How do I express 67, 69, 83, 84, 86, 87, 88, 93 with 2,0,2,2 only?

Question

Use $2, 0, 2, 2$ and some mathematical operators to express numbers from $1$ to $100$.

Operators: $+$, $-$, $\times $, $\div$, ( ), √ (square root), $^$ (power), $!$ (factorial), $!!$ (double factorial), nr (permutation) , nr (combination)

Update:88 done,7 remain

Note: exactly 3 twos and one zero can be used. You can use .2 as 0.2, and you can concatenate numbers like 202

I have done $93%$ of them, and I don’t know $7$ of them. Please help me. Here are a few examples:

$$1=2\times 0+2/2 \quad 2=-20+22 \quad 3=2\times 0!/2+2 \quad 4=2\times 0+2+2$$ $$5=2-0!+2+2 \quad 6=2+0+2+2 \quad 7=2+0!+2+2 \quad 8=2\times 2\times 2+0$$ $$9=2\times 2\times 2+0! \quad 10=\left(\sqrt{20/2}\right)^2 \quad 11=(2+2)!/2-0! \quad 12=(2+2)!/2-0$$ $$13=(2+2)!/2+0!$$ $$14=(2+2)!!+(2+0!)!$$ $$15=2^{2\times 2}-0!$$ $$16=2^{2+2}+0$$ $$17=2^{2+2}+0!$$ $$18=(2+2)!-(2+0!)!$$ $$19=22-2-0!$$ $$20=-2-0+22$$ $$21=-2+0!+22$$ $$22=2\times 0+22$$ $$23=(2+2)!-2+0!$$ $$24=2-0+22$$ $$25=(2+2)!+2-0!$$ $$26=(2+2)!+2+0$$ $$28=22+(2+0!)!$$ $$29=(.2)^{-0!}+(2+2)!$$ $$30=(2+2)!+(2+0!)!$$ $$31=(.2)^{-2}+(2+0!)!$$ $$32=2^{2+2+0!}$$ $$33=2^{\sqrt{(.2)^{-2}}}+0!$$ $$34=((2+0!)!)^2-2$$ $$35=\sqrt{(.(0!))^{-2}}+(.2)^{-2}$$ $$36=(20-2)\times 2$$ $$37=((.2)^{-0!})!!+22$$ $$38=20\times 2-2$$ $$39=((.2)^{-0!})!!+(2+2)!!$$ $$40=20\times \sqrt{2+2}$$ $$41=20\times 2+\underbrace{\sqrt{\sqrt{\dots2}}}_{\infty}$$ $$42=20+22$$ $$43=22\times 2-0!$$ $$44=22\times 2-0$$ $$45=22\times 2+0!$$ $$46=(22+0!)\times 2$$ $$47=(2+2+2)!!-0!$$ $$48=(2+2+2)!!+0$$ $$49=(2+2+2)!!+0!$$ $$50=(.2)^{-2}\times 2+0$$ $$51=(.2)^{-2}\times 2+0!$$ $$52=((.2)^{-2}+0!)\times 2$$ $$53=((2+0!)!)!!)+\sqrt{(.2)^{-2}}$$ $$70=(2+\sqrt{(.2)^{-2}})/(.(0!))$$ $$71=\sqrt{(2+\sqrt{(.2)^{-2}}))!+0!}$$ $$80=(2+2)!!\times \sqrt{(.(0!))^{2}}$$ $$81=(2+0!)^{2+2}$$ $$88=P^{\sqrt{(.(0!))^{-2}}}_2-2$$ $$89=P^{\sqrt{(.2)^{-2}}}_2-0!$$ $$90=((2+0!)!)!/(2+2)!!$$ $$91=P^{\sqrt{(.2)^{-2}}}_2+0!$$ $$92=P^{\sqrt{(.(0!))^{-2}}}_2+2$$ $$95=C^{20}_2 /2$$

Answer 1

Update: I have just figured out 88 $$88=P^{\sqrt{(.(0!))^{-2}}}_2-2$$

Answer 2

Edit: added alternate solutions with alternate operations (see comment discussion).

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It seems that this is impossible, given your current specification of rules.

I wrote a python program to solve this. It returned:

unknowns: 7 [67, 69, 83, 84, 86, 87, 93]

Below is the full output. (Notation: 2i = 1 = $\underbrace{\sqrt{\sqrt{\dots2}}}_{\infty}$.)

(To avoid issues with extremely large numbers, largest absolute values of numerators/denominators of arguments to $a^b$ and to $n!$ were bounded by $10^4$, and largest output value of any given binary operation was bounded by $\pm 10^{20}$.)

(WLOG we can use any subset of $\{2,0,2,2\}$ as "$+(0\cdot 2)$" absorbs excess elements.)

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

Notice that the output has more unary minuses than needed. I guess the sets were sorted from low to high, so those were the first found solutions.

Also notice that the square root was never needed, other than in the limit expression.

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Edit $1$: If you ban the infinite square root limit, then more numbers become impossible.

unknowns: 11 [54, 67, 68, 69, 79, 83, 84, 86, 87, 93, 97]

Full list of alternate solutions not using the limit 2i:

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

Where we can see that this time, the square root is actually useful.

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Edit $2$: If you allow subfactorial $!n$, gamma $\Gamma(n)$ and overline ($.\overline{2}=0.222\dots=2/9]$), then only one number remains impossible.

unknowns: 1 [93]

Below is the full list of alternate solutions (,n stands for overline .n̅).

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

Edit $3$: Allowing arbitrary roots solves the last number, $93$. See this answer on Puzzling.SE.