Puzzling Asked by Chris Morris on April 25, 2021
How can you make the numbers 1-100 using only the digits 2, 0, 0, 0? I’ve already found several, the ones I mostly need are 13, 26, 34-40, and 64-85, but I figure it would be interesting to have a record here for other potentially easier/simpler methods.
Addition, subtraction, multiplication, division, exponentiation, factorial, powers (as long as all powers are from the available numbers), sin, cos, tan (and the inverses of those three), and precedence adjustment (parentheses) are allowed, but not concatenation.
A more elegant way to get all the numbers from 1 to 100 if we allow logarithms and the use of lg as logarithm in base 10 is this (I know, I know, it looks like cheating a bit but it's beautiful)
$x = log_{frac{0!}{2}}left({lgunderbrace{sqrt{sqrt{dotssqrt{frac{0!}{0!%},},},}}_text{x+1 square roots}}right)$
This is equivalent
$x = log_{frac{0!}{2}}left({lgunderbrace{sqrt{sqrt{dotssqrt{frac{1}{frac{1}{100}},},},}}_text{x+1 square roots}}right)$
Moving on
$x = log_{frac{1}{2}}left({lgunderbrace{sqrt{sqrt{dotssqrt{10,},},}}_text{x square roots}}right)$
$x = log_{frac{1}{2}}left({lg{10^{frac{1}{2^x}}}}right)$
$x = log_{frac{1}{2}}frac{1}{2^x}$
But here is the brute force approach without log.
Operations used: + - $frac{a}{b}$ $times$ $sqrt{a}$ $a^b$ $lfloor a rfloor$ $lceil arceil$ $a%$ (wich is basically division by 100) and trigonometrical functions (sin, cos, ...)
$0 = 0 times 0 times 0 times 2$
$1 = 0 times 0 + 2^0$
$2 = 0 + 0 + 0 + 2$
$3 = 0 + 0 + 0! + 2$
$4 = 0 + 2^{0!+0!}$
$5 = 0! + 2^{0!+0!}$
$6 = (0! + 0!+0!) times 2 $
$7 = (0! + 2)! + 0! +0$
$8 = (0! + 2)! + 0! +0!$
$9 = (0! + 0! + 0!)^2$
$10 = frac{20}{0!+0!}$
$11 = lceil{sqrt{(2+0! +0!+0!)!}}rceil$
$12 = (0! + 0! + 0!)! times 2$
$13 = lfloor{sqrt{200}}rfloor - 0!$
$14 = lfloor{sqrt{200}}rfloor + 0$
$15 = lceil{sqrt{200}}rceil + 0$
$16 = lceil{sqrt{200}}rceil + 0!$
$17 = lfloorfrac{(2+0!)!}{0!%}rfloor + 0!$
$18 = 20 - 0! - 0!$
$19 = 20 + 0 - 0!$
$20 = 20 + 0 + 0$
$21 = 20 + 0! + 0$
$22 = 20 + 0! + 0!$
$23 = (2 + 0! + 0!)! - 0!$
$24 = (2 + 0! + 0!)! + 0$
$25 = (2 + 0! + 0!)! + 0!$
$26 = lfloor{sqrt{((2+0!)!)!}}rfloor + 0 + 0$
$27 = lceil{sqrt{((2+0!)!)!}}rceil + 0 + 0$
$28 = lceil{sqrt{((2+0!)!)!}}rceil + 0! + 0$
$29 = lceil{sqrt{((2+0!)!)!}}rceil + 0! + 0!$
$30 = sqrt{frac{0!}{0!%}} + 20$
$31 = lceilsqrt{sqrt{sqrt{sqrt{((2+0!+0!)!)!}}}}rceil + 0$
$32 = 2 ^{lfloorsqrt{sqrt{((0!+0!+0!)!)!}}rfloor}$
$33 = lfloor frac{frac{0!}{0!%}}{2+0!} rfloor$
$34 = lceil frac{frac{0!}{0!%}}{2+0!} rceil$
$35 = lfloor-frac{sqrt{frac{0!}{0!%}}}{sin((2+0!)!)}rfloor$
$36 = ((0! + 0! + 0!)! )^2$
$37 = sqrt{frac{0!}{0!%}} + lceilsqrt{(2+0!)!}rceil$
$38 = $
$39 = lfloorsqrt{sqrt{sqrt{sqrt{frac{0!}{0!%}}!}}} times (2+0!)!rfloor$
$40 = 20 times (0! + 0!)$
$41 = lfloorsqrt{sqrt{sqrt{frac{0!}{0!%}}!}}rfloor - 2 + 0$
$42 = lfloorsqrt{sqrt{sqrt{frac{0!}{0!%}}!}}rfloor - 2 ^ 0$
$43 = lfloorsqrt{sqrt{sqrt{frac{0!}{0!%}}!}}rfloor + 2 times 0$
$44 = lfloor{sqrt{2000}}rfloor$
$45 = lceil{sqrt{2000}}rceil$
$46 = lfloorsqrt{sqrt{sqrt{sqrt{lfloorsqrt{(2+0!)!}rfloor!}}}}rfloor + 0 +0$
$47 = lfloorsqrt{sqrt{sqrt{sqrt{lfloorsqrt{(2+0!)!}rfloor!}}}}rfloor + 0! +0$
$48 = frac{0!}{(0! + 0!)%} - 2$
$49 = frac{0!}{2 times 0!%} -0!$
$50 = frac{0!}{2 times 0!%} +0$
$51 = frac{0!}{2 times 0!%} +0!$
$52 = frac{0!}{(0! + 0!)%} + 2$
$53 = $
$54 = lceil{sqrt{((0! + 0! + 0!)!)!}}rceil times 2$
$55 = lceil{sqrt{((0! + 0! + 0!)!)!}} times 2rceil$
$56 = lfloorsqrt{sqrt{sqrt{sqrt{lfloorsqrt{(2+0!)!}rfloor!}}}}rfloor + sqrt{frac{0!}{0!%}}$
$57 = lfloorsqrt{sqrt{sqrt{sqrt{lceilsqrt{(2+0!)!}rceil!}}}}rfloor + 0 +0$
$58 = sqrt{sqrt{sqrt{sqrt{lceilsqrt{(2+0!)!}rceil!}}}} + 0! +0$
$59 = sqrt{sqrt{sqrt{sqrt{lceilsqrt{(2+0!)!}rceil!}}}} + 0! +0!$
$60 = sqrt{frac{0!}{0!%}} times (2 + 0!)!$
$61 = $
$62 = $
$63 = $
$64 = 2^{(0!+0!+0!)!}$
$65 = lceilfrac{frac{0!}{0!%}}{ctan(((2+0!)!)!)}rceil$
$66 = $
$67 = lfloorsqrt{sqrt{sqrt{sqrt{lceilsqrt{(2+0!)!}rceil!}}}}rfloor + sqrt{frac{0!}{0!%}}$
$68 = lceilsqrt{sqrt{sqrt{sqrt{lceilsqrt{(2+0!)!}rceil!}}}}rceil + sqrt{frac{0!}{0!%}}$
$69 = lfloor{sqrt{(((2+0!)!) + 0!)!}}rfloor - 0!$
$70 = lfloor{sqrt{(((2+0!)!) + 0!)!}}rfloor + 0$
$71 = lceil{sqrt{(((2+0!)!) + 0!)!}}rceil + 0$
$72 = lceil{sqrt{(((2+0!)!) + 0!)!}}rceil + 0!$
$73 = frac{0!}{0!%} - lceil{sqrt{((2 + 0!)!)!}}rceil$
$74 = frac{0!}{0!%} - lfloor{sqrt{((2 + 0!)!)!}}rfloor$
$75 = $
$76 = $
$77 = $
$78 = $
$79 = lfloorsqrt{sqrt{lceilsqrt{(2+0!+0!+0!)!}rceil!}}rfloor$
$80 = frac{0!}{0!%} - 20$
$81 = lceilsqrt{sqrt{sqrt{(sqrt{frac{0!}{0!%}} + 0!)!}}}rceil ^ 2$
$82 = $
$83 = -lfloorcos((2+0!)!)!) * frac{0!}{0!%}rfloor$
$84 = lfloorsqrt{(2+0!)!*sqrt{frac{0!}{0!%}}}rfloor$
$85 = lceilsqrt{(2+0!)!*sqrt{frac{0!}{0!%}}}rceil$
$86 = $
$87 = $
$88 = $
$89 = lfloorsin(2) times frac{0!}{0!%}rfloor - 0!$
$90 = lfloorsin(2) times frac{0!}{0!%}rfloor + 0$
$91 = lfloorsin(2) times frac{0!}{0!%}rfloor + 0!$
$92 = $
$93 = $
$94 = frac{0!}{0!%} - (2 + 0!)!$
$95 = lceilsqrt{sqrt{sqrt{(20 - 0! -0!)!}}}rceil$
$96 = lfloorfrac{0!}{0!%} + ctan((2+0!)!)rfloor$
$97 = frac{0!}{0!%} - 2 - 0!$
$98 = frac{0!}{0!%} - 2 + 0$
$99 = frac{0!}{0!%} - 2 + 0!$
$100 = frac{0!}{0!%} + 2 times 0$
Correct answer by Marius on April 25, 2021
Operators used: +, -, !, $sqrt{cdot}$, %, $lfloor cdot rfloor$, $lceil cdot rceil$. I also concatenate 2 to 0 to form 20, and get reciprocals with $a^{-(0!)}$ and use bracketing. Multiplication and division operators are not used in this answer.
Split the given 4 digits into two sets, {0,0} and {2,0}. We can form 1-7 just using either set:
begin{array}{clll} 1 & = 0! + 0 & = 2 - 0! 2 & = 0! + 0! & = 2 + 0 3 & = lfloor √sqrt{0!%^{-(0!)}} rfloor & = 2 + 0! & 4 & = lceil √sqrt{0!%^{-(0!)}} rceil & = lfloor sqrt{20} rfloor 5 & = lfloor √(lceil √sqrt{0!%^{-(0!)}} rceil !) rfloor & = lceil sqrt{20} rceil 6 & = lfloor √√√(sqrt{0!%^{-(0!)}}!) rfloor & = (2 + 0!)! 7 & = lceil √√√(sqrt{0!%^{-(0!)}}!) rceil & = lfloor sqrt{2%^{-(0!)}} rfloor end{array}
We now form 8-11 using either definition of 5 and 7 above:
begin{array}{cccc} 8 = lfloor √√(7!) rfloor & 9 = lceil √√(7!) rceil & 10 = lfloor √(5!) rfloor & 11 = lceil √(5!) rceil end{array}
Now that we can construct all the numbers from 1 to 11 using just {2,0} as well as using just {0,0}, if we have any integer $x$ constructed from either set, we can get all the integers $x + k$, where $-11 leq k leq 11$. If $x$ was formed from {2,0}, form $k$ from {0,0}, and vice versa.
Pick any of the above expressions for the numbers in the expansions below.
begin{array}{c|c|c} x & text{expansion} & text{covers} hline 11 & lceil √(5!) rceil & 0 - 22 24 & 4! & 13 - 35 44 & lceil √√(10!) rceil & 33 - 55 50 & 2%^{-1} & 39 - 61 71 & lceil √(7!) rceil & 60 - 82 80 & lceil √√(11!) rceil & 69 - 91 100 & 0!%^{-(0!)} & 89 - 111 end{array}
This produces all the numbers from 0 to 111. If it's required to use all 4 digits in {2,0,0,0}, observe that none of the combinations above use just the 3 zeros. They either use all 4 digits or at least one 0 is unused. If a 0 is unused, multiply that 0 by the sum of all the remaining unused digits, and add the result (also 0) to the original expression.
Answered by Lawrence on April 25, 2021
Using only the operations +, -, floor, ceiling, factorial (Gamma function extension), and square root:
1 = 2 - 0! + 0 + 0
2 = 2 + 0 + 0 + 0
3 = 2 + 0! + 0 + 0
4 = 2 + 0! + 0! + 0
5 = (2 + 0!)! - 0! + 0
6 = (2 + 0!)! + 0 + 0
7 = (2 + 0!)! + 0! + 0
8 = (2 + 0!)! + 0! + 0!
9 = ceil((sqrt(2)! + 0! + 0!)!) + 0
10 = floor((sqrt(2) + 0! + 0!)!) + 0
11 = ceil((sqrt(2) + 0! + 0!)!) + 0
12 = ceil((sqrt(2) + 0! + 0!)!) + 0!
13 = ceil((sqrt(2)! + 0!)!!!) + 0 + 0
14 = floor((sqrt((2 + 0!)! + 0!) + 0!)!)
15 = ceil((sqrt((2 + 0!)! + 0!) + 0!)!)
16 = floor((sqrt(2 + 0!) + 0! + 0!)!)
17 = ceil((sqrt(2 + 0!) + 0! + 0!)!)
18 = ceil((sqrt(2)!! + 0!)!!!!) + 0 + 0
19 = ceil((sqrt(sqrt((2 + 0!)! + 0!)) + 0!)!!)
20 = floor((sqrt((2 + 0! + 0!)!) - 0!)!)
21 = floor(sqrt((2 + 0!)! + 0!)!!) + 0
22 = ceil((sqrt(2) + 0!)!)! - 0! - 0!
23 = (2 + 0! + 0!)! - 0!
24 = (2 + 0! + 0!)! + 0
25 = (2 + 0! + 0!)! + 0!
26 = floor(sqrt((2 + 0!)!!)) + 0 + 0
27 = ceil(sqrt((2 + 0!)!!)) + 0 + 0
28 = ceil(sqrt((2 + 0!)!!)) + 0! + 0
29 = ceil(sqrt((2 + 0!)!!)) + 0! + 0!
30 = ceil((sqrt((2 + 0!)!)! + 0!)!) + 0
31 = floor(sqrt((sqrt(2) + 0!)!!!)) - 0! + 0
32 = floor((sqrt(sqrt(2)) + 0! + 0! + 0!)!)
33 = ceil((sqrt(sqrt(2)) + 0! + 0! + 0!)!)
34 = ceil(sqrt((sqrt(2) + 0!)!!!)) + 0! + 0
35 = floor((sqrt(2)! + 0! + 0! + 0!)!)
36 = ceil((sqrt(2)! + 0! + 0! + 0!)!)
37 = ceil(((sqrt(2)!! + 0!)! + 0! + 0!)!)
38 = floor((sqrt(2 + 0!) + 0!)!!) - 0!
39 = floor((sqrt(2 + 0!) + 0!)!!) + 0
40 = ceil((sqrt(2 + 0!) + 0!)!!) + 0
41 = ceil((sqrt(2 + 0!) + 0!)!!) + 0!
42 = floor((sqrt((sqrt(2) + 0!)!) + 0!)!!) + 0
43 = ceil((sqrt((sqrt(2) + 0!)!) + 0!)!!) + 0
44 = floor(((sqrt(sqrt(2)) + 0!)! + 0! + 0!)!)
45 = floor((sqrt(2) + 0! + 0! + 0!)!)
46 = ceil((sqrt(2) + 0! + 0! + 0!)!)
47 = ceil(sqrt(sqrt(((sqrt(sqrt(2)) + 0!)! + 0!)!!))) + 0
48 = floor((sqrt((2 + 0!)!) + 0! + 0!)!)
49 = ceil((sqrt((2 + 0!)!) + 0! + 0!)!)
50 = ceil(sqrt(((sqrt(sqrt(2)) + 0! + 0!)! - 0!)!))
51 = floor((sqrt((sqrt(2) + 0!)!!) + 0! + 0!)!)
52 = floor(sqrt((sqrt(sqrt(2)) + 0! + 0!)!)!!) + 0
53 = floor(sqrt(sqrt((sqrt(2) + 0! + 0!)!!))) + 0
54 = floor(((sqrt(sqrt(2) + 0!) + 0!)! + 0!)!)
55 = ceil(((sqrt(sqrt(2) + 0!) + 0!)! + 0!)!)
56 = floor(((sqrt(2)! + 0!)!! + 0!)!) + 0
57 = ceil(((sqrt(2)! + 0!)!! + 0!)!) + 0
58 = ceil(((sqrt(2)! + 0!)! + 0! + 0!)!)
59 = ceil(((sqrt(sqrt((2 + 0!)!)) + 0!)! + 0!)!)
60 = ceil(sqrt(floor(sqrt((2 + 0!)! + 0!)!!))!) + 0
61 = floor(sqrt(sqrt((2 + 0!)! + 0!)!!)!) + 0
62 = ceil(sqrt(sqrt((2 + 0!)! + 0!)!!)!) + 0
63 = floor((sqrt(sqrt(sqrt(2)) + 0! + 0!) + 0!)!!)
64 = ceil((sqrt(sqrt(sqrt(2)) + 0! + 0!) + 0!)!!)
65 = ceil(sqrt((sqrt((sqrt(2) + 0! + 0!)!)! - 0!)!))
66 = floor((sqrt(ceil((sqrt(2) + 0!)!!)) + 0! + 0!)!)
67 = floor(((sqrt(2 + 0!)! + 0!)! + 0!)!)
68 = ceil(((sqrt(2 + 0!)! + 0!)! + 0!)!)
69 = floor(sqrt(((2 + 0!)! + 0!)!)) - 0!
70 = floor(sqrt(((2 + 0!)! + 0!)!)) + 0
71 = ceil(sqrt(((2 + 0!)! + 0!)!)) + 0
72 = ceil(sqrt(((2 + 0!)! + 0!)!)) + 0!
73 = floor((sqrt(sqrt(sqrt(2) - 0!)) + 0! + 0!)!!)
74 = floor((sqrt(sqrt(2)! + 0! + 0!) + 0!)!!)
75 = ceil((sqrt(sqrt(2)! + 0! + 0!) + 0!)!!)
76 = ceil((((sqrt(2) - 0!)! + 0!)! + 0!)!!)
77 = floor(sqrt(sqrt(sqrt(((2 + 0!)! - 0!)!)!))) + 0
78 = ceil(sqrt(sqrt(sqrt(((2 + 0!)! - 0!)!)!))) + 0
79 = floor(sqrt(sqrt(ceil((sqrt(2) + 0! + 0!)!)!))) + 0
80 = ceil(sqrt(sqrt(ceil((sqrt(2) + 0! + 0!)!)!))) + 0
81 = floor(sqrt((sqrt(2)!! + 0! + 0!)!!)) + 0
82 = ceil(sqrt((sqrt(2)!! + 0! + 0!)!!)) + 0
83 = ceil(((sqrt(2)!! + 0!)!!! + 0!)!) + 0
84 = floor(sqrt(ceil((sqrt(2) + 0!)!)! - 0!)!) - 0!
85 = floor(sqrt((2 + 0! + 0!)! - 0!)!)
86 = ceil(sqrt((2 + 0! + 0!)! - 0!)!)
87 = floor(sqrt(((sqrt(2) + 0!)!! + 0!)!)) + 0
88 = ceil(sqrt(((sqrt(2) + 0!)!! + 0!)!)) + 0
89 = ceil(sqrt(sqrt((2 + 0!)!)!!!)) + 0 + 0
90 = ceil(sqrt(sqrt((2 + 0!)!)!!!)) + 0! + 0
91 = ceil(sqrt(sqrt((2 + 0!)!)!!!)) + 0! + 0!
92 = floor(((sqrt(sqrt(2) - 0!)! + 0!)! + 0!)!!)
93 = ceil(((sqrt(sqrt(2) - 0!)! + 0!)! + 0!)!!)
94 = floor(sqrt((2 + 0!)! + 0! + 0!)!!)
95 = ceil(sqrt((2 + 0!)! + 0! + 0!)!!)
96 = ceil(sqrt(floor((sqrt(2)! + 0! + 0!)!))!!) + 0!
97 = floor(sqrt(sqrt(((sqrt(2) + 0! + 0!)! + 0!)!)))
98 = ceil(sqrt(sqrt(((sqrt(2) + 0! + 0!)! + 0!)!)))
99 = floor(sqrt(((sqrt(2)! + 0! + 0!)! - 0!)!))
100 = floor(sqrt((2 + 0! + 0!)!)!) - 0!
It should be noted that copying one of the above formulas into WolframAlpha might not give the right answer. This is because it interprets '!!' as the "double factorial function": n!! = n(n-2)(n-4)..., where I use n!! to mean (n!)!. Putting spaces between the exclamation points should fix this.
Answered by benzene on April 25, 2021
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