mathematics - Make numbers 1 - 30 using the digits 2, 0, 1, 8

It's 2018 so let's repeat last year's challenge with new digits.

This is similar to the "Four fours" puzzle, but using the digits 2, 0, 1 and 8.

Rules:

• Use all four digits exactly once


• Allowed operations: +, -, x, ÷, ! (factorial), exponentiation, square root


• Parentheses and grouping (e.g. "21") are also allowed


• Squaring uses the digit 2 so expressions using multiple twos, like $2^2$ or $1^2 + 8^2$, are not allowed


• Keep the order "2 0 1 8" in at least 25 expressions (and more if you can!)


• The modulus operator is not allowed


• Rounding is not allowed (e.g. 201/8=25)

Good luck and Happy New Year!

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^{Similar question for 2016}

Answer

Found 29 solutions with the numbers in order. Found two almost acceptable cheats for the remaining one.

$30 = 21 + 0! + 8 = \frac{(2 + 0! + 1)!}{ .8 } = \sqrt{\frac{(2+0+1)!!}{.8}}$ (Cheaty McCheatface)
$29 = 20 + 1 + 8 $
$28 = 20 * 1 + 8 $
$27 = 20 -1 + 8$
$26 = 2 + \sqrt{\sqrt{(0! + 1)^8}} !$

$25 = \sqrt{\sqrt{((2 + 0!)!-1)^8}} $
$24 = (2+0!+1^8)!$
$23 = 20 + \sqrt{1+8}$
$22 = -2 + \sqrt{\sqrt{(0! + 1)^8}} !$
$21 = 20 + 1^8$
$20 = 20 * 1^8$
$19 = 20 - 1^8$
$18 = 2 * 0 + 18$
$17 = 20 - \sqrt{1+8}$
$16 = 2^{0!+\sqrt{1+8}}$

$15 = -2 - 0! + 18$
$14 = -(2 + 0) * (1 - 8)$
$13 = 20 + 1 -8 $
$12 = 20 * 1 - 8 $
$11 = 20 - 1 - 8 $
$10 = 2^0 + 1 + 8$
$9 = 2 * 0 + 1 + 8 $
$8 = 2 * 0 * 1 + 8 $
$7 = 2 * 0 - 1 + 8 $
$6 = (2 + 0) * \sqrt{1+8}$

$5 = -2 + 0 - 1 + 8$
$4 = 2 * (0! + 1^8)$
$3 = 2 + 0 + 1^8 $
$2 = 20 - 18$
$1 = 2 * 0 + 1^8$