=3\times(2^{4+1}-0!)$
$94=4!\times(3+0!)-1\times2$
$95=4!\times(3+0!)-1^2$
$96=4!\times(3+0!)\times1^2$
$97=4!\times(3+0!)+1^2$
$98=(4!+1)\times(3+0!)-2$
$99=3\times(2^{4+1}+0!)$
$100=(3!+4)^{2^{1^0}}$

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