Create the numbers 1 - 30 using the digits 2, 0, 1, 9 in this particular order!
Clash Royale CLAN TAG#URR8PPP
Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.
The rules haven't changed:
- Use all four digits exactly once in the order 2-0-1-9.
- Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.
- Parentheses and grouping (e.g. "19") are also allowed.
- Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.
- The modulus operator $(%, mod)$ is not allowed.
- Rounding (e.g. 201/9=22) is not allowed.
I'm curious to see your creative solutions!
May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!
Happy New Year and greetings from Germany!
André
formation-of-numbers number-theory
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
add a comment |
Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.
The rules haven't changed:
- Use all four digits exactly once in the order 2-0-1-9.
- Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.
- Parentheses and grouping (e.g. "19") are also allowed.
- Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.
- The modulus operator $(%, mod)$ is not allowed.
- Rounding (e.g. 201/9=22) is not allowed.
I'm curious to see your creative solutions!
May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!
Happy New Year and greetings from Germany!
André
formation-of-numbers number-theory
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
Will you be giving a green check to someone?
– flashstorm
Dec 31 '18 at 22:46
Of course I'll do.
– André
Jan 1 at 2:40
add a comment |
Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.
The rules haven't changed:
- Use all four digits exactly once in the order 2-0-1-9.
- Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.
- Parentheses and grouping (e.g. "19") are also allowed.
- Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.
- The modulus operator $(%, mod)$ is not allowed.
- Rounding (e.g. 201/9=22) is not allowed.
I'm curious to see your creative solutions!
May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!
Happy New Year and greetings from Germany!
André
formation-of-numbers number-theory
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the numbers 1 - 30.
The rules haven't changed:
- Use all four digits exactly once in the order 2-0-1-9.
- Allowed operations: $+, -, cdot, div, !$ (factorial), $!!$ (double factorial), square root, exponentiation.
- Parentheses and grouping (e.g. "19") are also allowed.
- Squaring uses the digit 2, so expressions using multiple 2's, e. g. $2^2$ or $1^2+2^9$, are not allowed.
- The modulus operator $(%, mod)$ is not allowed.
- Rounding (e.g. 201/9=22) is not allowed.
I'm curious to see your creative solutions!
May each day of 2019 bring happiness, good cheer, and sweet surprises to you and all your dear ones!
Happy New Year and greetings from Germany!
André
formation-of-numbers number-theory
formation-of-numbers number-theory
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
asked Dec 31 '18 at 18:28
AndréAndré
1,193717
1,193717
Will you be giving a green check to someone?
– flashstorm
Dec 31 '18 at 22:46
Of course I'll do.
– André
Jan 1 at 2:40
add a comment |
Will you be giving a green check to someone?
– flashstorm
Dec 31 '18 at 22:46
Of course I'll do.
– André
Jan 1 at 2:40
Will you be giving a green check to someone?
– flashstorm
Dec 31 '18 at 22:46
Will you be giving a green check to someone?
– flashstorm
Dec 31 '18 at 22:46
Of course I'll do.
– André
Jan 1 at 2:40
Of course I'll do.
– André
Jan 1 at 2:40
add a comment |
4 Answers
4
active
oldest
votes
1
1 = 2^(0*19)
2
2 = 2 + (0*19)
3
3 = 2 + 0!^19
4
4 = 2 ^ (0! + 1 ^ 9)
5
-((2 + 0! + 1) - 9)
6
-((2 + 0 + 1) - 9))
7
-((2 + 0*1 - 9))
8
-((2 - 01) - 9)
9
2*0*1 + 9
10
2*0 + 1 + 9
11
2 + 0*1 + 9
12
2 + 0 + 1 + 9
13
2 + 0! + 1 + 9
14
(2 + 0!)! - 1 + 9
15
(2 + 0 + 1)! + 9
16
(2 + 0!)! + 1 + 9
17
20 - (1 * sqrt(9))
18
(2 + (0 * 1)) * 9
19
20 - 1^9
20
20 * 1^9
21
20 + 1^9
22
2 + 0! + 19
23
20 + 1 * sqrt(9)
24
20 + 1 + sqrt(9)
25
2 || (0! + 1 + sqrt (9))
explained:
Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.
26
2 || ((0! + 1) * sqrt(9))
27
(2 + 0 + 1) * 9
28
((2 + 0!) || 1) - sqrt(9)
29
(2 + (0 * 1)) || 9
30
20 + 1 + 9
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
add a comment |
$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt9=-(2+0+1)!+9$$
$$4=2^0-1+sqrt9$$
$$5=20div (1+sqrt9)$$
$$6=(2cdot 0cdot 1+sqrt9)!$$
$$7=-2-0cdot 1+9$$
$$8=2^0cdot 1+sqrt9$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^0!+1+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^0+1+sqrt9$$
$$17=20-sqrt1cdot 9$$
$$18=20+1-sqrt9$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt1cdot 9$$
$$24=2^0-1+sqrt9!=20+1+sqrt9$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt9)!$$
$$27=(2+0+1)^sqrt9$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$
DONE!
edited Jan 1 at 3:26
Hugh
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
|
show 2 more comments
I wrote a program to determine all representable numbers between 1 and 1,000,000 following the rules, so this should be a comprehensive list.
0 through 30:
$$0 = 2times0timesleft(1+9right)$$
$$1 = 20-19$$
$$2 = 2+0timesleft(1+9right)$$
$$3 = 2+0+1^9$$
$$4 = 2-left(0+1-sqrt9right)$$
$$5 = frac201+sqrt9$$
$$6 = -2-left(0+1-9right)$$
$$7 = 2timesleft(0-1right)+9$$
$$8 = 2times0-left(1-9right)$$
$$9 = 2times0+1times9$$
$$10 = 2-left(0+1-9right)$$
$$11 = 2+0+1times9$$
$$12 = 2+0+1+9$$
$$13 = 20-left(1+left(sqrt9right)!right)$$
$$14 = 20-1timesleft(sqrt9right)!$$
$$15 = 20+1-left(sqrt9right)!$$
$$16 = 2timesleft(0-left(1-9right)right)$$
$$17 = -2+0+19$$
$$18 = 2timesleft(0+1times9right)$$
$$19 = 2times0+19$$
$$20 = 2timesleft(0+1+9right)$$
$$21 = 2+0+19$$
$$22 = 20-left(1-sqrt9right)$$
$$23 = 20+1timessqrt9$$
$$24 = 20+1+sqrt9$$
$$25 = 20-left(1-left(sqrt9right)!right)$$
$$26 = 20+1timesleft(sqrt9right)!$$
$$27 = left(2+0+1right)times9$$
$$28 = 20-left(1-9right)$$
$$29 = 20+1times9$$
$$30 = 20+1+9$$
31 through 100. Concatenation of expressions (denoted with ||) is only used when necessary. Interestingly enough, 31 is the smallest number that cannot be done without concatenation of expressions, and 75 is the smallest number that cannot be done in general.
$$31 = 2+left(left(0!+1right)||9right)$$
$$32 = sqrt2^0+1+9$$
$$33 = left(2+0!+1right)!+9$$
$$34 = left(2+0!right)||left(1+sqrt9right)$$
$$35 = 20+left(-1+left(sqrt9right)!right)!!$$
$$36 = 2timesleft(-0!+19right)$$
$$37 = left(2+0!right)||left(1+left(sqrt9right)!right)$$
$$38 = 2timesleft(0+19right)$$
$$39 = 20+19$$
$$40 = 20timesleft(-1+sqrt9right)$$
$$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt9right)!right)$$
$$42 = left(2+0!right)!timesleft(1+left(sqrt9right)!right)$$
$$43 = left(left(2+0!right)!right)!!+1-left(sqrt9right)!$$
$$44 = 20+left(1+sqrt9right)!$$
$$45 = left(left(2+0!right)!-1right)times9$$
$$46 = 2timesleft(-0!+left(1+sqrt9right)!right)$$
$$47 = 2times0-left(1-left(left(sqrt9right)!right)!!right)$$
$$48 = 2timesleft(0+left(1+sqrt9right)!right)$$
$$49 = 2-left(0+1-left(left(sqrt9right)!right)!!right)$$
$$50 = 2timesleft(0!+left(1+sqrt9right)!right)$$
$$51 = 2+0+1+left(left(sqrt9right)!right)!!$$
$$52 = 2+0!+1+left(left(sqrt9right)!right)!!$$
$$53 = left(2+0!right)!-left(1-left(left(sqrt9right)!right)!!right)$$
$$54 = left(2+0!right)!times1times9$$
$$55 = left(2+0!right)!+1+left(left(sqrt9right)!right)!!$$
$$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
$$57 = left(20-1right)timessqrt9$$
$$58 = left(left(2+0!right)!right)!!+1+9$$
$$59 = left(left(2+0!right)!-1right)||9$$
$$60 = 20times1timessqrt9$$
$$61 = left(2+0!right)!||left(1^9right)$$
$$62 = -2+left(0!+1right)^left(sqrt9right)!$$
$$63 = left(20+1right)timessqrt9$$
$$64 = 2^0+1timesleft(sqrt9right)!$$
$$65 = left(2+0!right)!||left(-1+left(sqrt9right)!right)$$
$$66 = 2+left(0!+1right)^left(sqrt9right)!$$
$$67 = frac201sqrt9$$
$$68 = 20+1timesleft(left(sqrt9right)!right)!!$$
$$69 = -2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$70 = left(left(2+0!right)!||1right)+9$$
$$71 = sqrt2^0+left(1+left(sqrt9right)!right)!$$
$$72 = left(2+0!right)timesleft(1+sqrt9right)!$$
$$73 = 2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$74 = 2timesleft(left(-0!||1right)+left(left(sqrt9right)!right)!!right)$$
$$76 = left(left(2+0!right)!+1right)||left(sqrt9right)!$$
$$78 = left(2+0!right)!timesleft(1||sqrt9right)$$
$$79 = left(left(2+0!right)!+1right)||9$$
$$80 = 20timesleft(1+sqrt9right)$$
$$81 = left(2+0!right)^1+sqrt9$$
$$83 = left(2+0!+1right)!!||sqrt9$$
$$84 = left(2||0!right)timesleft(1+sqrt9right)$$
$$85 = -20+left(1+left(sqrt9right)!right)!!$$
$$86 = left(2+0!+1right)!!||left(sqrt9right)!$$
$$89 = left(2+0!+1right)!!||9$$
$$90 = fracleft(left(2+0!right)!right)!-1+9$$
$$92 = 2timesleft(-0!-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$93 = left(left(2+0!right)||1right)timessqrt9$$
$$94 = 2timesleft(0-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt9right)!right)!!right)$$
$$96 = 2timesleft(0+1timesleft(left(sqrt9right)!right)!!right)$$
$$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt9right)!right)!!$$
$$98 = 2timesleft(0+1+left(left(sqrt9right)!right)!!right)$$
$$99 = left(left(2+0!right)!+1right)!!-left(sqrt9right)!$$
$$100 = 20timesleft(-1+left(sqrt9right)!right)$$
The rest of the numbers from 101 to 1,000,000 can be found here.
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
add a comment |
1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.
8:
$8 = sqrt(2^0!+1!!)!! / (sqrt9)!$
16:
$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$
17:
$17 = (2 + 0! + 1)!! + 9$
18:
$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^0+1 cdot 9$
19:
$19 = 2 cdot 0 + 19$
20:
$20 = 2^0 + 19 = 20! / 19!$
21:
$21 = 20 + 1^9$
22:
$22 = 20 - 1 + sqrt9$
23:
$23 = 20 + 1 cdot sqrt9$
24:
$24 = 2^0! + 1 cdot (sqrt9)! = 2^0! + 1!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$
25:
$25 = (2 + 0!)! + 19$
26:
$26 = 20 + 1 cdot (sqrt9)!$
27:
$27 = 2^0! + 1! + sqrt9$
28: Can't get one different from what I've already seen in other answers. Will maybe try later.
29:
$29 = 20 + 1 cdot 9$
30:
$30 = 2^0! + 1! + (sqrt9)!$
I know we're supposed to stop at 30, but I accidentally found this fun one:
32:
$32 = sqrt20!! / (1 + 9)!$
answered Dec 31 '18 at 19:13
tilpertilper
898514
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
1
1 = 2^(0*19)
2
2 = 2 + (0*19)
3
3 = 2 + 0!^19
4
4 = 2 ^ (0! + 1 ^ 9)
5
-((2 + 0! + 1) - 9)
6
-((2 + 0 + 1) - 9))
7
-((2 + 0*1 - 9))
8
-((2 - 01) - 9)
9
2*0*1 + 9
10
2*0 + 1 + 9
11
2 + 0*1 + 9
12
2 + 0 + 1 + 9
13
2 + 0! + 1 + 9
14
(2 + 0!)! - 1 + 9
15
(2 + 0 + 1)! + 9
16
(2 + 0!)! + 1 + 9
17
20 - (1 * sqrt(9))
18
(2 + (0 * 1)) * 9
19
20 - 1^9
20
20 * 1^9
21
20 + 1^9
22
2 + 0! + 19
23
20 + 1 * sqrt(9)
24
20 + 1 + sqrt(9)
25
2 || (0! + 1 + sqrt (9))
explained:
Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.
26
2 || ((0! + 1) * sqrt(9))
27
(2 + 0 + 1) * 9
28
((2 + 0!) || 1) - sqrt(9)
29
(2 + (0 * 1)) || 9
30
20 + 1 + 9
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
add a comment |
1
1 = 2^(0*19)
2
2 = 2 + (0*19)
3
3 = 2 + 0!^19
4
4 = 2 ^ (0! + 1 ^ 9)
5
-((2 + 0! + 1) - 9)
6
-((2 + 0 + 1) - 9))
7
-((2 + 0*1 - 9))
8
-((2 - 01) - 9)
9
2*0*1 + 9
10
2*0 + 1 + 9
11
2 + 0*1 + 9
12
2 + 0 + 1 + 9
13
2 + 0! + 1 + 9
14
(2 + 0!)! - 1 + 9
15
(2 + 0 + 1)! + 9
16
(2 + 0!)! + 1 + 9
17
20 - (1 * sqrt(9))
18
(2 + (0 * 1)) * 9
19
20 - 1^9
20
20 * 1^9
21
20 + 1^9
22
2 + 0! + 19
23
20 + 1 * sqrt(9)
24
20 + 1 + sqrt(9)
25
2 || (0! + 1 + sqrt (9))
explained:
Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.
26
2 || ((0! + 1) * sqrt(9))
27
(2 + 0 + 1) * 9
28
((2 + 0!) || 1) - sqrt(9)
29
(2 + (0 * 1)) || 9
30
20 + 1 + 9
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
add a comment |
1
1 = 2^(0*19)
2
2 = 2 + (0*19)
3
3 = 2 + 0!^19
4
4 = 2 ^ (0! + 1 ^ 9)
5
-((2 + 0! + 1) - 9)
6
-((2 + 0 + 1) - 9))
7
-((2 + 0*1 - 9))
8
-((2 - 01) - 9)
9
2*0*1 + 9
10
2*0 + 1 + 9
11
2 + 0*1 + 9
12
2 + 0 + 1 + 9
13
2 + 0! + 1 + 9
14
(2 + 0!)! - 1 + 9
15
(2 + 0 + 1)! + 9
16
(2 + 0!)! + 1 + 9
17
20 - (1 * sqrt(9))
18
(2 + (0 * 1)) * 9
19
20 - 1^9
20
20 * 1^9
21
20 + 1^9
22
2 + 0! + 19
23
20 + 1 * sqrt(9)
24
20 + 1 + sqrt(9)
25
2 || (0! + 1 + sqrt (9))
explained:
Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.
26
2 || ((0! + 1) * sqrt(9))
27
(2 + 0 + 1) * 9
28
((2 + 0!) || 1) - sqrt(9)
29
(2 + (0 * 1)) || 9
30
20 + 1 + 9
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
1
1 = 2^(0*19)
2
2 = 2 + (0*19)
3
3 = 2 + 0!^19
4
4 = 2 ^ (0! + 1 ^ 9)
5
-((2 + 0! + 1) - 9)
6
-((2 + 0 + 1) - 9))
7
-((2 + 0*1 - 9))
8
-((2 - 01) - 9)
9
2*0*1 + 9
10
2*0 + 1 + 9
11
2 + 0*1 + 9
12
2 + 0 + 1 + 9
13
2 + 0! + 1 + 9
14
(2 + 0!)! - 1 + 9
15
(2 + 0 + 1)! + 9
16
(2 + 0!)! + 1 + 9
17
20 - (1 * sqrt(9))
18
(2 + (0 * 1)) * 9
19
20 - 1^9
20
20 * 1^9
21
20 + 1^9
22
2 + 0! + 19
23
20 + 1 * sqrt(9)
24
20 + 1 + sqrt(9)
25
2 || (0! + 1 + sqrt (9))
explained:
Ok, this one needs explanation. The operation for "grouping" is known as concatenation and represented by ||. Basically this means push the digits together: 2 || 0 = 20. However, just like any operation, you can represent either side not by a number but by an equation on its own. So 0! + 1 + sqrt(9) = 5, meaning the above represents 2 || 5, qed.
26
2 || ((0! + 1) * sqrt(9))
27
(2 + 0 + 1) * 9
28
((2 + 0!) || 1) - sqrt(9)
29
(2 + (0 * 1)) || 9
30
20 + 1 + 9
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
edited Dec 31 '18 at 19:39
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
answered Dec 31 '18 at 18:40
flashstormflashstorm
8049
8049
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
add a comment |
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
edited for clarity
– flashstorm
Dec 31 '18 at 18:59
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
@flashstorm Um... $3ne 2+0^19$
– Frpzzd
Dec 31 '18 at 19:13
1
1
was missing an !
– flashstorm
Dec 31 '18 at 19:19
was missing an !
– flashstorm
Dec 31 '18 at 19:19
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
Ding! Fries are done :)
– flashstorm
Dec 31 '18 at 19:31
add a comment |
$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt9=-(2+0+1)!+9$$
$$4=2^0-1+sqrt9$$
$$5=20div (1+sqrt9)$$
$$6=(2cdot 0cdot 1+sqrt9)!$$
$$7=-2-0cdot 1+9$$
$$8=2^0cdot 1+sqrt9$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^0!+1+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^0+1+sqrt9$$
$$17=20-sqrt1cdot 9$$
$$18=20+1-sqrt9$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt1cdot 9$$
$$24=2^0-1+sqrt9!=20+1+sqrt9$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt9)!$$
$$27=(2+0+1)^sqrt9$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$
DONE!
edited Jan 1 at 3:26
Hugh
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
|
show 2 more comments
$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt9=-(2+0+1)!+9$$
$$4=2^0-1+sqrt9$$
$$5=20div (1+sqrt9)$$
$$6=(2cdot 0cdot 1+sqrt9)!$$
$$7=-2-0cdot 1+9$$
$$8=2^0cdot 1+sqrt9$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^0!+1+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^0+1+sqrt9$$
$$17=20-sqrt1cdot 9$$
$$18=20+1-sqrt9$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt1cdot 9$$
$$24=2^0-1+sqrt9!=20+1+sqrt9$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt9)!$$
$$27=(2+0+1)^sqrt9$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$
DONE!
edited Jan 1 at 3:26
Hugh
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
|
show 2 more comments
$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt9=-(2+0+1)!+9$$
$$4=2^0-1+sqrt9$$
$$5=20div (1+sqrt9)$$
$$6=(2cdot 0cdot 1+sqrt9)!$$
$$7=-2-0cdot 1+9$$
$$8=2^0cdot 1+sqrt9$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^0!+1+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^0+1+sqrt9$$
$$17=20-sqrt1cdot 9$$
$$18=20+1-sqrt9$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt1cdot 9$$
$$24=2^0-1+sqrt9!=20+1+sqrt9$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt9)!$$
$$27=(2+0+1)^sqrt9$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$
DONE!
edited Jan 1 at 3:26
Hugh
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
$$1=20-19$$
$$2=2+0cdot 19=20div(1+9)$$
$$3=2cdot 0cdot 1+sqrt9=-(2+0+1)!+9$$
$$4=2^0-1+sqrt9$$
$$5=20div (1+sqrt9)$$
$$6=(2cdot 0cdot 1+sqrt9)!$$
$$7=-2-0cdot 1+9$$
$$8=2^0cdot 1+sqrt9$$
$$9=2cdot 0cdot 1+9$$
$$10=2cdot 0+1+9=20-1-9$$
$$11=2+0cdot 1+9=20-1cdot 9=2^0+1+9$$
$$12=2+0+1+9=20+1-9$$
$$13=2+0!+1+9=2^0!+1+9$$
$$14=(2+0!)!-1+9$$
$$15=(2+0+1)!+9$$
$$16=2^0+1+sqrt9$$
$$17=20-sqrt1cdot 9$$
$$18=20+1-sqrt9$$
$$19=2cdot 0+19$$
$$20=2^0+19$$
$$21=20+1^9$$
$$22=2cdot (0!+1+9)$$
$$23=20+sqrt1cdot 9$$
$$24=2^0-1+sqrt9!=20+1+sqrt9$$
$$25=(2+0!)!+19$$
$$26=2+0+(1+sqrt9)!$$
$$27=(2+0+1)^sqrt9$$
$$28=20-1+9$$
$$29=20+1cdot 9=20cdot 1+9$$
$$30=20+1+9$$
DONE!
edited Jan 1 at 3:26
Hugh
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
edited Jan 1 at 3:26
Hugh
1,4781617
edited Jan 1 at 3:26
Hugh
1,4781617
edited Jan 1 at 3:26
Hugh
1,4781617
1,4781617
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
answered Dec 31 '18 at 19:21
FrpzzdFrpzzd
906121
906121
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
|
show 2 more comments
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
All finished now! :D
– Frpzzd
Dec 31 '18 at 19:48
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
Great :) But Spoiler-Tags would be nice ;)
– André
Dec 31 '18 at 21:23
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
@André Oh, sorry! I can't figure out how to get the mathjax to work inside of a spoiler tag. D:<
– Frpzzd
Dec 31 '18 at 21:26
1
1
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
@André Also: I've been trying to do 31, but it actually seems to be much harder than any of the previous ones! Looks like you picked just the right number to stop on! XD
– Frpzzd
Dec 31 '18 at 21:27
1
1
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
Your answer for 6 is incorrect; that expression comes out to 2. However, adding parentheses around part and another factorial will make it work.
– user1207177
Dec 31 '18 at 21:41
|
show 2 more comments
I wrote a program to determine all representable numbers between 1 and 1,000,000 following the rules, so this should be a comprehensive list.
0 through 30:
$$0 = 2times0timesleft(1+9right)$$
$$1 = 20-19$$
$$2 = 2+0timesleft(1+9right)$$
$$3 = 2+0+1^9$$
$$4 = 2-left(0+1-sqrt9right)$$
$$5 = frac201+sqrt9$$
$$6 = -2-left(0+1-9right)$$
$$7 = 2timesleft(0-1right)+9$$
$$8 = 2times0-left(1-9right)$$
$$9 = 2times0+1times9$$
$$10 = 2-left(0+1-9right)$$
$$11 = 2+0+1times9$$
$$12 = 2+0+1+9$$
$$13 = 20-left(1+left(sqrt9right)!right)$$
$$14 = 20-1timesleft(sqrt9right)!$$
$$15 = 20+1-left(sqrt9right)!$$
$$16 = 2timesleft(0-left(1-9right)right)$$
$$17 = -2+0+19$$
$$18 = 2timesleft(0+1times9right)$$
$$19 = 2times0+19$$
$$20 = 2timesleft(0+1+9right)$$
$$21 = 2+0+19$$
$$22 = 20-left(1-sqrt9right)$$
$$23 = 20+1timessqrt9$$
$$24 = 20+1+sqrt9$$
$$25 = 20-left(1-left(sqrt9right)!right)$$
$$26 = 20+1timesleft(sqrt9right)!$$
$$27 = left(2+0+1right)times9$$
$$28 = 20-left(1-9right)$$
$$29 = 20+1times9$$
$$30 = 20+1+9$$
31 through 100. Concatenation of expressions (denoted with ||) is only used when necessary. Interestingly enough, 31 is the smallest number that cannot be done without concatenation of expressions, and 75 is the smallest number that cannot be done in general.
$$31 = 2+left(left(0!+1right)||9right)$$
$$32 = sqrt2^0+1+9$$
$$33 = left(2+0!+1right)!+9$$
$$34 = left(2+0!right)||left(1+sqrt9right)$$
$$35 = 20+left(-1+left(sqrt9right)!right)!!$$
$$36 = 2timesleft(-0!+19right)$$
$$37 = left(2+0!right)||left(1+left(sqrt9right)!right)$$
$$38 = 2timesleft(0+19right)$$
$$39 = 20+19$$
$$40 = 20timesleft(-1+sqrt9right)$$
$$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt9right)!right)$$
$$42 = left(2+0!right)!timesleft(1+left(sqrt9right)!right)$$
$$43 = left(left(2+0!right)!right)!!+1-left(sqrt9right)!$$
$$44 = 20+left(1+sqrt9right)!$$
$$45 = left(left(2+0!right)!-1right)times9$$
$$46 = 2timesleft(-0!+left(1+sqrt9right)!right)$$
$$47 = 2times0-left(1-left(left(sqrt9right)!right)!!right)$$
$$48 = 2timesleft(0+left(1+sqrt9right)!right)$$
$$49 = 2-left(0+1-left(left(sqrt9right)!right)!!right)$$
$$50 = 2timesleft(0!+left(1+sqrt9right)!right)$$
$$51 = 2+0+1+left(left(sqrt9right)!right)!!$$
$$52 = 2+0!+1+left(left(sqrt9right)!right)!!$$
$$53 = left(2+0!right)!-left(1-left(left(sqrt9right)!right)!!right)$$
$$54 = left(2+0!right)!times1times9$$
$$55 = left(2+0!right)!+1+left(left(sqrt9right)!right)!!$$
$$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
$$57 = left(20-1right)timessqrt9$$
$$58 = left(left(2+0!right)!right)!!+1+9$$
$$59 = left(left(2+0!right)!-1right)||9$$
$$60 = 20times1timessqrt9$$
$$61 = left(2+0!right)!||left(1^9right)$$
$$62 = -2+left(0!+1right)^left(sqrt9right)!$$
$$63 = left(20+1right)timessqrt9$$
$$64 = 2^0+1timesleft(sqrt9right)!$$
$$65 = left(2+0!right)!||left(-1+left(sqrt9right)!right)$$
$$66 = 2+left(0!+1right)^left(sqrt9right)!$$
$$67 = frac201sqrt9$$
$$68 = 20+1timesleft(left(sqrt9right)!right)!!$$
$$69 = -2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$70 = left(left(2+0!right)!||1right)+9$$
$$71 = sqrt2^0+left(1+left(sqrt9right)!right)!$$
$$72 = left(2+0!right)timesleft(1+sqrt9right)!$$
$$73 = 2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$74 = 2timesleft(left(-0!||1right)+left(left(sqrt9right)!right)!!right)$$
$$76 = left(left(2+0!right)!+1right)||left(sqrt9right)!$$
$$78 = left(2+0!right)!timesleft(1||sqrt9right)$$
$$79 = left(left(2+0!right)!+1right)||9$$
$$80 = 20timesleft(1+sqrt9right)$$
$$81 = left(2+0!right)^1+sqrt9$$
$$83 = left(2+0!+1right)!!||sqrt9$$
$$84 = left(2||0!right)timesleft(1+sqrt9right)$$
$$85 = -20+left(1+left(sqrt9right)!right)!!$$
$$86 = left(2+0!+1right)!!||left(sqrt9right)!$$
$$89 = left(2+0!+1right)!!||9$$
$$90 = fracleft(left(2+0!right)!right)!-1+9$$
$$92 = 2timesleft(-0!-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$93 = left(left(2+0!right)||1right)timessqrt9$$
$$94 = 2timesleft(0-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt9right)!right)!!right)$$
$$96 = 2timesleft(0+1timesleft(left(sqrt9right)!right)!!right)$$
$$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt9right)!right)!!$$
$$98 = 2timesleft(0+1+left(left(sqrt9right)!right)!!right)$$
$$99 = left(left(2+0!right)!+1right)!!-left(sqrt9right)!$$
$$100 = 20timesleft(-1+left(sqrt9right)!right)$$
The rest of the numbers from 101 to 1,000,000 can be found here.
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
add a comment |
I wrote a program to determine all representable numbers between 1 and 1,000,000 following the rules, so this should be a comprehensive list.
0 through 30:
$$0 = 2times0timesleft(1+9right)$$
$$1 = 20-19$$
$$2 = 2+0timesleft(1+9right)$$
$$3 = 2+0+1^9$$
$$4 = 2-left(0+1-sqrt9right)$$
$$5 = frac201+sqrt9$$
$$6 = -2-left(0+1-9right)$$
$$7 = 2timesleft(0-1right)+9$$
$$8 = 2times0-left(1-9right)$$
$$9 = 2times0+1times9$$
$$10 = 2-left(0+1-9right)$$
$$11 = 2+0+1times9$$
$$12 = 2+0+1+9$$
$$13 = 20-left(1+left(sqrt9right)!right)$$
$$14 = 20-1timesleft(sqrt9right)!$$
$$15 = 20+1-left(sqrt9right)!$$
$$16 = 2timesleft(0-left(1-9right)right)$$
$$17 = -2+0+19$$
$$18 = 2timesleft(0+1times9right)$$
$$19 = 2times0+19$$
$$20 = 2timesleft(0+1+9right)$$
$$21 = 2+0+19$$
$$22 = 20-left(1-sqrt9right)$$
$$23 = 20+1timessqrt9$$
$$24 = 20+1+sqrt9$$
$$25 = 20-left(1-left(sqrt9right)!right)$$
$$26 = 20+1timesleft(sqrt9right)!$$
$$27 = left(2+0+1right)times9$$
$$28 = 20-left(1-9right)$$
$$29 = 20+1times9$$
$$30 = 20+1+9$$
31 through 100. Concatenation of expressions (denoted with ||) is only used when necessary. Interestingly enough, 31 is the smallest number that cannot be done without concatenation of expressions, and 75 is the smallest number that cannot be done in general.
$$31 = 2+left(left(0!+1right)||9right)$$
$$32 = sqrt2^0+1+9$$
$$33 = left(2+0!+1right)!+9$$
$$34 = left(2+0!right)||left(1+sqrt9right)$$
$$35 = 20+left(-1+left(sqrt9right)!right)!!$$
$$36 = 2timesleft(-0!+19right)$$
$$37 = left(2+0!right)||left(1+left(sqrt9right)!right)$$
$$38 = 2timesleft(0+19right)$$
$$39 = 20+19$$
$$40 = 20timesleft(-1+sqrt9right)$$
$$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt9right)!right)$$
$$42 = left(2+0!right)!timesleft(1+left(sqrt9right)!right)$$
$$43 = left(left(2+0!right)!right)!!+1-left(sqrt9right)!$$
$$44 = 20+left(1+sqrt9right)!$$
$$45 = left(left(2+0!right)!-1right)times9$$
$$46 = 2timesleft(-0!+left(1+sqrt9right)!right)$$
$$47 = 2times0-left(1-left(left(sqrt9right)!right)!!right)$$
$$48 = 2timesleft(0+left(1+sqrt9right)!right)$$
$$49 = 2-left(0+1-left(left(sqrt9right)!right)!!right)$$
$$50 = 2timesleft(0!+left(1+sqrt9right)!right)$$
$$51 = 2+0+1+left(left(sqrt9right)!right)!!$$
$$52 = 2+0!+1+left(left(sqrt9right)!right)!!$$
$$53 = left(2+0!right)!-left(1-left(left(sqrt9right)!right)!!right)$$
$$54 = left(2+0!right)!times1times9$$
$$55 = left(2+0!right)!+1+left(left(sqrt9right)!right)!!$$
$$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
$$57 = left(20-1right)timessqrt9$$
$$58 = left(left(2+0!right)!right)!!+1+9$$
$$59 = left(left(2+0!right)!-1right)||9$$
$$60 = 20times1timessqrt9$$
$$61 = left(2+0!right)!||left(1^9right)$$
$$62 = -2+left(0!+1right)^left(sqrt9right)!$$
$$63 = left(20+1right)timessqrt9$$
$$64 = 2^0+1timesleft(sqrt9right)!$$
$$65 = left(2+0!right)!||left(-1+left(sqrt9right)!right)$$
$$66 = 2+left(0!+1right)^left(sqrt9right)!$$
$$67 = frac201sqrt9$$
$$68 = 20+1timesleft(left(sqrt9right)!right)!!$$
$$69 = -2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$70 = left(left(2+0!right)!||1right)+9$$
$$71 = sqrt2^0+left(1+left(sqrt9right)!right)!$$
$$72 = left(2+0!right)timesleft(1+sqrt9right)!$$
$$73 = 2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$74 = 2timesleft(left(-0!||1right)+left(left(sqrt9right)!right)!!right)$$
$$76 = left(left(2+0!right)!+1right)||left(sqrt9right)!$$
$$78 = left(2+0!right)!timesleft(1||sqrt9right)$$
$$79 = left(left(2+0!right)!+1right)||9$$
$$80 = 20timesleft(1+sqrt9right)$$
$$81 = left(2+0!right)^1+sqrt9$$
$$83 = left(2+0!+1right)!!||sqrt9$$
$$84 = left(2||0!right)timesleft(1+sqrt9right)$$
$$85 = -20+left(1+left(sqrt9right)!right)!!$$
$$86 = left(2+0!+1right)!!||left(sqrt9right)!$$
$$89 = left(2+0!+1right)!!||9$$
$$90 = fracleft(left(2+0!right)!right)!-1+9$$
$$92 = 2timesleft(-0!-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$93 = left(left(2+0!right)||1right)timessqrt9$$
$$94 = 2timesleft(0-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt9right)!right)!!right)$$
$$96 = 2timesleft(0+1timesleft(left(sqrt9right)!right)!!right)$$
$$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt9right)!right)!!$$
$$98 = 2timesleft(0+1+left(left(sqrt9right)!right)!!right)$$
$$99 = left(left(2+0!right)!+1right)!!-left(sqrt9right)!$$
$$100 = 20timesleft(-1+left(sqrt9right)!right)$$
The rest of the numbers from 101 to 1,000,000 can be found here.
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
add a comment |
I wrote a program to determine all representable numbers between 1 and 1,000,000 following the rules, so this should be a comprehensive list.
0 through 30:
$$0 = 2times0timesleft(1+9right)$$
$$1 = 20-19$$
$$2 = 2+0timesleft(1+9right)$$
$$3 = 2+0+1^9$$
$$4 = 2-left(0+1-sqrt9right)$$
$$5 = frac201+sqrt9$$
$$6 = -2-left(0+1-9right)$$
$$7 = 2timesleft(0-1right)+9$$
$$8 = 2times0-left(1-9right)$$
$$9 = 2times0+1times9$$
$$10 = 2-left(0+1-9right)$$
$$11 = 2+0+1times9$$
$$12 = 2+0+1+9$$
$$13 = 20-left(1+left(sqrt9right)!right)$$
$$14 = 20-1timesleft(sqrt9right)!$$
$$15 = 20+1-left(sqrt9right)!$$
$$16 = 2timesleft(0-left(1-9right)right)$$
$$17 = -2+0+19$$
$$18 = 2timesleft(0+1times9right)$$
$$19 = 2times0+19$$
$$20 = 2timesleft(0+1+9right)$$
$$21 = 2+0+19$$
$$22 = 20-left(1-sqrt9right)$$
$$23 = 20+1timessqrt9$$
$$24 = 20+1+sqrt9$$
$$25 = 20-left(1-left(sqrt9right)!right)$$
$$26 = 20+1timesleft(sqrt9right)!$$
$$27 = left(2+0+1right)times9$$
$$28 = 20-left(1-9right)$$
$$29 = 20+1times9$$
$$30 = 20+1+9$$
31 through 100. Concatenation of expressions (denoted with ||) is only used when necessary. Interestingly enough, 31 is the smallest number that cannot be done without concatenation of expressions, and 75 is the smallest number that cannot be done in general.
$$31 = 2+left(left(0!+1right)||9right)$$
$$32 = sqrt2^0+1+9$$
$$33 = left(2+0!+1right)!+9$$
$$34 = left(2+0!right)||left(1+sqrt9right)$$
$$35 = 20+left(-1+left(sqrt9right)!right)!!$$
$$36 = 2timesleft(-0!+19right)$$
$$37 = left(2+0!right)||left(1+left(sqrt9right)!right)$$
$$38 = 2timesleft(0+19right)$$
$$39 = 20+19$$
$$40 = 20timesleft(-1+sqrt9right)$$
$$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt9right)!right)$$
$$42 = left(2+0!right)!timesleft(1+left(sqrt9right)!right)$$
$$43 = left(left(2+0!right)!right)!!+1-left(sqrt9right)!$$
$$44 = 20+left(1+sqrt9right)!$$
$$45 = left(left(2+0!right)!-1right)times9$$
$$46 = 2timesleft(-0!+left(1+sqrt9right)!right)$$
$$47 = 2times0-left(1-left(left(sqrt9right)!right)!!right)$$
$$48 = 2timesleft(0+left(1+sqrt9right)!right)$$
$$49 = 2-left(0+1-left(left(sqrt9right)!right)!!right)$$
$$50 = 2timesleft(0!+left(1+sqrt9right)!right)$$
$$51 = 2+0+1+left(left(sqrt9right)!right)!!$$
$$52 = 2+0!+1+left(left(sqrt9right)!right)!!$$
$$53 = left(2+0!right)!-left(1-left(left(sqrt9right)!right)!!right)$$
$$54 = left(2+0!right)!times1times9$$
$$55 = left(2+0!right)!+1+left(left(sqrt9right)!right)!!$$
$$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
$$57 = left(20-1right)timessqrt9$$
$$58 = left(left(2+0!right)!right)!!+1+9$$
$$59 = left(left(2+0!right)!-1right)||9$$
$$60 = 20times1timessqrt9$$
$$61 = left(2+0!right)!||left(1^9right)$$
$$62 = -2+left(0!+1right)^left(sqrt9right)!$$
$$63 = left(20+1right)timessqrt9$$
$$64 = 2^0+1timesleft(sqrt9right)!$$
$$65 = left(2+0!right)!||left(-1+left(sqrt9right)!right)$$
$$66 = 2+left(0!+1right)^left(sqrt9right)!$$
$$67 = frac201sqrt9$$
$$68 = 20+1timesleft(left(sqrt9right)!right)!!$$
$$69 = -2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$70 = left(left(2+0!right)!||1right)+9$$
$$71 = sqrt2^0+left(1+left(sqrt9right)!right)!$$
$$72 = left(2+0!right)timesleft(1+sqrt9right)!$$
$$73 = 2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$74 = 2timesleft(left(-0!||1right)+left(left(sqrt9right)!right)!!right)$$
$$76 = left(left(2+0!right)!+1right)||left(sqrt9right)!$$
$$78 = left(2+0!right)!timesleft(1||sqrt9right)$$
$$79 = left(left(2+0!right)!+1right)||9$$
$$80 = 20timesleft(1+sqrt9right)$$
$$81 = left(2+0!right)^1+sqrt9$$
$$83 = left(2+0!+1right)!!||sqrt9$$
$$84 = left(2||0!right)timesleft(1+sqrt9right)$$
$$85 = -20+left(1+left(sqrt9right)!right)!!$$
$$86 = left(2+0!+1right)!!||left(sqrt9right)!$$
$$89 = left(2+0!+1right)!!||9$$
$$90 = fracleft(left(2+0!right)!right)!-1+9$$
$$92 = 2timesleft(-0!-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$93 = left(left(2+0!right)||1right)timessqrt9$$
$$94 = 2timesleft(0-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt9right)!right)!!right)$$
$$96 = 2timesleft(0+1timesleft(left(sqrt9right)!right)!!right)$$
$$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt9right)!right)!!$$
$$98 = 2timesleft(0+1+left(left(sqrt9right)!right)!!right)$$
$$99 = left(left(2+0!right)!+1right)!!-left(sqrt9right)!$$
$$100 = 20timesleft(-1+left(sqrt9right)!right)$$
The rest of the numbers from 101 to 1,000,000 can be found here.
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
I wrote a program to determine all representable numbers between 1 and 1,000,000 following the rules, so this should be a comprehensive list.
0 through 30:
$$0 = 2times0timesleft(1+9right)$$
$$1 = 20-19$$
$$2 = 2+0timesleft(1+9right)$$
$$3 = 2+0+1^9$$
$$4 = 2-left(0+1-sqrt9right)$$
$$5 = frac201+sqrt9$$
$$6 = -2-left(0+1-9right)$$
$$7 = 2timesleft(0-1right)+9$$
$$8 = 2times0-left(1-9right)$$
$$9 = 2times0+1times9$$
$$10 = 2-left(0+1-9right)$$
$$11 = 2+0+1times9$$
$$12 = 2+0+1+9$$
$$13 = 20-left(1+left(sqrt9right)!right)$$
$$14 = 20-1timesleft(sqrt9right)!$$
$$15 = 20+1-left(sqrt9right)!$$
$$16 = 2timesleft(0-left(1-9right)right)$$
$$17 = -2+0+19$$
$$18 = 2timesleft(0+1times9right)$$
$$19 = 2times0+19$$
$$20 = 2timesleft(0+1+9right)$$
$$21 = 2+0+19$$
$$22 = 20-left(1-sqrt9right)$$
$$23 = 20+1timessqrt9$$
$$24 = 20+1+sqrt9$$
$$25 = 20-left(1-left(sqrt9right)!right)$$
$$26 = 20+1timesleft(sqrt9right)!$$
$$27 = left(2+0+1right)times9$$
$$28 = 20-left(1-9right)$$
$$29 = 20+1times9$$
$$30 = 20+1+9$$
31 through 100. Concatenation of expressions (denoted with ||) is only used when necessary. Interestingly enough, 31 is the smallest number that cannot be done without concatenation of expressions, and 75 is the smallest number that cannot be done in general.
$$31 = 2+left(left(0!+1right)||9right)$$
$$32 = sqrt2^0+1+9$$
$$33 = left(2+0!+1right)!+9$$
$$34 = left(2+0!right)||left(1+sqrt9right)$$
$$35 = 20+left(-1+left(sqrt9right)!right)!!$$
$$36 = 2timesleft(-0!+19right)$$
$$37 = left(2+0!right)||left(1+left(sqrt9right)!right)$$
$$38 = 2timesleft(0+19right)$$
$$39 = 20+19$$
$$40 = 20timesleft(-1+sqrt9right)$$
$$41 = left(left(2+0!right)!right)!!-left(1+left(sqrt9right)!right)$$
$$42 = left(2+0!right)!timesleft(1+left(sqrt9right)!right)$$
$$43 = left(left(2+0!right)!right)!!+1-left(sqrt9right)!$$
$$44 = 20+left(1+sqrt9right)!$$
$$45 = left(left(2+0!right)!-1right)times9$$
$$46 = 2timesleft(-0!+left(1+sqrt9right)!right)$$
$$47 = 2times0-left(1-left(left(sqrt9right)!right)!!right)$$
$$48 = 2timesleft(0+left(1+sqrt9right)!right)$$
$$49 = 2-left(0+1-left(left(sqrt9right)!right)!!right)$$
$$50 = 2timesleft(0!+left(1+sqrt9right)!right)$$
$$51 = 2+0+1+left(left(sqrt9right)!right)!!$$
$$52 = 2+0!+1+left(left(sqrt9right)!right)!!$$
$$53 = left(2+0!right)!-left(1-left(left(sqrt9right)!right)!!right)$$
$$54 = left(2+0!right)!times1times9$$
$$55 = left(2+0!right)!+1+left(left(sqrt9right)!right)!!$$
$$56 = left(left(2+0!right)!right)!!-left(1-9right)$$
$$57 = left(20-1right)timessqrt9$$
$$58 = left(left(2+0!right)!right)!!+1+9$$
$$59 = left(left(2+0!right)!-1right)||9$$
$$60 = 20times1timessqrt9$$
$$61 = left(2+0!right)!||left(1^9right)$$
$$62 = -2+left(0!+1right)^left(sqrt9right)!$$
$$63 = left(20+1right)timessqrt9$$
$$64 = 2^0+1timesleft(sqrt9right)!$$
$$65 = left(2+0!right)!||left(-1+left(sqrt9right)!right)$$
$$66 = 2+left(0!+1right)^left(sqrt9right)!$$
$$67 = frac201sqrt9$$
$$68 = 20+1timesleft(left(sqrt9right)!right)!!$$
$$69 = -2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$70 = left(left(2+0!right)!||1right)+9$$
$$71 = sqrt2^0+left(1+left(sqrt9right)!right)!$$
$$72 = left(2+0!right)timesleft(1+sqrt9right)!$$
$$73 = 2+sqrt0!+left(1+left(sqrt9right)!right)!$$
$$74 = 2timesleft(left(-0!||1right)+left(left(sqrt9right)!right)!!right)$$
$$76 = left(left(2+0!right)!+1right)||left(sqrt9right)!$$
$$78 = left(2+0!right)!timesleft(1||sqrt9right)$$
$$79 = left(left(2+0!right)!+1right)||9$$
$$80 = 20timesleft(1+sqrt9right)$$
$$81 = left(2+0!right)^1+sqrt9$$
$$83 = left(2+0!+1right)!!||sqrt9$$
$$84 = left(2||0!right)timesleft(1+sqrt9right)$$
$$85 = -20+left(1+left(sqrt9right)!right)!!$$
$$86 = left(2+0!+1right)!!||left(sqrt9right)!$$
$$89 = left(2+0!+1right)!!||9$$
$$90 = fracleft(left(2+0!right)!right)!-1+9$$
$$92 = 2timesleft(-0!-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$93 = left(left(2+0!right)||1right)timessqrt9$$
$$94 = 2timesleft(0-left(1-left(left(sqrt9right)!right)!!right)right)$$
$$95 = left(left(2+0!right)!right)!!-left(1-left(left(sqrt9right)!right)!!right)$$
$$96 = 2timesleft(0+1timesleft(left(sqrt9right)!right)!!right)$$
$$97 = left(left(2+0!right)!right)!!+1+left(left(sqrt9right)!right)!!$$
$$98 = 2timesleft(0+1+left(left(sqrt9right)!right)!!right)$$
$$99 = left(left(2+0!right)!+1right)!!-left(sqrt9right)!$$
$$100 = 20timesleft(-1+left(sqrt9right)!right)$$
The rest of the numbers from 101 to 1,000,000 can be found here.
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
edited Jan 1 at 6:21
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
answered Jan 1 at 5:50
The TurtleThe Turtle
1414
1414
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
add a comment |
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
Nice. I'm curious whether your program finds anything for 28 other than $20 - (1-9)$ and $20 - 1 +9$, etc.
– tilper
Jan 2 at 13:37
add a comment |
1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.
8:
$8 = sqrt(2^0!+1!!)!! / (sqrt9)!$
16:
$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$
17:
$17 = (2 + 0! + 1)!! + 9$
18:
$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^0+1 cdot 9$
19:
$19 = 2 cdot 0 + 19$
20:
$20 = 2^0 + 19 = 20! / 19!$
21:
$21 = 20 + 1^9$
22:
$22 = 20 - 1 + sqrt9$
23:
$23 = 20 + 1 cdot sqrt9$
24:
$24 = 2^0! + 1 cdot (sqrt9)! = 2^0! + 1!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$
25:
$25 = (2 + 0!)! + 19$
26:
$26 = 20 + 1 cdot (sqrt9)!$
27:
$27 = 2^0! + 1! + sqrt9$
28: Can't get one different from what I've already seen in other answers. Will maybe try later.
29:
$29 = 20 + 1 cdot 9$
30:
$30 = 2^0! + 1! + (sqrt9)!$
I know we're supposed to stop at 30, but I accidentally found this fun one:
32:
$32 = sqrt20!! / (1 + 9)!$
answered Dec 31 '18 at 19:13
tilpertilper
898514
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
add a comment |
1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.
8:
$8 = sqrt(2^0!+1!!)!! / (sqrt9)!$
16:
$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$
17:
$17 = (2 + 0! + 1)!! + 9$
18:
$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^0+1 cdot 9$
19:
$19 = 2 cdot 0 + 19$
20:
$20 = 2^0 + 19 = 20! / 19!$
21:
$21 = 20 + 1^9$
22:
$22 = 20 - 1 + sqrt9$
23:
$23 = 20 + 1 cdot sqrt9$
24:
$24 = 2^0! + 1 cdot (sqrt9)! = 2^0! + 1!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$
25:
$25 = (2 + 0!)! + 19$
26:
$26 = 20 + 1 cdot (sqrt9)!$
27:
$27 = 2^0! + 1! + sqrt9$
28: Can't get one different from what I've already seen in other answers. Will maybe try later.
29:
$29 = 20 + 1 cdot 9$
30:
$30 = 2^0! + 1! + (sqrt9)!$
I know we're supposed to stop at 30, but I accidentally found this fun one:
32:
$32 = sqrt20!! / (1 + 9)!$
answered Dec 31 '18 at 19:13
tilpertilper
898514
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
add a comment |
1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.
8:
$8 = sqrt(2^0!+1!!)!! / (sqrt9)!$
16:
$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$
17:
$17 = (2 + 0! + 1)!! + 9$
18:
$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^0+1 cdot 9$
19:
$19 = 2 cdot 0 + 19$
20:
$20 = 2^0 + 19 = 20! / 19!$
21:
$21 = 20 + 1^9$
22:
$22 = 20 - 1 + sqrt9$
23:
$23 = 20 + 1 cdot sqrt9$
24:
$24 = 2^0! + 1 cdot (sqrt9)! = 2^0! + 1!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$
25:
$25 = (2 + 0!)! + 19$
26:
$26 = 20 + 1 cdot (sqrt9)!$
27:
$27 = 2^0! + 1! + sqrt9$
28: Can't get one different from what I've already seen in other answers. Will maybe try later.
29:
$29 = 20 + 1 cdot 9$
30:
$30 = 2^0! + 1! + (sqrt9)!$
I know we're supposed to stop at 30, but I accidentally found this fun one:
32:
$32 = sqrt20!! / (1 + 9)!$
answered Dec 31 '18 at 19:13
tilpertilper
898514
1-16 were done as of the time I started this, so I wanted to focus only on 17-30, using answers different from ones that I've already seen. Others will be included if I find solutions that I like.
8:
$8 = sqrt(2^0!+1!!)!! / (sqrt9)!$
16:
$16 = ((2 + 0 + 1)!)!! / sqrt9 = 20 - 1 - sqrt9$
17:
$17 = (2 + 0! + 1)!! + 9$
18:
$18 = (2^0 + 1) cdot 9 = 2 cdot (0+1)cdot 9 = 2^0+1 cdot 9$
19:
$19 = 2 cdot 0 + 19$
20:
$20 = 2^0 + 19 = 20! / 19!$
21:
$21 = 20 + 1^9$
22:
$22 = 20 - 1 + sqrt9$
23:
$23 = 20 + 1 cdot sqrt9$
24:
$24 = 2^0! + 1 cdot (sqrt9)! = 2^0! + 1!! cdot sqrt9 = (2+0!+1)!! cdot sqrt9 = ((2 + 0!)! - 1)!! + 9$
25:
$25 = (2 + 0!)! + 19$
26:
$26 = 20 + 1 cdot (sqrt9)!$
27:
$27 = 2^0! + 1! + sqrt9$
28: Can't get one different from what I've already seen in other answers. Will maybe try later.
29:
$29 = 20 + 1 cdot 9$
30:
$30 = 2^0! + 1! + (sqrt9)!$
I know we're supposed to stop at 30, but I accidentally found this fun one:
32:
$32 = sqrt20!! / (1 + 9)!$
answered Dec 31 '18 at 19:13
tilpertilper
898514
edited Dec 31 '18 at 20:48
answered Dec 31 '18 at 19:13
tilpertilper
898514
answered Dec 31 '18 at 19:13
tilpertilper
898514
answered Dec 31 '18 at 19:13
tilpertilper
898514
898514
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
add a comment |
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
Well, there's a way to construct the number 31 satisfying the rules above (at least if we allow the subfactorial) :)
– André
Jan 3 at 1:43
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
@André do you mean without concatenation?
– tilper
Jan 3 at 1:44
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
Yes, without concatenation! Hint: Try using the "!!! faculty".
– André
Jan 3 at 2:14
add a comment |
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Dec 31 '18 at 22:46
Of course I'll do.
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Jan 1 at 2:40