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The title says (nearly) all. Write the digits $2$ $0$ $2$ $2$ in that order and apply mathematical operators to obtain all values from 0 to 9. What's allowed:

  • The four basic operators, $+ - * /$
  • Parenthesis to force the order of operations
  • Power (like $2^0$)
  • Factorial (like $2!$)
  • Concatenation (like $202$)
  • Square root ($\sqrt {}$)
  • Digital point (with or without leading 0, like $2.02$ or $.2$)
  • Periodic numbers (like $.\overline{2}$)

I did not need to use truncation.

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  • 4
    $\begingroup$ You are about two months early. $\endgroup$
    – Bass
    Nov 1, 2021 at 17:01
  • 1
    $\begingroup$ @Bass: I am letting people prepare in advance for puzzles for New Year's Eve $\endgroup$
    – mau
    Nov 1, 2021 at 17:19
  • $\begingroup$ Future Shock Syndrome... $\endgroup$
    – smci
    Nov 2, 2021 at 22:03

4 Answers 4

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0: 2*0*22
1: 2*0+(2/2)
2: 2+0+2-2
3: 2+0+(2/2)
4: 2*0+2+2
5: 2^0+2+2
6: 2+0+2+2
7: 2+0!+2+2
8: (2+0)*2*2
9: (2^0+2)^2

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7
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Since

2x0=0, we can add zero to 2-2=0, 2/2=1, sqrt(2x2)=2, sqrt(2/.2rec)=3, 2x2=4, 2/.2rec=9 to get 0,1,2,3,4,9.

Since

2-0!=1 and 2-0=2, we can add 1 or 2 instead of 0, or (-2+0+ etc.) subtract instead of adding, getting 5,6 as 1+4,2+4 and 8,7 as 9-1,9-2.

And we're done.

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Alternate solutions. Mostly.

$0 = 2 \times 0 \times 2 \times 2$
$1 = 2 + 0 - 2/2$
$2 = 2 + 0 \times 2/2$
$3 = 2 + 0 + 2 / 2$
$4 = 2 \times 0 + 2 \times 2$
$5 = 20/2/2$
$6 = 2 + 0 + 2 + 2$
$7 = 2/0.\overline{2} - 2$
$8 = 20/2 - 2$
$9 = (20 - 2)/2$

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    $\begingroup$ I completely missed the solution for 9... My own solution was $ 20/2.\overline{2}$ $\endgroup$
    – mau
    Nov 1, 2021 at 18:01
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A bit late but:

$2×0+2-2=0$
$2×0+2/2=1$
$2+0+2-2=2$
$2+0+2/2=3$
$2×0+2+2=4$
$2^0+2+2=5$
$2+0+2+2=6$
$-2+0+2/.\overline{2}=7$
$2×(0+2+2)=8$
$2×0+2/.\overline{2}=9$

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  • $\begingroup$ Very simple and elegant, esp. 7 and 9. If we assign points for using the more advanced, this would win. Btw. in 7 you used unary minus which the rules didn't spell out was allowed. $\endgroup$
    – smci
    Nov 5, 2021 at 0:18

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