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:

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

Comments are not for extended discussion; this conversation has been moved to chat. — quid, Apr 26 '21 at 12:35
Note that this was cross-posted on Puzzling.SE as Make 1-100 using 2 0 2 2. — Vepir, Jun 22 '21 at 13:23

伍逸朗 · Accepted Answer · 2021-04-25T14:20:07.540

3

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

edited Apr 25 '21 at 14:20

answered Apr 25 '21 at 14:12

伍逸朗

67

Vepir · Answer 2 · 2021-05-13T11:44:41.677

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

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.

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.

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.

Comments are not for extended discussion; this conversation has been moved to chat. — quid, Apr 26 '21 at 12:36
@伍逸朗 To find out which operations can allow 93 to be made in a non-trivial way, is actually a good question I decided to post on puzzling: Make 93 using 2, 0, 2, 2 and a least powerful set of operations. — Vepir, Apr 26 '21 at 12:55
That is a nice idea, to extend the square root to other roots. With this new rule, we can solve 93. You can maybe include the solution to 93 using that rule, in your post here or your answer here. (Btw, If you feel like this question has been solved, you can accept an answer by clicking the checkmark.) — Vepir, Apr 27 '21 at 14:04

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

2 Answers2