Write a function or a program to find the number of rotations required by a wheel to travel a given distance, given its radius.


Input can be 2 positive rational numbers and can be taken in any convenient format.

Both inputs are of same unit.

There must not be any digits 0-9 in your code.

The output will be an integer (in case of float, round to infinity)

This is code-golf so shortest code wins


distance radius  output
10       1       2
50       2       4
52.22    4       3
3.4      0.08    7
12.5663  0.9999  3
  • 4
    You probably should add that digits are also forbidden in compiler options (or anywhere else): if you limit this constraint to code only, with gcc we can do something like -DP=3.14 in compiler flags, that would define P as an approximation of pi, which is probably not what you intended – Annyo Nov 21 at 16:50

31 Answers 31

MathGolf, 5 4 bytes


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τ      Push tau (2*pi)
 /     Divide the first argument (total distance) by tau
  ╠    Reverse divide (computes (distance/tau)/radius)
   ü   Ceiling

APL+WIN, 9 bytes

Prompts for radius followed by distance:


Try it online! Courtesy of Dyalog Classic


○r+r←⎕ prompt for radius and double it and multiply by pie

⌈⎕÷ prompt for distance, divide by result above and take ceiling
  • ⌈⎕÷○+⍨⎕ works for 7 bytes. – J. Sallé Nov 22 at 13:04
  • @J.Sallé Thanks but unfortunately my ancient APL+WIN interpreter does not have the ⍨ operator – Graham Nov 22 at 15:51

Java 8, 32 30 bytes


Contains unprintable \u0002 between the single quotes.

Port of @jOKing's Perl 6 answer.

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  • Is that the digit '1' in your code? I think that might not be allowed. – ouflak Nov 21 at 14:18
  • 4
    @ouflak Looks like it can be fixed like this. – Erik the Outgolfer Nov 21 at 14:22
  • @ouflak Woops, that was a pretty stupid mistake.. Using the unprintable so I don't use the digit 2, and then just use digit 1... Luckily Erik is indeed right that a simple negative unary has the same effect as +1 (often used to get rid of parenthesis since the negative and unary have higher precedence than most other operators). – Kevin Cruijssen Nov 21 at 18:21

Perl 6, 15 12 bytes

-3 bytes tjanks to nwellnhof reminding me about tau


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Anonymous Whatever lambda that uses the formula (a/b/tau).floor+1. Tau is two times pi. The two anonymous variables $ are coerced to the number 0, which is used to floor the number +|0 (bitwise or 0) and add one +!$ (plus not zero).

  • There must not be any digits 0-9 in your code. – Titus Nov 21 at 13:35
  • @Titus I can't believe I forgot that. Thanks, fixed! – Jo King Nov 21 at 13:37
  • Are digits in exponents also allowed? – ouflak Nov 21 at 14:21

Python 2, 47 45 44 43 bytes

lambda l,r:l/(r+r)//math.pi+l/l
import math

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  • -2 bytes, thanks to flawr
  • -1 byte, thanks to Jonathan Allan
  • Since inputs have been guaranteed to be both (strictly) positive and rational we never hit the edge-case of requiring an exact number of rotations, so I think we can do l/(r+r)//pi+l/l and save a byte. – Jonathan Allan Nov 21 at 13:50
  • @JonathanAllan Thanks :) – TFeld Nov 21 at 14:16

05AB1E, 6 bytes


Port of @flawr's Python 2 comment.
Takes the input in the order radius,distance.

Try it online or verify all test cases.


·         # Double the first (implicit) input
 /        # Divide the second (implicit) input by it
  žq/     # Divide it by PI
     î    # Ceil it (and output implicitly)

C, 46 bytes

f(float a,float b){return ceil(a/(b+b)/M_PI);}

I'm new to PPCG, so I'm not sure wether I have to count other parts in the byte count, such as the

include <math.h>

needed for the ceil function, which will rise the count to 64 bytes

  • Welcome to PPCG! This is a nice first answer. Yes, you do need to count #include and the like towards your byte total. A link to an online test suite is always appreciated, here's one you are free to incorporate into your post: tio.run/… – O.O.Balance Nov 21 at 14:53
  • @O.O.Balance Digits are not allowed in the code for this challenge ;) – Annyo Nov 21 at 15:25
  • @Annyo I knew I was forgetting something :( – O.O.Balance Nov 21 at 15:26
  • Suggest b/2 instead of (b+b) – ceilingcat Dec 6 at 2:26

Catholicon, 8 bytes



  /ĊǓĊ    divide the first input by the doubled second input
 /    `Ė  divide that by pi
ċ         ceil

New version (pi builtin made one byte, division parameters swapped), 5 bytes


Stax, 5 bytes


Run and debug it

Vt*   multiply by tau (2pi)
/     divide
e     ceiling

MathGolf, 6 5 bytes


Semi-port of @flawr's Python 2 comment.
Takes the input in the order radius distance.

-1 byte because ceil builtin has just been added, replacing the floor+1.

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∞        # Double the first (implicit) input
 /       # Divide the second (implicit) input by it
  π/     # Divide it by PI
    ü    # Ceil (and output implicitly)

C (gcc), 45 47 45 bytes

f(d,r,R)float d,r;{R=ceil(d/r/'G'/'\n'*'q');}

A reasonable approximation of pi is 355/113. Since circumference C = 2 * r * PI, we can instead of pi use tau, which is then of course ~710/113. 710 happens to have the convenient factors 2 * 5 * 71, which is compactly expressed as 'G' * '\n'. We add one (r/r) to force rounding to infinity.

Edit: My trick was too clever for its own good: it of course made it fail if the distance was a multiple of the circumference.

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Julia 1.0, 20 bytes


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R, 39 32 bytes

-7 bytes Thanks to Giuseppe


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I feel like this could definitely be golfed, but I am a bit lazy right now to do anything about it

PHP, 47 bytes


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Jelly, 6 bytes


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Ruby, 29 bytes


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J, 10 9 bytes


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Japt, 7 bytes

/MT/V c

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JavaScript (Babel Node), 25 bytes

-2 bytes using @flawr comment =D. -1 from @Kevin. -7 from @Shaggy


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min, 16 bytes

/ tau / ceil int

Takes the distance and radius put on the stack in that order. Then divides by tau, rounds, and makes int.

Dart, 47 46 bytes


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  • -1 byte thanks to @Shaggy

Haskell, 25 bytes

f d r=ceiling(d/(r+r)/pi)
  • You can define an operator (!) instead of f and use ceiling$ instead of ceiling(..)which saves you 3 bytes: Try it online! – BMO Dec 8 at 15:15

Lua, 61 58 57 49 bytes

function(s,r)return math.ceil(s/(r+r)/math.pi)end

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Thanks to KirillL. -8 bytes.

  • I don't know much Lua (so maybe it's still too long), but it appears to be shorter as a function: 49 bytes – Kirill L. Nov 22 at 11:18
  • @KirillL., I'm still learning the rules here. The OP's challenge is pretty open on the input. So my question is, would we have to count your program call() against the byte count? If not, your's definitely shaves off a nice chunk. – ouflak Nov 22 at 11:43
  • A quite common style of submission here is an anonymous function (so that we don't have to count the name, unless it is recursive), which outputs by its return value. The footer section with function calls and actual printing to console is then basically used for visualizing the results and doesn't count towards your score. BTW, you may add more of the OP's test examples to the footer, so that they can be conveniently viewed all at once. Note that in some cases a full program may actually turn out to be golfier! – Kirill L. Nov 22 at 11:58

Common Lisp, 36 bytes

(lambda(a b)(ceiling(/ a(+ b b)pi)))

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Tcl, 50 bytes

proc N d\ r {expr ceil($d/(($r+$r)*acos(-$r/$r)))}

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Tcl, 53 bytes

proc N d\ r {expr ceil($d/(($r+$r)*acos(-[incr i])))}

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Lack of a pi constant or function makes me lose the golf competition!

  • Do I need to remove the .0 at end of each output? It would make me consume more bytes! – sergiol Nov 22 at 18:55
  • 1
    [incr i] is quite clever but I think you can use $d/$d or $r/$r instead. – david Nov 22 at 20:17
  • Saved some bytes thanks to @david's idea! – sergiol Nov 22 at 22:56

PowerShell, 53 52 51 bytes

-1 byte thanks to @mazzy
-1 byte after I realized I don't need a semicolon after the param() block


Takes input from two commandline parameters, distance -d and radius -r.

  • ? param($d,$r);($a=[math])::ceiling($d/($r+$r)/$a::pi) – mazzy Nov 24 at 5:54

JavaScript (Babel Node), 23 bytes


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  • 2
    There must not be any digits 0-9 in your code. – Dennis Nov 22 at 12:55

Clojure, 50 bytes

(fn[a b](int(Math/ceil(/ a Math/PI(count"  ")b))))

An anonymous function that accepts two integers a and b as arguments: the distance and the wheel's radius, respectively.

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(count " ") evaluates to 2, so this function implements \$\lceil \dfrac a{2\pi b} \rceil\$.

TI-Basic (83 series), 12 bytes


Takes input as a list of radius and distance in Ans: for example, {0.9999:12.5663:prgmX.

e^(ΔList(ln(Ans will take the ratio of those distances, and min( turns this into a number. Then we divide by Tmax, which is a graphing parameter that's equal to 2π by default. Finally, -int(- takes the ceiling.

Pari/GP, 23 bytes


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