# Create integers from 1 to 50 using only one integer and other functions

Your task is to create all the integers from 1 to 50 (inclusive) using only a single integer $x$ and...

• Addition ($+$), Subtraction($-$), Multiplication($\times$), and Division($\div$)
• Floor($\left \lfloor x \right \rfloor$) and Ceiling ($\left \lceil x \right \rceil$)
• Exponentation($x^a$) and Radicals($\sqrt[a]{x}$)
• Factorial($!$)
• Tau($\tau$, a.k.a. $2\pi$) and Phi($\phi$, a.k.a. the golden ratio)
• and Parentheses($()$).

MathJaX syntax:

+, -, \times, \div, \lfloor\rfloor, \lceil\rceil, x^a, \sqrt[a]{x}, !, \tau, \phi, () $+, -, \times, \div, \lfloor\rfloor, \lceil\rceil, x^a, \sqrt[a]{x}, !, \tau, \phi, ()$

This is a competition! The winner is the person with the lowest amount of points. A point is given when $x$ is used and when any function listed above is used. Examples($x$ is $3$):

• $1 = 3\div3$ (3 points)
• $6 = 3!$ (2 points)
• $19 = \left \lceil 3 \times \tau\right \rceil$ (4 points)

Good Luck!

EDIT: I don't think anyone is going to get all 50anytime soon, so until then the points will be scored by doing an average of all integers already found. So, for example, if you found 1, 4, 6, 23, 74, and 83, your score would be the sum divided by 6 until someone finds all 50.

EDIT2: Clarifications: $x^x$ scores 3 points, $(x)$ scores two, and you cannot combine digits (if $x$ is 2, then you cannot use $22$ in your answer). Also, x does not need to be used, nor do any of the functions.

• Would $2^2$ count as 2 or 3 points? And do parentheses cost a point? – Tryth Apr 13 '15 at 5:25
• Parentheses shouldn't cost points, if you ask me. – Joe Z. Apr 13 '15 at 5:36
• Also, are you allowed to concatenate digits ($7$ -> $77$)? – Tryth Apr 13 '15 at 9:23
• @Tryth clarifications are at the second edit of the question. – GamrCorps Apr 13 '15 at 14:49
• It's been 10 days without any updates so I think you should conclude this puzzle. – Ben Frankel Apr 23 '15 at 18:28

I can cover all of the numbers for 223 points, 4.46 average, with $x = 34$.

Note that this doesn't make use of radicals, factorials, or parentheses.

$1 = \lfloor\phi\rfloor$
$2 = \lceil\phi\rceil$
$3 = \lfloor\tau\div\phi\rfloor$
$4 = \lfloor\tau-\phi\rfloor$
$5 = \lceil\tau-\phi\rceil$
$6 = \lfloor\tau\rfloor$
$7 = \lceil\tau\rceil$
$8 = \lceil\tau+\phi\rceil$
$9 = \lceil\tau\rceil + \lceil\phi\rceil$
$10 = \lfloor\tau\times\phi\rfloor$
$11 = \lceil\tau\times\phi\rceil$
$12 = \lfloor\tau+\tau\rfloor$
$13 = \lceil\tau+\tau\rceil$
$14 = \lceil\tau\rceil\times\lceil\phi\rceil$
$15 = \lceil\phi^\tau-\tau\rceil$
$16 = \lfloor\tau\times\phi+\tau\rfloor$
$17 = 34\div\lceil\phi\rceil$
$18 = \lceil\phi^{\lfloor\tau\rfloor}\rceil$
$19 = \lfloor\tau^\phi\rfloor$
$20 = \lceil\tau^\phi\rceil$
$21 = \lceil\phi^\tau\rceil$
$22 = \lceil34\div\phi\rceil$
$23 = \lfloor\lceil\tau\rceil^\phi\rfloor$
$24 = \lceil\lceil\tau\rceil^\phi\rceil$
$25 = \lfloor\tau^\phi+\tau\rfloor$
$26 = \lfloor\phi^\tau+\tau\rfloor$
$27 = 34 - \lceil\tau\rceil$
$28 = 34 - \lfloor\tau\rfloor$
$29 = \lfloor\phi^{\lceil\tau\rceil}\rfloor$
$30 = \lceil\phi^{\lceil\tau\rceil}\rceil$
$31 = \lfloor\phi\times\tau^\phi\rfloor$
$32 = 34 - \lceil\phi\rceil$
$33 = 34 - \lfloor\phi\rfloor$
$34 = 34$
$35 = 34 + \lfloor\phi\rfloor$
$36 = 34 + \lceil\phi\rceil$
$37 = \lfloor\lfloor\tau\rfloor\times\tau\rfloor$
$38 = \lceil\lfloor\tau\rfloor\times\tau\rceil$
$39 = \lfloor\tau\times\tau\rfloor$
$40 = 34 + \lfloor\tau\rfloor$
$41 = 34 + \lceil\tau\rceil$
$42 = \lfloor\tau\rfloor\times\lceil\tau\rceil$
$43 = \lfloor\lceil\tau\rceil\times\tau\rfloor$
$44 = \lceil\lceil\tau\rceil\times\tau\rceil$
$45 = \lceil\phi^{\tau+\phi}\rceil$
$46 = \lceil\tau\times\tau+\tau\rceil$
$47 = 34 + \lceil\tau+\tau\rceil$
$48 = \lfloor34\times\phi-\tau\rfloor$
$49 = \lceil\tau\rceil^{\lceil\phi\rceil}$
$50 = \lfloor\lceil\tau\rceil\times\tau+\tau\rfloor$

Points per line:

2 2 4 4 4   2 2 4 5 4 : Sums to 33
4 4 4 5 6   6 4 5 4 4 : Sums to 46
4 4 5 5 6   6 4 4 5 5 : Sums to 48
6 4 4 1 4   4 5 5 4 4 : Sums to 41
4 5 5 5 6   6 6 6 5 7 : Sums to 55


Foundation: http://pastebin.com/AWKKzncy (this achieves 248 without use of $x$).

• looks like 7 might be useful. – Jodrell Apr 13 '15 at 13:23

$x=4$ totalling 264 points averaging 5.28 points per target integer.

I've assumed that you don't have to use x if its not really needed. Otherwise the answer for 1 is 4/4 the answer for 2 is $\lfloor4-\phi\rfloor$ and the other 18 numbers that don't include x will need $\times4\div4$ appended increasing the total points to 339 for an average of 6.78 (although I might have another go at finding better answers for those 18 if they are inadmissible.)

$1= \lfloor \phi \rfloor$ (2 points)

$2=\lceil \phi \rceil$ (2 points)

$3=4-\lfloor \phi \rfloor$ (4 points)

$4=4$ (1 point)

$5=4+\lceil \phi \rceil$ (4 points)

$6=\lfloor \phi \times 4 \rfloor$ (4 points)

$7=\lceil \phi \times 4\rceil$ (4 points)

$8=4 \times \lceil p\rceil$ (4 points)

$9=4+4+\lceil \phi \rceil$ (6 points)

$10=\lfloor 4 \times \phi \times \phi \rfloor$ (6 points)

$11=\lceil 4 \times \phi \times \phi \rceil$ (6 points)

$12=\lfloor \tau \times \lceil \phi \rceil \rfloor ($5 points)

$13=\lceil \tau \times \lceil \phi \rceil\rceil$ (5 points)

$14=4 \times 4- \lceil \phi \rceil$ (6 points)

$15=\lfloor \tau \times 4 \div \phi \rfloor$ (6 points)

$16=4 \times 4$ (3 points)

$17=4 \times 4+\lfloor \phi \rfloor$ (6 points)

$18=\lfloor \tau \times 4 \div \phi + \phi \rfloor$ (8 points)

$19=\lfloor \phi + \phi +4 \times 4 \rfloor$ (8 points)

$20=\lfloor \phi ^t \rfloor$ (4 points)

$21=\lceil \phi ^ \tau \rceil$ (4 points)

$22=\lceil \phi + \phi ^ \tau \rceil$ (6 points)

$23=\lceil \phi + \phi ^ \tau \rceil$ (6 points)

$24=4!$ (2 points)

$25=\lfloor \tau \times 4 \rfloor$ (4 points)

$26=\lceil \tau \times 4 \rceil$ (4 points)

$27=\lceil \tau \times 4+ \phi \rceil$ (6 points)

$28=4 \times \lceil \phi \times 4 \rceil$ (6 points)

$29=\lfloor \phi +4 \times \phi ^4 \rfloor$ (8 points)

$30=\lfloor 4^{4\div\phi}\rfloor$ ( 6 points)

$31=\lceil 4^{4\div \phi }\rceil$ (6 points)

$32=4 \times 4 \times \lceil \phi \rceil$ (6 points)

$33=\lfloor \phi \times \phi ^ \tau \rfloor$ (6 points)

$34=\lceil \phi \times \phi ^ \tau \rceil$ (6 points)

$35=\lceil \phi + \phi \times \phi ^ \tau \rceil$ (8 points)

$36=\lfloor (4!- \phi ) \times \phi \rfloor$ (8 points)

$37=\lceil (4!- \phi ) \times \phi \rceil$ (8 points)

$38=\lfloor (4+4+ \phi ) \times 4 \rfloor$ (9 points)

$39=\lfloor \tau \times \tau \rfloor$ ( 4 points)

$40=\lceil \tau \times \tau \rceil$ (4 points)

$41=\lceil 4 \times \tau \times \phi \rceil$ (6 points)

$42=\lceil \phi \times \phi \times 4 \times 4 \rceil$ ( 8 points)

$43=\lfloor \phi ^{ \phi + \tau }- \phi \rfloor$ ( 6 points)

$44=\lfloor \phi ^{ \phi + \tau }\rfloor$ ( 6 points)

$45=\lceil \phi ^{ \phi + \tau }\rceil$ ( 6 points)

$46=\lfloor \phi + \phi ^{ \phi + \tau }\rfloor$ ( 8 points)

$47=\lceil \phi + \phi ^{ \phi + \tau }\rceil$ ( 8 points)

$48=\lfloor \phi + \phi + \phi ^{ \phi + \tau }\rfloor$ ( 10 points)

$49=\lceil \phi + \phi + \phi ^{ \phi + \tau }\rceil$ ( 10 points)

$50=\lfloor 4 \times \tau \times \lceil \phi \rceil\rfloor$ ( 7 points)

Suppose the integer we use is 2. Then, in reverse Polish notation (to make counting operators and numbers easier), the equations can be expressed as:

 1 = 2 2 /                    ( 3 pts.)
2 = 2                        ( 1 pt. )
3 = 2 2 2 / +                ( 5 pts.)
4 = 2 2 *                    ( 4 pts.)
5 = 2 2 * 2 2 / +            ( 7 pts.)
6 = 2 2 * 2 +                ( 5 pts.)
7 = 2 2 2 2 2 / + + +        ( 9 pts.)
8 = 2 2 2 * *                ( 5 pts.)
9 = 2 2 / 2 * 2 ^            ( 7 pts.)
10 = 2 2 2 * * 2 +            ( 7 pts.)
11 = 22 2 /                   ( 4 pts.)
12 = 2 2 + ! 2 /              ( 6 pts.)
13 = 22 2 / 2 +               ( 6 pts.)
14 = 2 2 + ! 2 / 2 +          ( 8 pts.)
15 = 2 2 2 * ^ 2 2 / -        ( 9 pts.)
16 = 2 2 2 * ^                ( 5 pts.)
17 = 2 2 2 * ^ 2 2 / +        ( 9 pts.)
18 = 22 2 2 + -               ( 6 pts.)
19 = 22 2 2 2 / + -           ( 8 pts.)
20 = 22 2 -                   ( 4 pts.)
21 = 22 2 2 / -               ( 6 pts.)
22 = 22                       ( 2 pts.)
23 = 22 2 2 / +               ( 6 pts.)
24 = 2 2 + !                  ( 4 pts.)
25 = 2 2 + ! 2 2 / +          ( 8 pts.)
26 = 2 2 + ! 2 +              ( 6 pts.)
27 = 22 2 2 2 2 / + + +       (10 pts.)
28 = 2 2 + ! 2 2 * +          ( 8 pts.)
29 = 2 2 + ! 2 2 + + 2 2 / +  (12 pts.)
30 = 22 2 2 2 * * +           ( 8 pts.)
31 = 2 2 2 * ^ 2 * 2 2 / -    (11 pts.)
32 = 2 2 2 * ^ 2 *            ( 7 pts.)
33 = 2 2 2 * ^ 2 * 2 2 / +    (11 pts.)
34 = 2 2 2 * ^ 2 * 2 +        ( 9 pts.)
35 = 2 2 + ! 22 2 / +         ( 9 pts.)
36 = 2 2 2 + + 2 2 2 + + *    (11 pts.)


The above solutions don't make any use of the provided constants $\tau$ and $\phi$, as generally those constants aren't allowed as tokens in such challenges. I'll post answers with constants below if they outscore the above answers.