Make numbers 1-50 using $\pi$ and its digits (but with some penalties) [closed]

Question

In this problem, you will be allowed to use some operations and additional digits from the basic approximation of $\pi=3.14$ having some penalties, as follows:

Operations:

• Using basic operations and parenthesis ($ +, -, *, /, (),$) gives you 4 points per number:

$$ 3+1+4 = 8 $$

• Using additional operations without numbers (! - factorial, $\sqrt{}$ - square root, unary minus,...) gives you 3 points $$ -3 + \sqrt{1+4!} = 22 $$

• Using additional operations with numbers ($x^2, x^3,...$ - exponentiation, other roots, etc) gives you 2 points $$ \sqrt[3]{3^2 - 1} + 4 = 6 $$

• Concatenation of original digits gives you 1 point.

$$ cat(3,1)+4 = 35 $$

this results in a partial score $S_p$:

$$ S_p = \sum_{i=0}^{50}{S_i} $$

Number of decimal digits of pi:

The previous partial score must be divided by the number of decimal digits used, starting from $3.14$, which contains 2 decimal digits. For example, if $\pi = 3.14$ is used, the total score $S_T$ is $S_p/2$. But if $\pi = 3.14159$ is used, $S_T$ is $S_p/5$.

Valid options are (from 2 to 5 digits): 3.14, 3.142, 3.1416, 3.14159.

Rules:

• All digits should be used once and in their order
• Exponents must be integers (positive and negative ;))
• Operations can be done on groups with parenthesis, e.g. $(3+1)!+4$ or $3+(1+4)^2$
• Operations/operators must be defined before expanding, this is, it is not allowed to add operators after solving partially. e.g. This is not allowed:
  • $$ concat((3+1),4) = 44) $$
  • $$ 3^4+1+4 = 81+1+4 = 8+1+1+4 = 14 $$
• Stacked operators are valid (Double factorial, multiple exponents, etc)
• Unary minus $-$ is allowed, e.g. $ -3+1+4 = 2 $

Question:

Having a theoretical max score of 100, How close can you get to it?

What is the highest score you can achieve?

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P.D.

• Making $0$ is valid for bonus points ;)
• Partial answers allowed (For populating the leaderboard)

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Leaderboard:

*

Answer 1

Per comments, this answer is invalid - it doesn't follow the "in order" rule.

Current score: 50

Using: 3.1415, for a divisor 4
Basic operations only - 4 points per number, 200 partial score.

Working:

1=1
2=1+1
3=3
4=4
5=5
6=5+1
7=3+4
8=4*(3-1)
9=3*(4-1)
10=5*(3-1)
11=(3*4)-1
12=3*4
13=3*4+1
14=(3+4)*(1+1)
15=3*5
16=3*5+1
17=3*5+1+1
18=3*(5+1)
19=(4*5)-1
20=4*5
21=(3+4)*(5-1-1)
22=(4*5)+1+1
23=(4*5)+3
24=(5+1)4
25=5(1+4)
26=(5*(1+4))+1
27=3*(4+5)
28=(3+4)(5-1)
29=(3+4)(5-1)+1
30=3*(4+1+5)
31=(3*(4+1+5))+1
32=4*(3+5)
33=3*(1+4+1+5)
34=((3+4)*5)-1
35=(3+4)*5
36=3*4*(5-1-1)
37= ((3+4)5)+1+1
38=(3-1)((4*5)-1)
39=((3+1+4)*5)-1
40=4*5*(3-1)
41=4*5*(3-1)+1
42=3*(((4-1)*5)-1)
43=(4*5*(1+1))+3
44=((3*(5-1)-1)4
45=3(4-1)5
46=(3(4-1)*5)+1
47=(3*4*(5-1))-1
48=3*4*(5-1)
49=(3+4)(5+1+1)
50=5((3*4)-1-1)

Bonus:

0=3-1+4-1-5

Answer 2

My solution

using an approximation $\pi\approx3.14$ with the score of 70.5 ($=(4\times7+3\times27+2\times16)/2$)

Note

I believe that operators like $[x]$ (integer part of $x$) are allowed and classified as "additional operations without numbers" (note the "..." and "etc" in the question, so nothing disallows using it).

Here we go (numbers in parentheses indicate point count):

$1=(3+1)/4\ (4)$
$2=-3+1+4\ (3)$
$3=3!+1-4\ (3)$
$4=3-1+\sqrt4\ (3)$
$5=3\times1+\sqrt4\ (3)$
$6=3-1+4\ (4)$
$7=3\times1+4\ (4)$
$8=3+1+4\ (4)$
$9=3!-1+4\ (3)$
$10=3!\times1+4\ (3)$
$11=3!+1+4\ (3)$
$12=3\times1\times4\ (4)$
$13=3^2\times1+4\ (2)$
$14=3^2+1+4\ (2)$
$15=3\times(1+4)\ (4)$
$16=(3+1)\times4\ (4)$
$17=-3!-1+4!\ (3)$
$18=-3!\times1+4!\ (3)$
$19=-3!+1+4!\ (3)$
$20=-3-1+4!\ (3)$
$21=-3\times1+4!\ (3)$
$22=-3!+1+4!\ (3)$
$23=-[\sqrt3]\times1+4!\ (3)$
$24=[\sqrt3]-1+4!\ (3)$
$25=[\sqrt3]\times1+4!\ (3)$
$26=3-1+4!\ (3)$
$27=3\times1+4!\ (3)$
$28=3+1+4!\ (3)$
$29=3!-1+4!\ (3)$
$30=3!\times1+4!\ (3)$
$31=3!+1+4!\ (3)$
$32=(3+1+4)!!!!\ (3)$(quadruple factorial means $8\times4$)
$33=[\sqrt{3-1}\times4!]\ (3)$
$34=3^2+1+4!\ (2)$
$35=3\times1+\sqrt{4^5}\ (2)$
$36=3!\times(-(1-4))!\ (3)$
$37=-3^3\times1+4^3\ (2)$
$38=-3^3+1+4^3\ (2)$
$39=3!+1+\sqrt{4^5}\ (2)$
$40=3^2-1+\sqrt{4^5}\ (2)$
$41=[\sqrt3\times1\times4!]\ (3)$
$42=3^3-1+4^2\ (2)$
$43=3^3\times1+4^2\ (2)$
$44=3^3+1+4^2\ (2)$
$45=3^2\times(1+4)\ (2)$
$46=-3^4-1+\sqrt{4^7}\ (2)$
$47=-3^4\times1+\sqrt{4^7}\ (2)$
$48=(3-1)\times4!\ (3)$
$49=(3\times1+4)^2\ (2)$
$50=3^3-1+4!\ (2)$

Bonus:

$0=3+1-4$