Naive solution for a twin riddle

entwin.py

"""
Provides a naive solution to the following problem:

Given a couple (a,b), use the transformations

L: (a,b)->( 2a, b+1 )
R: (a,b)->( a+1, 2b )

to come to two equal integers.

Source:
http://www.reddit.com/r/mathriddles/comments/2v6eaj/doubling_and_adding_1/

This script gives all the solutions in 13 transformations or less, 
for all couples 0 <= a,b < 10, using a recursion to try all
combinations of transformations.
"""


from itertools import product

operations = {"L": lambda a,b : (2*a, b+1),
              "R": lambda a,b : (a+1, 2*b)}


def recursive(a,b, seq=[], max_depth=13):
    
    if a==b:
        print "%s -> %s"%(" ".join(seq), str((a,b)))
        return
    elif max_depth == 0:
        return
    
    for side in ["L", "R"]:
        new_a, new_b = operations[side](a,b)
        recursive(new_a, new_b, seq=seq+[side], max_depth=max_depth-1)

for a,b in product(range(10), range(10)):
    print "\n~~~~~~~ a=%d, b=%d ~~~~~~~"%(a,b)
    recursive(a,b)

gistfile1.txt

(Output of the script above)

~~~~~~~ a=0, b=0 ~~~~~~~
 -> (0, 0)

~~~~~~~ a=0, b=1 ~~~~~~~
L L L L L L R L L L R L L -> (36, 36)
L L L L L L R R L L L L -> (32, 32)
L L L L L R L L L L -> (16, 16)
L L L R L L R L R L L -> (44, 44)
L L L R R L L R L R L L -> (76, 76)
L L R R R L L R L R L L -> (108, 108)
L R L L R R L R L L -> (52, 52)
L R L R L L -> (12, 12)
L R L R R R L L R L R L L -> (172, 172)
L R R L L R R L R L L -> (84, 84)
L R R L R L L -> (20, 20)
L R R R R L L R L R L L -> (140, 140)
R L L -> (4, 4)
R L R R L L L L R L R R -> (132, 132)
R L R R R R L L R L R L L -> (204, 204)
R R L L L L R L R R -> (68, 68)
R R L R R L L L L R L R R -> (196, 196)
R R R L L L L R L R R -> (100, 100)
R R R R L L R R L R L L -> (148, 148)
R R R R L R L L -> (36, 36)

~~~~~~~ a=0, b=2 ~~~~~~~
L L L L L R L L L R L L -> (36, 36)
L L L L L R L R R L L L L -> (64, 64)
L L L L L R R L L L L -> (32, 32)
L L L L R L L L L -> (16, 16)
L L R L L R L R L L -> (44, 44)
L L R R L L R L R L L -> (76, 76)
L R L L R L L L L R L R R -> (164, 164)
L R R R L L R L R L L -> (108, 108)
R L L R R L L R R L R L L -> (212, 212)
R L L R R L R L L -> (52, 52)
R L R L L -> (12, 12)
R L R R R L L R L R L L -> (172, 172)
R R L L R R L R L L -> (84, 84)
R R L R L L -> (20, 20)
R R R R L L R L R L L -> (140, 140)

~~~~~~~ a=0, b=3 ~~~~~~~
L L L L R L L L R L L -> (36, 36)
L L L L R L R R L L L L -> (64, 64)
L L L L R R L L L L -> (32, 32)
L L L R L L L L -> (16, 16)
L L R L L L L L L R L R R -> (132, 132)
L R L L R L L R R L R L L -> (180, 180)
L R L L R L R L L -> (44, 44)
L R R L L R L R L L -> (76, 76)
R L L R L L L L R L R R -> (164, 164)
R R R L L R L R L L -> (108, 108)

~~~~~~~ a=0, b=4 ~~~~~~~
L L L R L L L R L L -> (36, 36)
L L L R L R L R R L L L L -> (128, 128)
L L L R L R R L L L L -> (64, 64)
L L L R R L L L L -> (32, 32)
L L R L L L L -> (16, 16)
L R L L L L L L R L R R -> (132, 132)
L R L L R R L L R L R L L -> (204, 204)
R L L R L L R R L R L L -> (180, 180)
R L L R L R L L -> (44, 44)
R R L L R L L R R L R L L -> (308, 308)
R R L L R L R L L -> (76, 76)

~~~~~~~ a=0, b=5 ~~~~~~~
L L R L L L L R R L R L L -> (148, 148)
L L R L L L R L L -> (36, 36)
L L R L R L R R L L L L -> (128, 128)
L L R L R R L L L L -> (64, 64)
L L R R L L L L -> (32, 32)
L R L L L L -> (16, 16)
R L L L L L L R L R R -> (132, 132)
R L L R R L L R L R L L -> (204, 204)

~~~~~~~ a=0, b=6 ~~~~~~~
L R L L L L R R L R L L -> (148, 148)
L R L L L R L L -> (36, 36)
L R L R L R L R R L L L L -> (256, 256)
L R L R L R R L L L L -> (128, 128)
L R L R R L L L L -> (64, 64)
L R R L L L L -> (32, 32)
R L L L L -> (16, 16)

~~~~~~~ a=0, b=7 ~~~~~~~
L L L L R L R L L L R L L -> (100, 100)
L L R L L R L L R L R L L -> (172, 172)
R L L L L R R L R L L -> (148, 148)
R L L L R L L -> (36, 36)
R L R L R L R R L L L L -> (256, 256)
R L R L R R L L L L -> (128, 128)
R L R R L L L L -> (64, 64)
R R L L L L -> (32, 32)

~~~~~~~ a=0, b=8 ~~~~~~~
L L L R L R L L L R L L -> (100, 100)
L R L L R L L R L R L L -> (172, 172)
R R R L R R L L R L L L L -> (528, 528)

~~~~~~~ a=0, b=9 ~~~~~~~
L L R L R L L L R L L -> (100, 100)
R L L R L L R L R L L -> (172, 172)

~~~~~~~ a=1, b=0 ~~~~~~~
L L L L R L R R -> (36, 36)
L L L L R R L L R L R R -> (148, 148)
L L L R R R R L R L L -> (100, 100)
L L R L L R R R R L R L L -> (196, 196)
L L R R R R L R L L -> (68, 68)
L R L L L L R R L R L R R -> (204, 204)
L R L L R R R R L R L L -> (132, 132)
L R R -> (4, 4)
R L L L L R R L R L R R -> (140, 140)
R L L R L R R -> (20, 20)
R L L R R L L R L R R -> (84, 84)
R L R L L L R R L R L R R -> (172, 172)
R L R L R R -> (12, 12)
R L R R L L R L R R -> (52, 52)
R R L L L R R L R L R R -> (108, 108)
R R R L L R R L R L R R -> (76, 76)
R R R L R R L R L R R -> (44, 44)
R R R R R L R R R R -> (16, 16)
R R R R R R L L R R R R -> (32, 32)
R R R R R R L R R R L R R -> (36, 36)

~~~~~~~ a=1, b=1 ~~~~~~~
 -> (1, 1)

~~~~~~~ a=1, b=2 ~~~~~~~
L L -> (4, 4)
L R L R R L L L L R L R R -> (260, 260)
L R R L L L L R L R R -> (132, 132)
L R R R R L L R L R L L -> (204, 204)
R L L L L R L R R -> (68, 68)
R L L L L R R L L R L R R -> (276, 276)
R L R R L L L L R L R R -> (196, 196)
R R L L L L R L R R -> (100, 100)
R R L R R R L L R L R L L -> (300, 300)
R R R L L R R L R L L -> (148, 148)
R R R L R L L -> (36, 36)

~~~~~~~ a=1, b=3 ~~~~~~~
L L L L L R L R R -> (68, 68)
L L L L L R R L L R L R R -> (276, 276)
L L R R L L L L R L R R -> (196, 196)
L R L L L L R L R R -> (100, 100)
L R L R R R L L R L R L L -> (300, 300)
L R R L L R R L R L L -> (148, 148)
L R R L R L L -> (36, 36)
R L L -> (8, 8)
R L R R R L L R L R L L -> (236, 236)
R R L L R R L R L L -> (116, 116)
R R L R L L -> (28, 28)

~~~~~~~ a=1, b=4 ~~~~~~~
L L L L L R R R R L R L L -> (292, 292)
L L R R L L R R L R L L -> (212, 212)
L L R R L R L L -> (52, 52)
L R L L -> (12, 12)
L R R R L L R L R L L -> (172, 172)
R L L R R L L R R L R L L -> (340, 340)
R L L R R L R L L -> (84, 84)
R L R L L -> (20, 20)
R R R L L R L R L L -> (140, 140)

~~~~~~~ a=1, b=5 ~~~~~~~
L L L -> (8, 8)
L L R R R L L R L R L L -> (236, 236)
L R L L R R L R L L -> (116, 116)
L R L R L L -> (28, 28)

~~~~~~~ a=1, b=6 ~~~~~~~
L L L R R L L L L R L R R -> (324, 324)
L L R L L L L R L R R -> (164, 164)
R R L L R L L R R L R L L -> (436, 436)
R R L L R L R L L -> (108, 108)

~~~~~~~ a=1, b=7 ~~~~~~~
L L L R R L L R R L R L L -> (340, 340)
L L L R R L R L L -> (84, 84)
L L R L L -> (20, 20)
L R R L L R L R L L -> (140, 140)
R L L R L L L L R L R R -> (292, 292)

~~~~~~~ a=1, b=8 ~~~~~~~
L L L R R R L L R L R L L -> (364, 364)
L L R L L R R L R L L -> (180, 180)
L L R L R L L -> (44, 44)
R L L R L L R R L R L L -> (308, 308)
R L L R L R L L -> (76, 76)

~~~~~~~ a=1, b=9 ~~~~~~~
R L L R R L L R L R L L -> (332, 332)

~~~~~~~ a=2, b=0 ~~~~~~~
L L L L R R L R L R R -> (140, 140)
L L R L R R -> (20, 20)
L L R R L L R L R R -> (84, 84)
L R L L L R R L R L R R -> (172, 172)
L R L R R -> (12, 12)
L R R L L R L R R -> (52, 52)
L R R L L R R L L R L R R -> (212, 212)
R L L L R R L R L R R -> (108, 108)
R L R R L R R R R L R L L -> (164, 164)
R R L L R R L R L R R -> (76, 76)
R R L R R L R L R R -> (44, 44)
R R R R L R R R R -> (16, 16)
R R R R R L L R R R R -> (32, 32)
R R R R R L R L L R R R R -> (64, 64)
R R R R R L R R R L R R -> (36, 36)

~~~~~~~ a=2, b=1 ~~~~~~~
L L L R L R R -> (36, 36)
L L L R R L L R L R R -> (148, 148)
L L R L L L R R L R L R R -> (300, 300)
L L R R R R L R L L -> (100, 100)
L R L L R R R R L R L L -> (196, 196)
L R R R R L L R R L R L L -> (276, 276)
L R R R R L R L L -> (68, 68)
R L L L L R R L R L R R -> (204, 204)
R L L R R R R L R L L -> (132, 132)
R L R L L R R R R L R L L -> (260, 260)
R R -> (4, 4)

~~~~~~~ a=2, b=2 ~~~~~~~
 -> (2, 2)

~~~~~~~ a=2, b=3 ~~~~~~~
L -> (4, 4)
R L R R L L L L R L R R -> (260, 260)
R R L L L L R L R R -> (132, 132)
R R R R L L R L R L L -> (204, 204)

~~~~~~~ a=2, b=4 ~~~~~~~
L L L L R L R R -> (68, 68)
L L L L R R L L R L R R -> (276, 276)
L L R R R R L L R L R L L -> (396, 396)
L R L R R L L L L R L R R -> (388, 388)
L R R L L L L R L R R -> (196, 196)
R L L L L R L R R -> (100, 100)
R L L L L R R L L R L R R -> (404, 404)
R L R R R L L R L R L L -> (300, 300)
R R L L R R L R L L -> (148, 148)
R R L R L L -> (36, 36)

~~~~~~~ a=2, b=5 ~~~~~~~
L L L L R R R R L R L L -> (292, 292)
L R L R R R L L R L R L L -> (428, 428)
L R R L L R R L R L L -> (212, 212)
L R R L R L L -> (52, 52)
R L L -> (12, 12)
R R R L L R L R L L -> (172, 172)

~~~~~~~ a=2, b=6 ~~~~~~~
L L -> (8, 8)
L R R R L L R L R L L -> (236, 236)
R L L R R L L R R L R L L -> (468, 468)
R L L R R L R L L -> (116, 116)
R L R L L -> (28, 28)

~~~~~~~ a=2, b=7 ~~~~~~~
L L R R L L L L R L R R -> (324, 324)
L R L L L L R L R R -> (164, 164)

~~~~~~~ a=2, b=8 ~~~~~~~
L L R R L L R R L R L L -> (340, 340)
L L R R L R L L -> (84, 84)
L R L L -> (20, 20)
R R L L R L L R R L R L L -> (564, 564)
R R L L R L R L L -> (140, 140)

~~~~~~~ a=2, b=9 ~~~~~~~
L L R R R L L R L R L L -> (364, 364)
L R L L R R L R L L -> (180, 180)
L R L R L L -> (44, 44)

~~~~~~~ a=3, b=0 ~~~~~~~
L L L R R L R L R R -> (108, 108)
L R R L R R R R L R L L -> (164, 164)
R L L R R L R L R R -> (76, 76)
R L R R L R L R R -> (44, 44)
R L R R L R R L L R L R R -> (180, 180)
R R L R R R R R R L R L L -> (132, 132)
R R R L R R R R -> (16, 16)
R R R R L L R R R R -> (32, 32)
R R R R L R L L R R R R -> (64, 64)
R R R R L R R R L R R -> (36, 36)

~~~~~~~ a=3, b=1 ~~~~~~~
L L R L R R -> (28, 28)
L L R R L L R L R R -> (116, 116)
L R L L L R R L R L R R -> (236, 236)
L R R -> (8, 8)
R L L R L R R -> (36, 36)
R L L R R L L R L R R -> (148, 148)
R L R L L L R R L R L R R -> (300, 300)
R L R R R R L R L L -> (100, 100)
R R L L R R R R L R L L -> (196, 196)
R R R R R L L R R L R L L -> (276, 276)
R R R R R L R L L -> (68, 68)

~~~~~~~ a=3, b=2 ~~~~~~~
L L L L R R L R L R R -> (204, 204)
L L R R R R L R L L -> (132, 132)
L R L L R R R R L R L L -> (260, 260)
R -> (4, 4)

~~~~~~~ a=3, b=3 ~~~~~~~
 -> (3, 3)

~~~~~~~ a=3, b=4 ~~~~~~~
L L L R R R R L R L L -> (228, 228)
L L R L L R R R R L R L L -> (452, 452)
L R L L L L R R L R L R R -> (460, 460)
L R R R R L L R L R L L -> (332, 332)
R L R R L L L L R L R R -> (324, 324)
R R L L L L R L R R -> (164, 164)

~~~~~~~ a=3, b=5 ~~~~~~~
L -> (6, 6)
R L R R R L L R L R L L -> (364, 364)
R R L L R R L R L L -> (180, 180)
R R L R L L -> (44, 44)

~~~~~~~ a=3, b=6 ~~~~~~~
L L R R R R L L R L R L L -> (524, 524)
L R L R R L L L L R L R R -> (516, 516)
L R R L L L L R L R R -> (260, 260)
R L L L L R L R R -> (132, 132)
R L L L L R R L L R L R R -> (532, 532)
R R R L L R L R L L -> (204, 204)

~~~~~~~ a=3, b=7 ~~~~~~~
L L L L L R R L R L R R -> (396, 396)
L R L R R R L L R L R L L -> (556, 556)
L R R L L R R L R L L -> (276, 276)
L R R L R L L -> (68, 68)
R L L -> (16, 16)

~~~~~~~ a=3, b=8 ~~~~~~~
L L L L R L R R -> (100, 100)
L L L L R R L L R L R R -> (404, 404)
L R R R L L R L R L L -> (300, 300)
R L L R R L L R R L R L L -> (596, 596)
R L L R R L R L L -> (148, 148)
R L R L L -> (36, 36)

~~~~~~~ a=3, b=9 ~~~~~~~
L L L L R R R R L R L L -> (420, 420)

~~~~~~~ a=4, b=0 ~~~~~~~
L L R R L R L R R -> (76, 76)
L L R R L R R L L R L R R -> (308, 308)
L R R L R L R R -> (44, 44)
L R R L R R L L R L R R -> (180, 180)
R L R R L L R R L R L R R -> (204, 204)
R L R R R R R R L R L L -> (132, 132)
R R L R R R R -> (16, 16)
R R R L L R R R R -> (32, 32)
R R R L R L L R R R R -> (64, 64)
R R R L R L R L L R R R R -> (128, 128)
R R R L R R R L R R -> (36, 36)

~~~~~~~ a=4, b=1 ~~~~~~~
L L L R R L R L R R -> (140, 140)
L R L R R -> (20, 20)
L R R L L R L R R -> (84, 84)
L R R L L R R L L R L R R -> (340, 340)
R L L L R R L R L R R -> (172, 172)
R L R R -> (12, 12)
R R L L R L R R -> (52, 52)
R R L L R R L L R L R R -> (212, 212)
R R R R R L L L L R L R R -> (292, 292)

~~~~~~~ a=4, b=2 ~~~~~~~
L L R L R R -> (36, 36)
L L R R L L R L R R -> (148, 148)
L R L L L R R L R L R R -> (300, 300)
L R R R R L L R R L R L L -> (404, 404)
L R R R R L R L L -> (100, 100)
R L L R R R R L R L L -> (196, 196)
R L R L L R R R R L R L L -> (388, 388)
R R L L L L R R L R L R R -> (396, 396)
R R R R L L R R L R L L -> (276, 276)
R R R R L R L L -> (68, 68)

~~~~~~~ a=4, b=3 ~~~~~~~
L L R R R R L R L L -> (164, 164)
L R L L R R R R L R L L -> (324, 324)
R L L L L R R L R L R R -> (332, 332)
R L R R R R L L R L R L L -> (460, 460)
R R L R R L L L L R L R R -> (452, 452)
R R R L L L L R L R R -> (228, 228)

~~~~~~~ a=4, b=4 ~~~~~~~
 -> (4, 4)

~~~~~~~ a=4, b=5 ~~~~~~~
L L L R L R R -> (68, 68)
L L L R R L L R L R R -> (276, 276)
L L R L L L R R L R L R R -> (556, 556)
L R R R R L L R L R L L -> (396, 396)
R L R R L L L L R L R R -> (388, 388)
R R L L L L R L R R -> (196, 196)

~~~~~~~ a=4, b=6 ~~~~~~~
L L L R R R R L R L L -> (292, 292)
L L R L L R R R R L R L L -> (580, 580)
L R L L L L R R L R L R R -> (588, 588)
R L R R R L L R L R L L -> (428, 428)
R R L L R R L R L L -> (212, 212)
R R L R L L -> (52, 52)

~~~~~~~ a=4, b=7 ~~~~~~~
L -> (8, 8)
R R R L L R L R L L -> (236, 236)

~~~~~~~ a=4, b=8 ~~~~~~~
L L R R R R L L R L R L L -> (652, 652)
L R L R R L L L L R L R R -> (644, 644)
L R R L L L L R L R R -> (324, 324)
R L L L L R L R R -> (164, 164)
R L L L L R R L L R L R R -> (660, 660)

~~~~~~~ a=4, b=9 ~~~~~~~
L R L R R R L L R L R L L -> (684, 684)
L R R L L R R L R L L -> (340, 340)
L R R L R L L -> (84, 84)
R L L -> (20, 20)

~~~~~~~ a=5, b=0 ~~~~~~~
L R R L L R R L R L R R -> (204, 204)
L R R R R R R L R L L -> (132, 132)
R L R R R R -> (16, 16)
R R L L R R R R -> (32, 32)
R R L R L L R R R R -> (64, 64)
R R L R L R L L R R R R -> (128, 128)
R R L R R R L R R -> (36, 36)
R R L R R R R L L R L R R -> (148, 148)

~~~~~~~ a=5, b=1 ~~~~~~~
R L R L R R -> (28, 28)
R L R R L L R L R R -> (116, 116)
R R L L L R R L R L R R -> (236, 236)
R R R -> (8, 8)

~~~~~~~ a=5, b=2 ~~~~~~~
L L L R R L R L R R -> (172, 172)
L R R -> (12, 12)
R L L R L R R -> (52, 52)
R L L R R L L R L R R -> (212, 212)
R L R L L L R R L R L R R -> (428, 428)
R R R R L L L L R L R R -> (292, 292)

~~~~~~~ a=5, b=3 ~~~~~~~
L L R L R R -> (44, 44)
L L R R L L R L R R -> (180, 180)
L R L L L R R L R L R R -> (364, 364)
R -> (6, 6)

~~~~~~~ a=5, b=4 ~~~~~~~
L L R R R R L R L L -> (196, 196)
L R L L R R R R L R L L -> (388, 388)
R L L L L R R L R L R R -> (396, 396)
R R L R R R L L R L R L L -> (556, 556)
R R R L L R R L R L L -> (276, 276)
R R R L R L L -> (68, 68)

~~~~~~~ a=5, b=5 ~~~~~~~
 -> (5, 5)

~~~~~~~ a=5, b=6 ~~~~~~~
L L L L R R L R L R R -> (332, 332)
L R R R R L L R L R L L -> (460, 460)
R L R R L L L L R L R R -> (452, 452)
R R L L L L R L R R -> (228, 228)

~~~~~~~ a=5, b=7 ~~~~~~~
L L L R L R R -> (84, 84)
L L L R R L L R L R R -> (340, 340)
L L R L L L R R L R L R R -> (684, 684)
R L R R R L L R L R L L -> (492, 492)
R R L L R R L R L L -> (244, 244)
R R L R L L -> (60, 60)

~~~~~~~ a=5, b=8 ~~~~~~~
L L L R R R R L R L L -> (356, 356)
L L R L L R R R R L R L L -> (708, 708)
L R L L L L R R L R L R R -> (716, 716)
R R R L L R L R L L -> (268, 268)

~~~~~~~ a=5, b=9 ~~~~~~~
L -> (10, 10)

~~~~~~~ a=6, b=0 ~~~~~~~
L R R R R -> (16, 16)
R L L R R R R -> (32, 32)
R L R L L R R R R -> (64, 64)
R L R L R L L R R R R -> (128, 128)
R L R L R L R L L R R R R -> (256, 256)
R L R R R L R R -> (36, 36)
R L R R R R L L R L R R -> (148, 148)

~~~~~~~ a=6, b=1 ~~~~~~~
L L R R L R L R R -> (108, 108)
L L R R L R R L L R L R R -> (436, 436)
R R L R R R R L R L L -> (164, 164)
R R R L L R R R R L R L L -> (324, 324)

~~~~~~~ a=6, b=2 ~~~~~~~
L R L R R -> (28, 28)
L R R L L R L R R -> (116, 116)
L R R L L R R L L R L R R -> (468, 468)
R L L L R R L R L R R -> (236, 236)
R R -> (8, 8)

~~~~~~~ a=6, b=3 ~~~~~~~
L L L R R L R L R R -> (204, 204)
L R R R R L L R R L R L L -> (532, 532)
L R R R R L R L L -> (132, 132)
R L L R R R R L R L L -> (260, 260)
R L R L L R R R R L R L L -> (516, 516)
R R L L L L R R L R L R R -> (524, 524)

~~~~~~~ a=6, b=4 ~~~~~~~
L L R L R R -> (52, 52)
L L R R L L R L R R -> (212, 212)
L R L L L R R L R L R R -> (428, 428)
R L R R R R L L R L R L L -> (588, 588)
R R L R R L L L L R L R R -> (580, 580)
R R R L L L L R L R R -> (292, 292)

~~~~~~~ a=6, b=5 ~~~~~~~
L L R R R R L R L L -> (228, 228)
L R L L R R R R L R L L -> (452, 452)
R L L L L R R L R L R R -> (460, 460)
R R R R L L R L R L L -> (332, 332)

~~~~~~~ a=6, b=6 ~~~~~~~
 -> (6, 6)

~~~~~~~ a=6, b=7 ~~~~~~~
L R R R R L L R L R L L -> (524, 524)
R L R R L L L L R L R R -> (516, 516)
R R L L L L R L R R -> (260, 260)

~~~~~~~ a=6, b=8 ~~~~~~~
L L L L R R L R L R R -> (396, 396)
R L R R R L L R L R L L -> (556, 556)
R R L L R R L R L L -> (276, 276)
R R L R L L -> (68, 68)

~~~~~~~ a=6, b=9 ~~~~~~~
L L L R L R R -> (100, 100)
L L L R R L L R L R R -> (404, 404)
L L R L L L R R L R L R R -> (812, 812)
R R R L L R L R L L -> (300, 300)

~~~~~~~ a=7, b=0 ~~~~~~~
L L R R R R -> (32, 32)
L R L L R R R R -> (64, 64)
L R L R L L R R R R -> (128, 128)
L R L R L R L L R R R R -> (256, 256)
L R R R L R R -> (36, 36)
L R R R R L L R L R R -> (148, 148)
R R L R R L R R L R L R R -> (172, 172)
R R R R L R L R R R L R R -> (100, 100)

~~~~~~~ a=7, b=1 ~~~~~~~
L R R L R R R R L R L L -> (292, 292)
R L L R R L R L R R -> (140, 140)
R R L R R -> (20, 20)
R R R L L R L R R -> (84, 84)
R R R L L R R L L R L R R -> (340, 340)

~~~~~~~ a=7, b=2 ~~~~~~~
R L R R R R L R L L -> (164, 164)
R R L L R R R R L R L L -> (324, 324)

~~~~~~~ a=7, b=3 ~~~~~~~
L R R -> (16, 16)
R L L R L R R -> (68, 68)
R L L R R L L R L R R -> (276, 276)
R L R L L L R R L R L R R -> (556, 556)
R R R R R L L R L R L L -> (396, 396)

~~~~~~~ a=7, b=4 ~~~~~~~
L L L R R L R L R R -> (236, 236)
R -> (8, 8)

~~~~~~~ a=7, b=5 ~~~~~~~
L L R L R R -> (60, 60)
L L R R L L R L R R -> (244, 244)
L R L L L R R L R L R R -> (492, 492)
R R L R R R L L R L R L L -> (684, 684)
R R R L L R R L R L L -> (340, 340)
R R R L R L L -> (84, 84)

~~~~~~~ a=7, b=6 ~~~~~~~
L L R R R R L R L L -> (260, 260)
L R L L R R R R L R L L -> (516, 516)
R L L L L R R L R L R R -> (524, 524)

~~~~~~~ a=7, b=7 ~~~~~~~
 -> (7, 7)

~~~~~~~ a=7, b=8 ~~~~~~~
L R R R R L L R L R L L -> (588, 588)
R L R R L L L L R L R R -> (580, 580)
R R L L L L R L R R -> (292, 292)

~~~~~~~ a=7, b=9 ~~~~~~~
R L R R R L L R L R L L -> (620, 620)
R R L L R R L R L L -> (308, 308)
R R L R L L -> (76, 76)

~~~~~~~ a=8, b=0 ~~~~~~~
L L L R L L R R L R R R R -> (528, 528)
R L R R L R R L R L R R -> (172, 172)
R R R L R L R R R L R R -> (100, 100)

~~~~~~~ a=8, b=1 ~~~~~~~
L R R L R L R R -> (76, 76)
L R R L R R L L R L R R -> (308, 308)
R R L R L R R -> (44, 44)
R R L R R L L R L R R -> (180, 180)
R R R L L L R R L R L R R -> (364, 364)

~~~~~~~ a=8, b=2 ~~~~~~~
L L R R L R L R R -> (140, 140)
L L R R L R R L L R L R R -> (564, 564)
R L R R -> (20, 20)
R R L L R L R R -> (84, 84)
R R L L R R L L R L R R -> (340, 340)

~~~~~~~ a=8, b=3 ~~~~~~~
L R L R R -> (36, 36)
L R R L L R L R R -> (148, 148)
L R R L L R R L L R L R R -> (596, 596)
R L L L R R L R L R R -> (300, 300)
R R R R L L R R L R L L -> (404, 404)
R R R R L R L L -> (100, 100)

~~~~~~~ a=8, b=4 ~~~~~~~
L R R R R L L R R L R L L -> (660, 660)
L R R R R L R L L -> (164, 164)
R L L R R R R L R L L -> (324, 324)
R L R L L R R R R L R L L -> (644, 644)
R R L L L L R R L R L R R -> (652, 652)

~~~~~~~ a=8, b=5 ~~~~~~~
L L L R R L R L R R -> (268, 268)
R L R R R R L L R L R L L -> (716, 716)
R R L R R L L L L R L R R -> (708, 708)
R R R L L L L R L R R -> (356, 356)

~~~~~~~ a=8, b=6 ~~~~~~~
L L R L R R -> (68, 68)
L L R R L L R L R R -> (276, 276)
L R L L L R R L R L R R -> (556, 556)
R R R R L L R L R L L -> (396, 396)

~~~~~~~ a=8, b=7 ~~~~~~~
L L R R R R L R L L -> (292, 292)
L R L L R R R R L R L L -> (580, 580)
R L L L L R R L R L R R -> (588, 588)

~~~~~~~ a=8, b=8 ~~~~~~~
 -> (8, 8)

~~~~~~~ a=8, b=9 ~~~~~~~
L R R R R L L R L R L L -> (652, 652)
R L R R L L L L R L R R -> (644, 644)
R R L L L L R L R R -> (324, 324)

~~~~~~~ a=9, b=0 ~~~~~~~
L R R L R R L R L R R -> (172, 172)
R R L R L R R R L R R -> (100, 100)

~~~~~~~ a=9, b=1 ~~~~~~~
L R R L L R R L R L R R -> (332, 332)

~~~~~~~ a=9, b=2 ~~~~~~~
R L R L R R -> (44, 44)
R L R R L L R L R R -> (180, 180)
R R L L L R R L R L R R -> (364, 364)

~~~~~~~ a=9, b=3 ~~~~~~~
R R R R L L L L R L R R -> (420, 420)

~~~~~~~ a=9, b=4 ~~~~~~~
L R R -> (20, 20)
R L L R L R R -> (84, 84)
R L L R R L L R L R R -> (340, 340)
R L R L L L R R L R L R R -> (684, 684)

~~~~~~~ a=9, b=5 ~~~~~~~
R -> (10, 10)

~~~~~~~ a=9, b=6 ~~~~~~~
L L L R R L R L R R -> (300, 300)
R R L R R R L L R L R L L -> (812, 812)
R R R L L R R L R L L -> (404, 404)
R R R L R L L -> (100, 100)

~~~~~~~ a=9, b=7 ~~~~~~~
L L R L R R -> (76, 76)
L L R R L L R L R R -> (308, 308)
L R L L L R R L R L R R -> (620, 620)

~~~~~~~ a=9, b=8 ~~~~~~~
L L R R R R L R L L -> (324, 324)
L R L L R R R R L R L L -> (644, 644)
R L L L L R R L R L R R -> (652, 652)

~~~~~~~ a=9, b=9 ~~~~~~~
 -> (9, 9)