# Make numbers 1-30 using 2, 0, 1, 9

This is very similar to the 2, 0, 1, 8 problem. Just try to make all numbers 1-30 using the digits 2, 0, 1, 9.

Rules:

• Use all four digits exactly once
• Allowed operations: +, -, x, ÷, ! (factorial), exponentiation, square root
• Parentheses and concatenation are allowed (e.g. $20 + (1 * 9)$)
• Squaring uses the digit 2 so expressions using multiple twos, like 22 or $12+92$, are not allowed
• Keep the order "2 0 1 9" in all expressions
• The modulus operator is not allowed
• Rounding is not allowed (e.g. $201/9=22$)
• but.. but... how could you? we are not in 2019 yet! – Kepotx Apr 17 '18 at 17:07
• Is "0 1 9 2" still considered in to be in order? – infinitezero Apr 17 '18 at 18:55
• @infinitezero no – jkl3699 Apr 17 '18 at 19:19
• If squaring uses the two, why would square root not use both the one and the two? – T James Apr 17 '18 at 22:17
• Just for fun, $(-2^{-(0!)})!^{-1+\sqrt9}=\pi$, if you use the gamma function for the factorial – LegionMammal978 Apr 17 '18 at 23:24

$30 = 20 + 1 + 9$
$29 = 20^1 + 9$
$28 = 20 - 1 + 9$
$27 =( 2 + 0 + 1)* 9$
$26 = 2 + 0 + (1 + \sqrt{9})!$
$25 = 2^0 + (1 + \sqrt{9})!$
$24 = (2 + 0 - 1 + \sqrt{9})!$
$23 = -2^0 + (1 + \sqrt{9})!$
$22 = -2 + 0 + (1 + \sqrt{9})!$
$21 = 2 + 0 + 19$
$20 = (2 + 0)(1 + 9)$
$19 = 20 - 1^9$
$18 = 2^{(0+1)} * 9$
$17 = -2 + 0 + 19$
$16 = 20 - (1 + \sqrt{9})$
$15 = (2 + 0 + 1)! + 9$
$14 = 20^1 - (\sqrt{9})!$
$13 = 20 - 1 - (\sqrt{9})!$
$12 = 2 + 0 + 1 + 9$
$11 = 2^{(0 + 1)} + 9$
$10 = 2 + 0 - (1 - 9)$
$9 = (2 + 0 + 1)! + \sqrt{9}$
$8 = (2 + 0)(1 + \sqrt{9})$
$7 = -2^{(0 + 1)} + 9$
$6 = 2 + 0 + 1 + \sqrt{9}$
$5 = 2^0 + 1 + \sqrt{9}$
$4 = 2 + 0 - 1 + \sqrt{9}$
$3 = (2 + 0 - 1) * \sqrt{9}$
$2 = (2 + 0) * 1^9$
$1 = (2 - 0 - 1)^9$

$1 = 20 - 19$
$2 = 2 + 0 * 1 * 9$
$3 = 2 + 0 + 1^9$
$4 = 2 + 0! + 1^9$
$5 = (2 + 0!)! - 1^9$
$6 = (2 + 0 + 1^9)!$
$7 = (2 + 0!)! + 1^9$
$8 = 2 * 0 - 1 + 9$
$9 = 2 * 0 * 1 + 9$
$10 = 2 * 0 + 1 + 9$
$11 = 2 + 0 * 1 +9$
$12 = 2 * 0! + 1 + 9$
$13 = 2 * (0! + 1) + 9$
$14 = (2+0!)!-1+9$
$15 = (2+0!)!+1*9$
$16 = (2+0!)!+1+9$
$17 = 20-\sqrt{1*9}$
$18 = 2 * 0! * 1 *9$
$19 = 20 - 1^9$
$20 = 20 * 1^9$
$21 = 20 + 1^9$
$22 = 20 - 1 + \sqrt{9}$
$23 = 20 + \sqrt{1 * 9}$
$24 = (2 + 0! + 1^9)!$
$25 = 20 -1 + \sqrt{9}!$
$26 = 20 + \sqrt{1 * 9}!$
$27 = (2 + 0 + 1) * 9$
$28 = 20 - 1 + 9$
$29 = 20^1 + 9$
$30 = 20 + 1 + 9$

PARTIAL:

19:

$2*0+19$

21:

$2+0+19$

29:

$20+1*9$

30:

$20+1+9$

more incoming

00

$2 * 0 * 1 * 9$

01

$2 * 0 + 1^9$

02

$2 + 0 * 1 * 9$

03

$2 + 0 + 1 ^ 9$

04

$2 * 0 + 1 + \sqrt{9}$

05

$2 + 0 * 1 + \sqrt{9}$

06

$2 + 0 + 1 + \sqrt{9}$

07

$(20 + 1) / \sqrt{9}$

08

$2 * 0 + ( -1 + 9)$

09

$2 * 0 * 1 + 9$

10

$2 * 0 + 1 + 9$

11

$2 + 0 * 1 + 9$

12

$2 + 0 + 1 + 9$

13

$20 - 1 - \sqrt{9}!$

14

2$0 - 1 * \sqrt{9}!$

15

$20 + 1 - \sqrt{9}!$

16

$20 - 1 - \sqrt{9}$

17

$-2 + 0 + 19$

18

$-(2^0) + 19$

19

$2 * 0 + 19$

20

$2 ^ 0 + 19$

21

$2 + 0 + 19$

22

$20 - 1 + \sqrt{9}$

23

$20 + 1 * \sqrt{9}$

24

$20 + 1 + \sqrt{9}$

25

$20 - 1 + \sqrt{9}!$

26

$20 + 1 * \sqrt{9}!$

27

$20 + 1 + \sqrt{9}!$

28

$20 - 1 + 9$

29

$20 + 1 * 9$

30

$20 + 1 + 9$

1. 20 - 19 = 1
OR
SQRT(20 - 19) = 1
OR
(20 - 19)! = 1

2. 2 + 0 * 19 = 2
OR
20 / (1 + 9) = 2

3. 2 + 0 + 1^9 = 3

4. 2 * 0 + 1 + SQRT(9) = 4

5. 2 + 0 * 1 + SQRT(9) = 5

6. - 2 + 0 - 1 + 9 = 6
OR
2 + 0 + 1 + SQRT(9) = 6

7. - 2 + 0 * 1 + 9 = 7
OR (20 + 1) / SQRT(9) = 7

8. 2 * 0 - 1 + 9 = 8

9. 2 * 0 * 1 + 9 = 9
OR
2^0 - 1 + 9 = 9

10. 20 - 1 - 9 = 10
OR
20 - (1 + 9) = 10
OR
2 * 0 + 1 + 9 = 10

11. 2 + 0 * 1 + 9 = 11
OR
20^1 - 9 = 11
OR
2 * 01 + 9 = 11
OR
20 * 1 - 9 = 11

12. 20 + 1 - 9 = 12
OR
2 + 0 + 1 + 9 = 12
OR
2 - 0 + 1 + 9 = 12

13. 20 - 1 - (SQRT(9))! = 13

14. 20 - 1 * (SQRT(9))! = 14

15. 20 + 1 - (SQRT(9))! = 15

16. 20 - 1 - SQRT(9) = 16
OR
20 - (1 + SQRT(9)) = 16

17. 20 - 1 * SQRT(9) = 17

18. 20 + 1 - SQRT(9) = 18
OR
(2 + 0) * 1 * 9 = 18
OR
(2 + 0) / (1 / 9) = 18

19. 2 * 0 + 19 = 19

20. 2^0 + 19 = 20

21. 2 + 0 + 19 = 21

22. 20 - 1 + SQRT(9) = 22

23. 20 * 1 + SQRT(9) = 23

24. 20 + 1 + SQRT(9) = 24

25. 20 - 1 + (SQRT(9))! = 25

26. 20^1 + (SQRT(9))! = 26

27. 20 + 1 + (SQRT(9))! = 27

28. 20 - 1 + 9 = 28

29. 20 * 1 + 9 = 29

30. 20 + 1 + 9 = 30

• I really like how you gave "or" options. That said, you missed a few more that make 10. – sirjonsnow Apr 18 '18 at 14:11
• The ORs were as I found them, and not intended to be a complete solution. I have added a few for 10. – NetJohn Apr 18 '18 at 15:17

Here they are, as simple and neat as I could make them!

$1-6:$

$$\small\begin{array}{c|c}1&2&3&4&5&6\\\hline2\cdot0+1^9&2+0\cdot1\cdot9&2\cdot0\cdot1+\sqrt9&2+0!+1^9&-2-0!-1+9&-2+0-1+9\end{array}$$

$7-12:$

$$\small\begin{array}{c|c}7&8&9&10&11&12\\\hline-2+0\cdot1+9&-2+0+1+9&2\cdot0\cdot1+9&2+0-1+9&2+0\cdot1+9&2+0+1+9\end{array}$$

$13-18:$

$$\small\begin{array}{c|c}13&14&15&16&17&18\\\hline2+0!+1+9&(2+0!)!-1+9&20+1-\sqrt9!&(2+0!)!+1+9&-2+0+19&2\cdot(0+1)\cdot9\end{array}$$

$19-25:$

$$\small\begin{array}{c|c}19&20&21&22&23&24&25\\\hline2\cdot0+19&2^0+19&20+1^9&2+0!+19&20\cdot1+\sqrt9&20+1+\sqrt9&(2+0!)!+19\end{array}$$

$26-30:$

$$\small\begin{array}{c|c}26&27&28&29&30\\\hline20+1\cdot\sqrt9!&20+1+\sqrt9!&20-1+9&20\cdot1+9&20+1+9\end{array}$$

• @Warlord099 Sorted! – TheSimpliFire Apr 17 '18 at 19:48

Full solution

$20-19 = 1$
$2 + 0 * 19 = 2$
$2 * 0 * 1 + \sqrt{9} = 3$
$2 * 0 + 1 + \sqrt{9} = 4$
$2 + 0 * 1 + \sqrt{9} = 5$
$-2 + 0 - 1 + 9 = 6$
$-2 + 0 * 1 + 9 = 7$
$2 * 0 - 1 + 9 = 8$
$2 * 0 * 1 + 9 = 9$
$20 - 1 - 9 = 10$
$20^1 - 9 = 11$
$20 + 1 - 9 = 12$
$20 - 1 - (\sqrt{9})! = 13$
$2 * (0 + 1 + (\sqrt{9})! = 14$
$(2 + 0 + 1)! + 9 = 15$
$2 * (0 - 1 + 9) = 16$
$-2 + 0 + 19 = 17$
$-2^0 + 19 = 18$
$2 * 0 + 19 = 19$
$20! / 19! = 20$
$2 + 0 + 19 = 21$
$20 - 1 + \sqrt{9} = 22$
$20 * 1 + \sqrt{9} = 23$
$20 + 1 + \sqrt{9} = 24$
$20 - 1 + (\sqrt{9})! = 25$
$20 + 1 * (\sqrt{9})! = 26$
$(2 + 0 + 1) * 9 = 27$
$20 - 1 + 9 = 28$
$20 ^1 + 9 = 29$
$20 + 1 + 9 = 30$

1

$20-19$

2

$2+(0*19)$

3

$(2*0*1)+√9$

4

$(2*0)+√(9)+1$

5

$2+(0*1)+√(9)$

6

$2+0+1+√(9)$

7

$-2+0+(1*9)$

8

$(2*0)-1+9$

9

$(2*0*1)+9$

10

$(2*0)+1+9$

11

$2+(0*1)+9$

12

$2+0+1+9$

13

$20-1-(√9)!$

14

$20*1-(√9)!$

15

$(2+0+1)!+9$

16

$2^(0+1+√9)$

17

$20-(1*√9)$

18

$(2+0*1)*9$

19

$(2*0)+19$

20

$(2^0)+19$

21

$2+0+19$

22

$20-1+√9$

23

$(20*1)+√9$

24

$20+1+√9$

25

$20-1+(√9)!$

26

$(20*1)+(√9)!$

27

$(2+0+1)*9$

28

$20-1+9$

29

$(20*1)+9$

30

$20+1+9$

• 1= ((2 + 0 + 1) / √9)!
• 2= 2 + 0 * 1 * 9
• 3 = 2 + (0 * 1 * 9)!
• 4 = 2^(0 - 1 + √9)
• 5 = 2 + 0 * 1 + √9
• 6 = -(2 + 0 + 1) + 9
• 7 = -2 + 0 * 1 + 9
• 8 = -2 + 0 + 1 + 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 = -2 + 0 + 19
• 18 = 2 * (0 * 1 + 9)
• 19 = 2 * 0 + 19
• 20 = 20 * 1^9
• 21 = 20 + 1^9
• 22 = 20 - 1 + √9
• 23 = 20 * 1 + √9
• 24 = 20 + 1 + √9
• Welcome to PSE! Please use spoilers '>!' to hide your answer. Happy puzzling = ) – Ak19 Jul 13 at 11:27

## Solutions for 25-36

• 25 = 2 - 0! + (1 + √9)!

• 26 = 2 + 0 + (1 + √9)!

• 27 = (2 + 0 + 1) * 9

• 28 = 20 - 1 + 9

• 29 = 20 + 1 * 9

• 30 = 20 + 1 + 9

• 31 = 2 || 0! + 1 + 9

• 32 = 2 ^ (0! + 1 + √9)

• 33 = (2 - 0!) || 1 * √9

• 34 = 2 || 0! + 1 || √9

• 35 = —not here yet—

• 36 = (2 + 0!)! ^ (-1 + 9)

• I believe you should merge this into your previous answer of solutions from 1-24, but nice answer nevertheless. – Feeds Jul 14 at 2:41