# $\pi$ Day puzzle one to twenty

Create the numbers from 1 to 20

Using

• $\pi$
• Normal arithmetic operation $+ - * /$
• Square root $\surd$
• Exponential $(X^Y)$
• Negative() minus sign $-$
• Floor() function, express between $[ x ]$

$[ 3.0 ]$ and $[ 3.9 ] = 3$
$[-3.1 ]$ and $[-3.9 ] = -4$

Forbidden

• Factorial
• Log
• More than four $\pi$

Doing the 20 numbers takes time but is easy.
Now the challenge is to create all 20 numbers using the minimal number of $\pi$.

For example:

1 = $\pi$ /$\pi$ use two $\pi$
1 = [ $\surd\pi$ ] only need one $\pi$

2 = $[ \surd\pi ] + [ \surd\pi ]$ use two $\pi$
2 = $-[ - \surd\pi ]$ use one $\pi$

I give you an initial target of 50 $\pi$ for all 20 numbers. But I know can be lower.

• In the exponential can you use any number as exponent or just something else derived by π? Mar 25, 2015 at 18:00
• Are parentheses allowed? Mar 25, 2015 at 18:13
• @ChrisCudmore The exponential rule seems to use them. Mar 25, 2015 at 18:19
• @leol2 exponential just derived by $\pi$ Mar 25, 2015 at 18:26
• @ChrisCudmore Parentheses allowed Mar 25, 2015 at 18:26

I managed to do it in 42 $\pi$s.

• 1: $\lfloor\sqrt\pi\rfloor$
• 2: $\sqrt{-\lfloor-\pi\rfloor}$
• 3: $\lfloor\pi\rfloor$
• 4: $-\lfloor-\pi\rfloor$
• 5: $\pi-\lfloor-\sqrt\pi\rfloor$
• 6: $\lfloor\pi\rfloor+\lfloor\pi\rfloor$
• 7: $\lfloor\pi\rfloor-\lfloor-\pi\rfloor$
• 8: $-\lfloor-\pi\rfloor-\lfloor-\pi\rfloor$
• 9: $\lfloor\pi\rfloor\times\lfloor\pi\rfloor$
• 10: $-\lfloor-\pi\times\pi\rfloor$
• 11: $\lfloor\lfloor-\lfloor-\pi\rfloor\rfloor^{\sqrt\pi}\rfloor$
• 12: $-\lfloor-\pi\rfloor\times\lfloor\pi\rfloor$
• 13: $\lfloor-\lfloor-\pi\rfloor\times\pi\rfloor$
• 14: $-\lfloor-\pi\times\pi\rfloor-\lfloor-\pi\rfloor$
• 15: $-\lfloor-\pi\rfloor\times\lfloor\pi\rfloor+\lfloor\pi\rfloor$
• 16: $\lfloor-\pi\rfloor\times\lfloor-\pi\rfloor$
• 17: $\lfloor-\lfloor-\pi\rfloor\times\pi\rfloor-\lfloor-\pi\rfloor$
• 18: $\sqrt{-\lfloor-\pi\rfloor}\times\lfloor\pi\rfloor\times\lfloor\pi\rfloor$
• 19: $-\lfloor-\sqrt{-\lfloor-\pi\rfloor}\times\pi\times\lfloor\pi\rfloor\rfloor$
• 20: $\lfloor\pi^\pi/\sqrt\pi\rfloor$
• Ceiling function is not allowed. Mar 25, 2015 at 18:13
• wow 20 blow my mind. Please check "2" I will check the rest later. But i see you only use 2 $\pi$ for 11 and 20 and Ian use 3 on both so your count should be 41 unless you overspend in other place. Mar 25, 2015 at 21:10
• His 20 still uses 3.... And I swear this answer wasn't here as I was typing mine.... Mar 25, 2015 at 21:12
• Nevermind you also use 3 on 20 didnt count the exponent Mar 25, 2015 at 21:12
• THE ANSWER IS 42 !!! @JuanCarlosOropeza has found the ultimate question to Life, the Universe, and Everything! Mar 25, 2015 at 22:09

I did it with 43.

43 = $⌊\pi⌋^{⌊\pi⌋}-⌊-\pi⌋*-⌊-\pi⌋$

There might be a smaller number that this could be done with, but I got lazy after 13. I also don't use anything other than $\surd$, $\pi$, $⌊$ $⌋$, $-$, $+$, $*$.

Exponents and parentheses are for suckers!
(or maybe for people that get a lower score than me)

1 = $⌊\surd\pi⌋$
2 = $-⌊-\surd\pi⌋$
3 = $⌊\pi⌋$
4 = $-⌊-\pi⌋$
5 = $⌊\pi⌋-⌊-\surd\pi⌋$
6 = $⌊\pi+\pi⌋$
7 = $⌊\pi⌋-⌊-\pi⌋$
8 = $-⌊-\pi⌋-⌊-\pi⌋$
9 = $⌊\pi * \pi⌋$
10 = $-⌊-\pi * \pi⌋$
11 = $-⌊-\pi * \pi⌋ + ⌊\surd\pi⌋$
12 = $⌊\pi * -⌊-\pi⌋⌋$
13 = $-⌊-\pi * -⌊-\pi⌋⌋$
14 = $-⌊-\surd\pi⌋ * ⌊⌊\pi⌋ - ⌊-\pi⌋⌋$
15 = $-⌊-\pi⌋ * -⌊-\pi⌋ - ⌊\surd\pi⌋$
16 = $-⌊-\pi⌋ * -⌊-\pi⌋$
17 = $-⌊-\pi⌋ * -⌊-\pi⌋ + ⌊\surd\pi⌋$
18 = $-⌊-\pi⌋ * -⌊-\pi⌋ - ⌊-\surd\pi⌋$
19 = $-⌊-\pi⌋ * -⌊-\pi⌋ + ⌊\pi⌋$
20 = $-⌊-\pi⌋ * -⌊-\pi⌋ - ⌊-\pi⌋$

• ROFL on the hilarious observation Mar 25, 2015 at 18:35
• I'm pretty sure this is a optimal solution, let's wait for rand'al thor and Gamow though, eheh! Mar 25, 2015 at 18:37
• maybe parentheses are for suckers but in 14 and 15 you use floor as parentheses :P Mar 25, 2015 at 18:46
• I sure do! But I don't use parentheses! ;) Mar 25, 2015 at 18:56
• Please Ian can you double check 15. I dont think is ok. I reduce it to −[−√π∗7] and that is 13 Mar 25, 2015 at 19:00

OK, so I got 42

Most of my answers are similar to Ian MacDonald's with the exception of 11 where I was a "sucker" and used an exponent...

1 = $⌊√π⌋$
2 = $√(−⌊−π⌋)$
3 = $⌊π⌋$
4 = $−⌊−π⌋$
5 = $⌊π*√π⌋$
6 = $⌊π⌋+⌊π⌋$
7 = $⌊π⌋−⌊−π⌋$
8 = $−⌊−π⌋−⌊−π⌋$
9 = $⌊π⌋∗⌊π⌋$
10 = $−⌊−π∗π⌋$
11 = $⌊(-⌊-π⌋)^{√π}⌋$
12 = $-⌊π⌋∗⌊−π⌋$
13 = $−⌊−π∗−⌊−π⌋⌋$
14 = $−⌊−√π⌋∗⌊⌊π⌋−⌊−π⌋⌋$
15 = $⌊−π⌋∗⌊−π⌋−⌊√π⌋$
16 = $⌊−π⌋∗⌊−π⌋$
17 = $⌊−π⌋∗⌊−π⌋+⌊√π⌋$
18 = $⌊−π⌋∗⌊−π⌋−⌊−√π⌋$
19 = $⌊−π⌋∗⌊−π⌋+⌊π⌋$
20 = $⌊−π⌋∗⌊−π⌋−⌊−π⌋$

• Yes, his answer was there. I was already checking it while your appear but nice try Mar 25, 2015 at 21:13
• It wasn't when I started. For whatever reason formatting takes time... Mar 25, 2015 at 21:15
• Btw Exponent doesnt suck. And I dont know why parentheses suck, but use $\lfloor x \rfloor$ to replace it doesnt. Mar 25, 2015 at 21:15
• @GOTO I realized that after the fact, but at first I saw the "answered 3 hours ago" and I thought I was going crazy.... Mar 25, 2015 at 21:21
• THE ANSWER IS 42 !!! @JuanCarlosOropeza has found the ultimate question to Life, the Universe, and Everything! Mar 25, 2015 at 22:08