clockpuzzle.py

#!/usr/bin/python
from fractions import Fraction
from math import floor

# "time" datatype is [0,1) where 0 represents 0:00:00 and 1 represents 12:00:00
# "angle" datatype is [0,1) where 1 represents 360 degrees

def str_time(t):
	h, m, s = time_to_hands(t)
	return f"{floor(h*12)}:{floor(m*60):02d}:{s*60:05.2f}"

def time_to_hands(t):
	return t%1, (t*12)%1, (t*720)%1

def time_to_angles(t):
	h, m, s = time_to_hands(t)
	return (m-h)%1, (s-h)%1

def hm_angle_to_times(x):
	for i in range(11):
		yield (x + i)/11

def solve(hm_angle, hs_angle):
	for t in hm_angle_to_times(hm_angle):
		hm_res, hs_res = time_to_angles(t)
		assert hm_res == hm_angle
		if hs_res == hs_angle:
			yield t

def solve_all(angle1, angle2):
	yield from solve(angle1, (angle1 + angle2) % 1)
	yield from solve((angle1 + angle2) % 1, angle1)
	yield from solve(angle2, (-angle1) % 1)
	yield from solve((-angle1) % 1, angle2)
	yield from solve((-angle2) % 1, (-angle1 - angle2) % 1)
	yield from solve((-angle1 - angle2) % 1, (-angle2) % 1)

def main():
	for t in solve_all(Fraction(155,360), Fraction(60,360)):
		print(str_time(t))

if __name__ == "__main__":
	main()