# Playing Billiards

In this code golf, you will have to determine the direction of the shortest shot that hits exactly n cushions before falling into a pocket.

The billiard table is a 6 pocket pool table with the following characteristics:

• Dimensions are variable (a x b)
• No friction : the ball will roll forever until it falls into a pocket
• Pockets and ball sizes are almost zero. This means that the ball will fall in the pocket only if they have same position.
• The ball is placed at the bottom left hole at the beginning (but doesn't fall in it)

Create a full program or function that takes the dimensions (a,b) of the table and a the number of cushions to hit n as input and returns the angle in degrees of the shortest path hitting exactly n cushions before falling into a pocket.

• a > 0
• b > 0
• 0 <= n < 10000000
• 0 < alpha < 90 (in degrees) precision : at least 10^-6

## examples :

with a = 2, b = 1, n = 1 there are three possible paths : (1) (2) (3) on the following figure. the number (1) is the shortest so the output should be atan(2) = 63.43494882292201 degrees

The solution for a = 2, b = 1, n = 4 is atan(4/3) = 53.13010235415598 degrees

## test samples :

a = 2,    b = 1,    n = 1,       -> alpha = 63.43494882292201
a = 2,    b = 1,    n = 2,       -> alpha = 71.56505117707799
a = 2,    b = 1,    n = 3,       -> alpha = 75.96375653207353
a = 2,    b = 1,    n = 4,       -> alpha = 53.13010235415598
a = 2,    b = 1,    n = 5,       -> alpha = 59.03624346792648
a = 2,    b = 1,    n = 6,       -> alpha = 81.86989764584403
a = 4.76, b = 3.64, n = 27,      -> alpha = 48.503531644784466
a = 2,    b = 1,    n = 6,       -> alpha = 81.86989764584403
a = 8,    b = 3,    n = 33,      -> alpha = 73.24425107080101
a = 43,   b = 21,   n = 10005,   -> alpha = 63.97789961246943


This is code/billiard golf : shortest code wins!

• Does the ball have to hit exactly n cushions, or at least n cushions? – Peter Taylor Dec 16 '15 at 19:40
• @PeterTaylor exactly n cushions – Damien Dec 16 '15 at 19:41
• isn´t the shortest path always back and forth between the left side top and bottom and then into one of the middle holes? – Eumel Dec 17 '15 at 8:55
• no, look at the 2 1 4 example. This path is sqrt(25) = 5 long whereas your solution is sqrt(26) – Damien Dec 17 '15 at 9:49

## Python 2.7, 352344 281 bytes

from math import*
def l(a,b,n):
a*=1.;b*=1.
r=set()
for i in range(1,n+3):
t=[]
for k in range(1,i):
for h in[0,.5]:
x=(i-k-h)
d=(a*n+1)**2+(b*n+1)**2
for x,y in t:
if x*x+y*y<d:d=x*x+y*y;o=degrees(atan(y/x))
return o

• -16 bytes thanks to @Dschoni

Explanation: instead calculating the cushions hits, I'm adding n tables and taking the new holes as valid : Black border/holes is the original, green border/holes is the valid for n=1, red border/holes is the valid for n=2 and so on. Then I remove the invalid holes (e.g. the blue arrow for n=1). I'll have a list of valid holes and their coordinates, then I calculate their distance from initial point, and then the angle of the smaller distance.
Notes:
a=4.76, b=3.64, n=27 - give 52.66286, trying to figure out why fixed, and saved 8 bytes in the process =D
a=43, b=21, n=10005 - takes ~80 seconds (but gives the right angle)

from math import *
def bill(a,b,n):
a=float(a)
b=float(b)
ratios = set()
for i in range(0,n+2): # Create the new boards
outter = []
j=i+1
for k in range(1,j): # Calculate the new holes for each board
#y=k
for hole_offset in [0,0.5]:
x=(j-k-hole_offset)
if (x/k) not in ratios:
outter.append((x*a,k*b))
min_dist = (a*n+1)**2+(b*n+1)**2
for x,y in outter:
if x*x+y*y<min_dist:
min_dist = x*x+y*y
min_alpha=degrees(atan(y/x))
return min_alpha

• You can save a byte by removing the space in : degrees – Morgan Thrapp Dec 16 '15 at 19:21
• I have no idea how your answer works (mathwise) but I think you can gain 1 byte by removing the space after the colon. :) (What @MorganThrapp said) – basile-henry Dec 16 '15 at 19:23
• This path is valid, but it's not the shortest in all cases, for example with 2 1 4.. – Damien Dec 16 '15 at 20:13
• This also assumes that b < a. That could easily fixed by getting the minimum/maximum of a and b though. – user81655 Dec 16 '15 at 21:31
• fixed  ( sorta ) – Rod Dec 17 '15 at 12:02

z=toEnum