*Monte Carlo Programming Assignment, ECE 4271 Spring 2022
Michelle Davies (mdd94) Monte Carlo - Metropolis Implementation.*
This is the file which verifies, interprets, reports and returns a data report on the runs done on the class in mdd94_monte_carlo.py.
The plan is to run 3 trials for each N runs (N=10000, N=50000 and N=100000) and analyze how those results compare. Then, there will be a 4th trial for each N runs where instead of looking at the nine nearest neighbors, we will look at step sizes / offsets of 1 and 2.

In each trial, we will get the results of the Monte Carlo runs and compare them to the SLLN approach as well as what we expect from the Metropolic algorithm.

The neighbors of state (2, 49) are [(1, 50), (2, 50), (3, 50), (1, 49), (2, 49), (3, 49), (1, 48), (2, 48), (3, 48)]

The neighbors of state (3, 50) are [(2, 51), (3, 51), (4, 51), (2, 50), (3, 50), (4, 50), (2, 49), (3, 49), (4, 49)]

The neighbors of state (4, 50) are [(3, 51), (4, 51), (5, 51), (3, 50), (4, 50), (5, 50), (3, 49), (4, 49), (5, 49)]

The neighbors of state (5, 50) are [(4, 51), (5, 51), (6, 51), (4, 50), (5, 50), (6, 50), (4, 49), (5, 49), (6, 49)]

The neighbors of state (6, 50) are [(5, 51), (6, 51), (7, 51), (5, 50), (6, 50), (7, 50), (5, 49), (6, 49), (7, 49)]

The neighbors of state (7, 50) are [(6, 51), (7, 51), (8, 51), (6, 50), (7, 50), (8, 50), (6, 49), (7, 49), (8, 49)]

The neighbors of state (8, 50) are [(7, 51), (8, 51), (9, 51), (7, 50), (8, 50), (9, 50), (7, 49), (8, 49), (9, 49)]

The neighbors of state (9, 50) are [(8, 51), (9, 51), (10, 51), (8, 50), (9, 50), (10, 50), (8, 49), (9, 49), (10, 49)]

The neighbors of state (10, 50) are [(9, 51), (10, 51), (11, 51), (9, 50), (10, 50), (11, 50), (9, 49), (10, 49), (11, 49)]

The neighbors of state (11, 50) are [(10, 51), (11, 51), (12, 51), (10, 50), (11, 50), (12, 50), (10, 49), (11, 49), (12, 49)]

The neighbors of state (12, 50) are [(11, 51), (12, 51), (13, 51), (11, 50), (12, 50), (13, 50), (11, 49), (12, 49), (13, 49)]

The neighbors of state (13, 50) are [(12, 51), (13, 51), (14, 51), (12, 50), (13, 50), (14, 50), (12, 49), (13, 49), (14, 49)]

The neighbors of state (14, 50) are [(13, 51), (14, 51), (15, 51), (13, 50), (14, 50), (15, 50), (13, 49), (14, 49), (15, 49)]

The neighbors of state (15, 50) are [(14, 51), (15, 51), (16, 51), (14, 50), (15, 50), (16, 50), (14, 49), (15, 49), (16, 49)]

The neighbors of state (16, 50) are [(15, 51), (16, 51), (17, 51), (15, 50), (16, 50), (17, 50), (15, 49), (16, 49), (17, 49)]

The neighbors of state (17, 50) are [(16, 51), (17, 51), (18, 51), (16, 50), (17, 50), (18, 50), (16, 49), (17, 49), (18, 49)]

The neighbors of state (18, 50) are [(17, 51), (18, 51), (19, 51), (17, 50), (18, 50), (19, 50), (17, 49), (18, 49), (19, 49)]

The neighbors of state (19, 50) are [(18, 51), (19, 51), (20, 51), (18, 50), (19, 50), (20, 50), (18, 49), (19, 49), (20, 49)]

The neighbors of state (20, 50) are [(19, 51), (20, 51), (21, 51), (19, 50), (20, 50), (21, 50), (19, 49), (20, 49), (21, 49)]

The neighbors of state (21, 50) are [(20, 51), (21, 51), (22, 51), (20, 50), (21, 50), (22, 50), (20, 49), (21, 49), (22, 49)]

The neighbors of state (22, 50) are [(21, 51), (22, 51), (23, 51), (21, 50), (22, 50), (23, 50), (21, 49), (22, 49), (23, 49)]

The neighbors of state (23, 50) are [(22, 51), (23, 51), (24, 51), (22, 50), (23, 50), (24, 50), (22, 49), (23, 49), (24, 49)]

The neighbors of state (24, 50) are [(23, 51), (24, 51), (25, 51), (23, 50), (24, 50), (25, 50), (23, 49), (24, 49), (25, 49)]

The neighbors of state (25, 50) are [(24, 51), (25, 51), (26, 51), (24, 50), (25, 50), (26, 50), (24, 49), (25, 49), (26, 49)]

The neighbors of state (26, 50) are [(25, 51), (26, 51), (27, 51), (25, 50), (26, 50), (27, 50), (25, 49), (26, 49), (27, 49)]

The neighbors of state (27, 50) are [(26, 51), (27, 51), (28, 51), (26, 50), (27, 50), (28, 50), (26, 49), (27, 49), (28, 49)]

The neighbors of state (28, 50) are [(27, 51), (28, 51), (29, 51), (27, 50), (28, 50), (29, 50), (27, 49), (28, 49), (29, 49)]

The neighbors of state (29, 50) are [(28, 51), (29, 51), (30, 51), (28, 50), (29, 50), (30, 50), (28, 49), (29, 49), (30, 49)]

The neighbors of state (30, 50) are [(29, 51), (30, 51), (31, 51), (29, 50), (30, 50), (31, 50), (29, 49), (30, 49), (31, 49)]

The neighbors of state (31, 50) are [(30, 51), (31, 51), (32, 51), (30, 50), (31, 50), (32, 50), (30, 49), (31, 49), (32, 49)]

The neighbors of state (32, 50) are [(31, 51), (32, 51), (33, 51), (31, 50), (32, 50), (33, 50), (31, 49), (32, 49), (33, 49)]

The neighbors of state (33, 50) are [(32, 51), (33, 51), (34, 51), (32, 50), (33, 50), (34, 50), (32, 49), (33, 49), (34, 49)]

The neighbors of state (34, 50) are [(33, 51), (34, 51), (35, 51), (33, 50), (34, 50), (35, 50), (33, 49), (34, 49), (35, 49)]

The neighbors of state (35, 50) are [(34, 51), (35, 51), (36, 51), (34, 50), (35, 50), (36, 50), (34, 49), (35, 49), (36, 49)]

The neighbors of state (36, 50) are [(35, 51), (36, 51), (37, 51), (35, 50), (36, 50), (37, 50), (35, 49), (36, 49), (37, 49)]

The neighbors of state (37, 50) are [(36, 51), (37, 51), (38, 51), (36, 50), (37, 50), (38, 50), (36, 49), (37, 49), (38, 49)]

The neighbors of state (38, 50) are [(37, 51), (38, 51), (39, 51), (37, 50), (38, 50), (39, 50), (37, 49), (38, 49), (39, 49)]

The neighbors of state (39, 50) are [(38, 51), (39, 51), (40, 51), (38, 50), (39, 50), (40, 50), (38, 49), (39, 49), (40, 49)]

The neighbors of state (40, 50) are [(39, 51), (40, 51), (41, 51), (39, 50), (40, 50), (41, 50), (39, 49), (40, 49), (41, 49)]

The neighbors of state (41, 50) are [(40, 51), (41, 51), (42, 51), (40, 50), (41, 50), (42, 50), (40, 49), (41, 49), (42, 49)]

The neighbors of state (42, 50) are [(41, 51), (42, 51), (43, 51), (41, 50), (42, 50), (43, 50), (41, 49), (42, 49), (43, 49)]

The neighbors of state (43, 50) are [(42, 51), (43, 51), (44, 51), (42, 50), (43, 50), (44, 50), (42, 49), (43, 49), (44, 49)]

The neighbors of state (44, 50) are [(43, 51), (44, 51), (45, 51), (43, 50), (44, 50), (45, 50), (43, 49), (44, 49), (45, 49)]

The neighbors of state (45, 50) are [(44, 51), (45, 51), (46, 51), (44, 50), (45, 50), (46, 50), (44, 49), (45, 49), (46, 49)]

The neighbors of state (46, 50) are [(45, 51), (46, 51), (47, 51), (45, 50), (46, 50), (47, 50), (45, 49), (46, 49), (47, 49)]

The neighbors of state (47, 50) are [(46, 51), (47, 51), (48, 51), (46, 50), (47, 50), (48, 50), (46, 49), (47, 49), (48, 49)]

The neighbors of state (48, 50) are [(47, 51), (48, 51), (49, 51), (47, 50), (48, 50), (49, 50), (47, 49), (48, 49), (49, 49)]

The neighbors of state (49, 50) are [(48, 51), (49, 51), (50, 51), (48, 50), (49, 50), (50, 50), (48, 49), (49, 49), (50, 49)]

The neighbors of state (50, 50) are [(49, 51), (50, 51), (51, 51), (49, 50), (50, 50), (51, 50), (49, 49), (50, 49), (51, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 76.46

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

Compared to the SLLN method's computed time average, the Monte Carlo method yielded a larger time average.

*Time of program run = 22.62 seconds*

The neighbors of state (74, 88) are [(73, 89), (74, 89), (75, 89), (73, 88), (74, 88), (75, 88), (73, 87), (74, 87), (75, 87)]

The neighbors of state (73, 87) are [(72, 88), (73, 88), (74, 88), (72, 87), (73, 87), (74, 87), (72, 86), (73, 86), (74, 86)]

The neighbors of state (72, 86) are [(71, 87), (72, 87), (73, 87), (71, 86), (72, 86), (73, 86), (71, 85), (72, 85), (73, 85)]

The neighbors of state (71, 85) are [(70, 86), (71, 86), (72, 86), (70, 85), (71, 85), (72, 85), (70, 84), (71, 84), (72, 84)]

The neighbors of state (70, 84) are [(69, 85), (70, 85), (71, 85), (69, 84), (70, 84), (71, 84), (69, 83), (70, 83), (71, 83)]

The neighbors of state (69, 83) are [(68, 84), (69, 84), (70, 84), (68, 83), (69, 83), (70, 83), (68, 82), (69, 82), (70, 82)]

The neighbors of state (68, 82) are [(67, 83), (68, 83), (69, 83), (67, 82), (68, 82), (69, 82), (67, 81), (68, 81), (69, 81)]

The neighbors of state (67, 81) are [(66, 82), (67, 82), (68, 82), (66, 81), (67, 81), (68, 81), (66, 80), (67, 80), (68, 80)]

The neighbors of state (66, 80) are [(65, 81), (66, 81), (67, 81), (65, 80), (66, 80), (67, 80), (65, 79), (66, 79), (67, 79)]

The neighbors of state (65, 79) are [(64, 80), (65, 80), (66, 80), (64, 79), (65, 79), (66, 79), (64, 78), (65, 78), (66, 78)]

The neighbors of state (64, 78) are [(63, 79), (64, 79), (65, 79), (63, 78), (64, 78), (65, 78), (63, 77), (64, 77), (65, 77)]

The neighbors of state (63, 77) are [(62, 78), (63, 78), (64, 78), (62, 77), (63, 77), (64, 77), (62, 76), (63, 76), (64, 76)]

The neighbors of state (62, 76) are [(61, 77), (62, 77), (63, 77), (61, 76), (62, 76), (63, 76), (61, 75), (62, 75), (63, 75)]

The neighbors of state (61, 75) are [(60, 76), (61, 76), (62, 76), (60, 75), (61, 75), (62, 75), (60, 74), (61, 74), (62, 74)]

The neighbors of state (60, 74) are [(59, 75), (60, 75), (61, 75), (59, 74), (60, 74), (61, 74), (59, 73), (60, 73), (61, 73)]

The neighbors of state (59, 73) are [(58, 74), (59, 74), (60, 74), (58, 73), (59, 73), (60, 73), (58, 72), (59, 72), (60, 72)]

The neighbors of state (58, 72) are [(57, 73), (58, 73), (59, 73), (57, 72), (58, 72), (59, 72), (57, 71), (58, 71), (59, 71)]

The neighbors of state (57, 71) are [(56, 72), (57, 72), (58, 72), (56, 71), (57, 71), (58, 71), (56, 70), (57, 70), (58, 70)]

The neighbors of state (56, 70) are [(55, 71), (56, 71), (57, 71), (55, 70), (56, 70), (57, 70), (55, 69), (56, 69), (57, 69)]

The neighbors of state (55, 69) are [(54, 70), (55, 70), (56, 70), (54, 69), (55, 69), (56, 69), (54, 68), (55, 68), (56, 68)]

The neighbors of state (54, 68) are [(53, 69), (54, 69), (55, 69), (53, 68), (54, 68), (55, 68), (53, 67), (54, 67), (55, 67)]

The neighbors of state (53, 67) are [(52, 68), (53, 68), (54, 68), (52, 67), (53, 67), (54, 67), (52, 66), (53, 66), (54, 66)]

The neighbors of state (52, 66) are [(51, 67), (52, 67), (53, 67), (51, 66), (52, 66), (53, 66), (51, 65), (52, 65), (53, 65)]

The neighbors of state (51, 65) are [(50, 66), (51, 66), (52, 66), (50, 65), (51, 65), (52, 65), (50, 64), (51, 64), (52, 64)]

The neighbors of state (50, 64) are [(49, 65), (50, 65), (51, 65), (49, 64), (50, 64), (51, 64), (49, 63), (50, 63), (51, 63)]

The neighbors of state (50, 63) are [(49, 64), (50, 64), (51, 64), (49, 63), (50, 63), (51, 63), (49, 62), (50, 62), (51, 62)]

The neighbors of state (50, 62) are [(49, 63), (50, 63), (51, 63), (49, 62), (50, 62), (51, 62), (49, 61), (50, 61), (51, 61)]

The neighbors of state (50, 61) are [(49, 62), (50, 62), (51, 62), (49, 61), (50, 61), (51, 61), (49, 60), (50, 60), (51, 60)]

The neighbors of state (50, 60) are [(49, 61), (50, 61), (51, 61), (49, 60), (50, 60), (51, 60), (49, 59), (50, 59), (51, 59)]

The neighbors of state (50, 59) are [(49, 60), (50, 60), (51, 60), (49, 59), (50, 59), (51, 59), (49, 58), (50, 58), (51, 58)]

The neighbors of state (50, 58) are [(49, 59), (50, 59), (51, 59), (49, 58), (50, 58), (51, 58), (49, 57), (50, 57), (51, 57)]

The neighbors of state (50, 57) are [(49, 58), (50, 58), (51, 58), (49, 57), (50, 57), (51, 57), (49, 56), (50, 56), (51, 56)]

The neighbors of state (50, 56) are [(49, 57), (50, 57), (51, 57), (49, 56), (50, 56), (51, 56), (49, 55), (50, 55), (51, 55)]

The neighbors of state (50, 55) are [(49, 56), (50, 56), (51, 56), (49, 55), (50, 55), (51, 55), (49, 54), (50, 54), (51, 54)]

The neighbors of state (50, 54) are [(49, 55), (50, 55), (51, 55), (49, 54), (50, 54), (51, 54), (49, 53), (50, 53), (51, 53)]

The neighbors of state (50, 53) are [(49, 54), (50, 54), (51, 54), (49, 53), (50, 53), (51, 53), (49, 52), (50, 52), (51, 52)]

The neighbors of state (50, 52) are [(49, 53), (50, 53), (51, 53), (49, 52), (50, 52), (51, 52), (49, 51), (50, 51), (51, 51)]

The neighbors of state (50, 51) are [(49, 52), (50, 52), (51, 52), (49, 51), (50, 51), (51, 51), (49, 50), (50, 50), (51, 50)]

The neighbors of state (50, 50) are [(49, 51), (50, 51), (51, 51), (49, 50), (50, 50), (51, 50), (49, 49), (50, 49), (51, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 126.025

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

Compared to the SLLN method's computed time average, the Monte Carlo method yielded a larger time average.

*Time of program run = 23.01 seconds*

The neighbors of state (70, 70) are [(69, 71), (70, 71), (71, 71), (69, 70), (70, 70), (71, 70), (69, 69), (70, 69), (71, 69)]

The neighbors of state (69, 69) are [(68, 70), (69, 70), (70, 70), (68, 69), (69, 69), (70, 69), (68, 68), (69, 68), (70, 68)]

The neighbors of state (68, 68) are [(67, 69), (68, 69), (69, 69), (67, 68), (68, 68), (69, 68), (67, 67), (68, 67), (69, 67)]

The neighbors of state (67, 67) are [(66, 68), (67, 68), (68, 68), (66, 67), (67, 67), (68, 67), (66, 66), (67, 66), (68, 66)]

The neighbors of state (66, 66) are [(65, 67), (66, 67), (67, 67), (65, 66), (66, 66), (67, 66), (65, 65), (66, 65), (67, 65)]

The neighbors of state (65, 65) are [(64, 66), (65, 66), (66, 66), (64, 65), (65, 65), (66, 65), (64, 64), (65, 64), (66, 64)]

The neighbors of state (64, 64) are [(63, 65), (64, 65), (65, 65), (63, 64), (64, 64), (65, 64), (63, 63), (64, 63), (65, 63)]

The neighbors of state (63, 63) are [(62, 64), (63, 64), (64, 64), (62, 63), (63, 63), (64, 63), (62, 62), (63, 62), (64, 62)]

The neighbors of state (62, 62) are [(61, 63), (62, 63), (63, 63), (61, 62), (62, 62), (63, 62), (61, 61), (62, 61), (63, 61)]

The neighbors of state (61, 61) are [(60, 62), (61, 62), (62, 62), (60, 61), (61, 61), (62, 61), (60, 60), (61, 60), (62, 60)]

The neighbors of state (60, 60) are [(59, 61), (60, 61), (61, 61), (59, 60), (60, 60), (61, 60), (59, 59), (60, 59), (61, 59)]

The neighbors of state (59, 59) are [(58, 60), (59, 60), (60, 60), (58, 59), (59, 59), (60, 59), (58, 58), (59, 58), (60, 58)]

The neighbors of state (58, 58) are [(57, 59), (58, 59), (59, 59), (57, 58), (58, 58), (59, 58), (57, 57), (58, 57), (59, 57)]

The neighbors of state (57, 57) are [(56, 58), (57, 58), (58, 58), (56, 57), (57, 57), (58, 57), (56, 56), (57, 56), (58, 56)]

The neighbors of state (56, 56) are [(55, 57), (56, 57), (57, 57), (55, 56), (56, 56), (57, 56), (55, 55), (56, 55), (57, 55)]

The neighbors of state (55, 55) are [(54, 56), (55, 56), (56, 56), (54, 55), (55, 55), (56, 55), (54, 54), (55, 54), (56, 54)]

The neighbors of state (54, 54) are [(53, 55), (54, 55), (55, 55), (53, 54), (54, 54), (55, 54), (53, 53), (54, 53), (55, 53)]

The neighbors of state (53, 53) are [(52, 54), (53, 54), (54, 54), (52, 53), (53, 53), (54, 53), (52, 52), (53, 52), (54, 52)]

The neighbors of state (52, 52) are [(51, 53), (52, 53), (53, 53), (51, 52), (52, 52), (53, 52), (51, 51), (52, 51), (53, 51)]

The neighbors of state (51, 51) are [(50, 52), (51, 52), (52, 52), (50, 51), (51, 51), (52, 51), (50, 50), (51, 50), (52, 50)]

The neighbors of state (50, 50) are [(49, 51), (50, 51), (51, 51), (49, 50), (50, 50), (51, 50), (49, 49), (50, 49), (51, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 120.0

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

Compared to the SLLN method's computed time average, the Monte Carlo method yielded a larger time average.

*Time of program run = 22.71 seconds*

The neighbors of state (67, 67) are [(65, 69), (67, 69), (69, 69), (65, 67), (67, 67), (69, 67), (65, 65), (67, 65), (69, 65)]

The neighbors of state (65, 65) are [(63, 67), (65, 67), (67, 67), (63, 65), (65, 65), (67, 65), (63, 63), (65, 63), (67, 63)]

The neighbors of state (63, 63) are [(61, 65), (63, 65), (65, 65), (61, 63), (63, 63), (65, 63), (61, 61), (63, 61), (65, 61)]

The neighbors of state (61, 61) are [(59, 63), (61, 63), (63, 63), (59, 61), (61, 61), (63, 61), (59, 59), (61, 59), (63, 59)]

The neighbors of state (59, 59) are [(57, 61), (59, 61), (61, 61), (57, 59), (59, 59), (61, 59), (57, 57), (59, 57), (61, 57)]

The neighbors of state (57, 57) are [(55, 59), (57, 59), (59, 59), (55, 57), (57, 57), (59, 57), (55, 55), (57, 55), (59, 55)]

The neighbors of state (55, 55) are [(53, 57), (55, 57), (57, 57), (53, 55), (55, 55), (57, 55), (53, 53), (55, 53), (57, 53)]

The neighbors of state (53, 53) are [(51, 55), (53, 55), (55, 55), (51, 53), (53, 53), (55, 53), (51, 51), (53, 51), (55, 51)]

The neighbors of state (51, 51) are [(49, 53), (51, 53), (53, 53), (49, 51), (51, 51), (53, 51), (49, 49), (51, 49), (53, 49)]

The neighbors of state (51, 49) are [(49, 51), (51, 51), (53, 51), (49, 49), (51, 49), (53, 49), (49, 47), (51, 47), (53, 47)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 76.46

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 22.92 seconds*

The neighbors of state (75, 25) are [(74, 26), (75, 26), (76, 26), (74, 25), (75, 25), (76, 25), (74, 24), (75, 24), (76, 24)]

The neighbors of state (74, 26) are [(73, 27), (74, 27), (75, 27), (73, 26), (74, 26), (75, 26), (73, 25), (74, 25), (75, 25)]

The neighbors of state (73, 27) are [(72, 28), (73, 28), (74, 28), (72, 27), (73, 27), (74, 27), (72, 26), (73, 26), (74, 26)]

The neighbors of state (72, 28) are [(71, 29), (72, 29), (73, 29), (71, 28), (72, 28), (73, 28), (71, 27), (72, 27), (73, 27)]

The neighbors of state (71, 29) are [(70, 30), (71, 30), (72, 30), (70, 29), (71, 29), (72, 29), (70, 28), (71, 28), (72, 28)]

The neighbors of state (70, 30) are [(69, 31), (70, 31), (71, 31), (69, 30), (70, 30), (71, 30), (69, 29), (70, 29), (71, 29)]

The neighbors of state (69, 31) are [(68, 32), (69, 32), (70, 32), (68, 31), (69, 31), (70, 31), (68, 30), (69, 30), (70, 30)]

The neighbors of state (68, 32) are [(67, 33), (68, 33), (69, 33), (67, 32), (68, 32), (69, 32), (67, 31), (68, 31), (69, 31)]

The neighbors of state (67, 33) are [(66, 34), (67, 34), (68, 34), (66, 33), (67, 33), (68, 33), (66, 32), (67, 32), (68, 32)]

The neighbors of state (66, 34) are [(65, 35), (66, 35), (67, 35), (65, 34), (66, 34), (67, 34), (65, 33), (66, 33), (67, 33)]

The neighbors of state (65, 35) are [(64, 36), (65, 36), (66, 36), (64, 35), (65, 35), (66, 35), (64, 34), (65, 34), (66, 34)]

The neighbors of state (64, 36) are [(63, 37), (64, 37), (65, 37), (63, 36), (64, 36), (65, 36), (63, 35), (64, 35), (65, 35)]

The neighbors of state (63, 37) are [(62, 38), (63, 38), (64, 38), (62, 37), (63, 37), (64, 37), (62, 36), (63, 36), (64, 36)]

The neighbors of state (62, 38) are [(61, 39), (62, 39), (63, 39), (61, 38), (62, 38), (63, 38), (61, 37), (62, 37), (63, 37)]

The neighbors of state (61, 39) are [(60, 40), (61, 40), (62, 40), (60, 39), (61, 39), (62, 39), (60, 38), (61, 38), (62, 38)]

The neighbors of state (60, 40) are [(59, 41), (60, 41), (61, 41), (59, 40), (60, 40), (61, 40), (59, 39), (60, 39), (61, 39)]

The neighbors of state (59, 41) are [(58, 42), (59, 42), (60, 42), (58, 41), (59, 41), (60, 41), (58, 40), (59, 40), (60, 40)]

The neighbors of state (58, 42) are [(57, 43), (58, 43), (59, 43), (57, 42), (58, 42), (59, 42), (57, 41), (58, 41), (59, 41)]

The neighbors of state (57, 43) are [(56, 44), (57, 44), (58, 44), (56, 43), (57, 43), (58, 43), (56, 42), (57, 42), (58, 42)]

The neighbors of state (56, 44) are [(55, 45), (56, 45), (57, 45), (55, 44), (56, 44), (57, 44), (55, 43), (56, 43), (57, 43)]

The neighbors of state (55, 45) are [(54, 46), (55, 46), (56, 46), (54, 45), (55, 45), (56, 45), (54, 44), (55, 44), (56, 44)]

The neighbors of state (54, 46) are [(53, 47), (54, 47), (55, 47), (53, 46), (54, 46), (55, 46), (53, 45), (54, 45), (55, 45)]

The neighbors of state (53, 47) are [(52, 48), (53, 48), (54, 48), (52, 47), (53, 47), (54, 47), (52, 46), (53, 46), (54, 46)]

The neighbors of state (52, 48) are [(51, 49), (52, 49), (53, 49), (51, 48), (52, 48), (53, 48), (51, 47), (52, 47), (53, 47)]

The neighbors of state (51, 49) are [(50, 50), (51, 50), (52, 50), (50, 49), (51, 49), (52, 49), (50, 48), (51, 48), (52, 48)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 100.0

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 111.58 seconds*

The neighbors of state (24, 90) are [(23, 91), (24, 91), (25, 91), (23, 90), (24, 90), (25, 90), (23, 89), (24, 89), (25, 89)]

The neighbors of state (25, 89) are [(24, 90), (25, 90), (26, 90), (24, 89), (25, 89), (26, 89), (24, 88), (25, 88), (26, 88)]

The neighbors of state (26, 88) are [(25, 89), (26, 89), (27, 89), (25, 88), (26, 88), (27, 88), (25, 87), (26, 87), (27, 87)]

The neighbors of state (27, 87) are [(26, 88), (27, 88), (28, 88), (26, 87), (27, 87), (28, 87), (26, 86), (27, 86), (28, 86)]

The neighbors of state (28, 86) are [(27, 87), (28, 87), (29, 87), (27, 86), (28, 86), (29, 86), (27, 85), (28, 85), (29, 85)]

The neighbors of state (29, 85) are [(28, 86), (29, 86), (30, 86), (28, 85), (29, 85), (30, 85), (28, 84), (29, 84), (30, 84)]

The neighbors of state (30, 84) are [(29, 85), (30, 85), (31, 85), (29, 84), (30, 84), (31, 84), (29, 83), (30, 83), (31, 83)]

The neighbors of state (31, 83) are [(30, 84), (31, 84), (32, 84), (30, 83), (31, 83), (32, 83), (30, 82), (31, 82), (32, 82)]

The neighbors of state (32, 82) are [(31, 83), (32, 83), (33, 83), (31, 82), (32, 82), (33, 82), (31, 81), (32, 81), (33, 81)]

The neighbors of state (33, 81) are [(32, 82), (33, 82), (34, 82), (32, 81), (33, 81), (34, 81), (32, 80), (33, 80), (34, 80)]

The neighbors of state (34, 80) are [(33, 81), (34, 81), (35, 81), (33, 80), (34, 80), (35, 80), (33, 79), (34, 79), (35, 79)]

The neighbors of state (35, 79) are [(34, 80), (35, 80), (36, 80), (34, 79), (35, 79), (36, 79), (34, 78), (35, 78), (36, 78)]

The neighbors of state (36, 78) are [(35, 79), (36, 79), (37, 79), (35, 78), (36, 78), (37, 78), (35, 77), (36, 77), (37, 77)]

The neighbors of state (37, 77) are [(36, 78), (37, 78), (38, 78), (36, 77), (37, 77), (38, 77), (36, 76), (37, 76), (38, 76)]

The neighbors of state (38, 76) are [(37, 77), (38, 77), (39, 77), (37, 76), (38, 76), (39, 76), (37, 75), (38, 75), (39, 75)]

The neighbors of state (39, 75) are [(38, 76), (39, 76), (40, 76), (38, 75), (39, 75), (40, 75), (38, 74), (39, 74), (40, 74)]

The neighbors of state (40, 74) are [(39, 75), (40, 75), (41, 75), (39, 74), (40, 74), (41, 74), (39, 73), (40, 73), (41, 73)]

The neighbors of state (41, 73) are [(40, 74), (41, 74), (42, 74), (40, 73), (41, 73), (42, 73), (40, 72), (41, 72), (42, 72)]

The neighbors of state (42, 72) are [(41, 73), (42, 73), (43, 73), (41, 72), (42, 72), (43, 72), (41, 71), (42, 71), (43, 71)]

The neighbors of state (43, 71) are [(42, 72), (43, 72), (44, 72), (42, 71), (43, 71), (44, 71), (42, 70), (43, 70), (44, 70)]

The neighbors of state (44, 70) are [(43, 71), (44, 71), (45, 71), (43, 70), (44, 70), (45, 70), (43, 69), (44, 69), (45, 69)]

The neighbors of state (45, 69) are [(44, 70), (45, 70), (46, 70), (44, 69), (45, 69), (46, 69), (44, 68), (45, 68), (46, 68)]

The neighbors of state (46, 68) are [(45, 69), (46, 69), (47, 69), (45, 68), (46, 68), (47, 68), (45, 67), (46, 67), (47, 67)]

The neighbors of state (47, 67) are [(46, 68), (47, 68), (48, 68), (46, 67), (47, 67), (48, 67), (46, 66), (47, 66), (48, 66)]

The neighbors of state (48, 66) are [(47, 67), (48, 67), (49, 67), (47, 66), (48, 66), (49, 66), (47, 65), (48, 65), (49, 65)]

The neighbors of state (49, 65) are [(48, 66), (49, 66), (50, 66), (48, 65), (49, 65), (50, 65), (48, 64), (49, 64), (50, 64)]

The neighbors of state (50, 64) are [(49, 65), (50, 65), (51, 65), (49, 64), (50, 64), (51, 64), (49, 63), (50, 63), (51, 63)]

The neighbors of state (50, 63) are [(49, 64), (50, 64), (51, 64), (49, 63), (50, 63), (51, 63), (49, 62), (50, 62), (51, 62)]

The neighbors of state (50, 62) are [(49, 63), (50, 63), (51, 63), (49, 62), (50, 62), (51, 62), (49, 61), (50, 61), (51, 61)]

The neighbors of state (50, 61) are [(49, 62), (50, 62), (51, 62), (49, 61), (50, 61), (51, 61), (49, 60), (50, 60), (51, 60)]

The neighbors of state (50, 60) are [(49, 61), (50, 61), (51, 61), (49, 60), (50, 60), (51, 60), (49, 59), (50, 59), (51, 59)]

The neighbors of state (50, 59) are [(49, 60), (50, 60), (51, 60), (49, 59), (50, 59), (51, 59), (49, 58), (50, 58), (51, 58)]

The neighbors of state (50, 58) are [(49, 59), (50, 59), (51, 59), (49, 58), (50, 58), (51, 58), (49, 57), (50, 57), (51, 57)]

The neighbors of state (50, 57) are [(49, 58), (50, 58), (51, 58), (49, 57), (50, 57), (51, 57), (49, 56), (50, 56), (51, 56)]

The neighbors of state (50, 56) are [(49, 57), (50, 57), (51, 57), (49, 56), (50, 56), (51, 56), (49, 55), (50, 55), (51, 55)]

The neighbors of state (50, 55) are [(49, 56), (50, 56), (51, 56), (49, 55), (50, 55), (51, 55), (49, 54), (50, 54), (51, 54)]

The neighbors of state (50, 54) are [(49, 55), (50, 55), (51, 55), (49, 54), (50, 54), (51, 54), (49, 53), (50, 53), (51, 53)]

The neighbors of state (50, 53) are [(49, 54), (50, 54), (51, 54), (49, 53), (50, 53), (51, 53), (49, 52), (50, 52), (51, 52)]

The neighbors of state (50, 52) are [(49, 53), (50, 53), (51, 53), (49, 52), (50, 52), (51, 52), (49, 51), (50, 51), (51, 51)]

The neighbors of state (50, 51) are [(49, 52), (50, 52), (51, 52), (49, 51), (50, 51), (51, 51), (49, 50), (50, 50), (51, 50)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 111.16667

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 110.2 seconds*

The neighbors of state (82, 22) are [(81, 23), (82, 23), (83, 23), (81, 22), (82, 22), (83, 22), (81, 21), (82, 21), (83, 21)]

The neighbors of state (81, 23) are [(80, 24), (81, 24), (82, 24), (80, 23), (81, 23), (82, 23), (80, 22), (81, 22), (82, 22)]

The neighbors of state (80, 24) are [(79, 25), (80, 25), (81, 25), (79, 24), (80, 24), (81, 24), (79, 23), (80, 23), (81, 23)]

The neighbors of state (79, 25) are [(78, 26), (79, 26), (80, 26), (78, 25), (79, 25), (80, 25), (78, 24), (79, 24), (80, 24)]

The neighbors of state (78, 26) are [(77, 27), (78, 27), (79, 27), (77, 26), (78, 26), (79, 26), (77, 25), (78, 25), (79, 25)]

The neighbors of state (77, 27) are [(76, 28), (77, 28), (78, 28), (76, 27), (77, 27), (78, 27), (76, 26), (77, 26), (78, 26)]

The neighbors of state (76, 28) are [(75, 29), (76, 29), (77, 29), (75, 28), (76, 28), (77, 28), (75, 27), (76, 27), (77, 27)]

The neighbors of state (75, 29) are [(74, 30), (75, 30), (76, 30), (74, 29), (75, 29), (76, 29), (74, 28), (75, 28), (76, 28)]

The neighbors of state (74, 30) are [(73, 31), (74, 31), (75, 31), (73, 30), (74, 30), (75, 30), (73, 29), (74, 29), (75, 29)]

The neighbors of state (73, 31) are [(72, 32), (73, 32), (74, 32), (72, 31), (73, 31), (74, 31), (72, 30), (73, 30), (74, 30)]

The neighbors of state (72, 32) are [(71, 33), (72, 33), (73, 33), (71, 32), (72, 32), (73, 32), (71, 31), (72, 31), (73, 31)]

The neighbors of state (71, 33) are [(70, 34), (71, 34), (72, 34), (70, 33), (71, 33), (72, 33), (70, 32), (71, 32), (72, 32)]

The neighbors of state (70, 34) are [(69, 35), (70, 35), (71, 35), (69, 34), (70, 34), (71, 34), (69, 33), (70, 33), (71, 33)]

The neighbors of state (69, 35) are [(68, 36), (69, 36), (70, 36), (68, 35), (69, 35), (70, 35), (68, 34), (69, 34), (70, 34)]

The neighbors of state (68, 36) are [(67, 37), (68, 37), (69, 37), (67, 36), (68, 36), (69, 36), (67, 35), (68, 35), (69, 35)]

The neighbors of state (67, 37) are [(66, 38), (67, 38), (68, 38), (66, 37), (67, 37), (68, 37), (66, 36), (67, 36), (68, 36)]

The neighbors of state (66, 38) are [(65, 39), (66, 39), (67, 39), (65, 38), (66, 38), (67, 38), (65, 37), (66, 37), (67, 37)]

The neighbors of state (65, 39) are [(64, 40), (65, 40), (66, 40), (64, 39), (65, 39), (66, 39), (64, 38), (65, 38), (66, 38)]

The neighbors of state (64, 40) are [(63, 41), (64, 41), (65, 41), (63, 40), (64, 40), (65, 40), (63, 39), (64, 39), (65, 39)]

The neighbors of state (63, 41) are [(62, 42), (63, 42), (64, 42), (62, 41), (63, 41), (64, 41), (62, 40), (63, 40), (64, 40)]

The neighbors of state (62, 42) are [(61, 43), (62, 43), (63, 43), (61, 42), (62, 42), (63, 42), (61, 41), (62, 41), (63, 41)]

The neighbors of state (61, 43) are [(60, 44), (61, 44), (62, 44), (60, 43), (61, 43), (62, 43), (60, 42), (61, 42), (62, 42)]

The neighbors of state (60, 44) are [(59, 45), (60, 45), (61, 45), (59, 44), (60, 44), (61, 44), (59, 43), (60, 43), (61, 43)]

The neighbors of state (59, 45) are [(58, 46), (59, 46), (60, 46), (58, 45), (59, 45), (60, 45), (58, 44), (59, 44), (60, 44)]

The neighbors of state (58, 46) are [(57, 47), (58, 47), (59, 47), (57, 46), (58, 46), (59, 46), (57, 45), (58, 45), (59, 45)]

The neighbors of state (57, 47) are [(56, 48), (57, 48), (58, 48), (56, 47), (57, 47), (58, 47), (56, 46), (57, 46), (58, 46)]

The neighbors of state (56, 48) are [(55, 49), (56, 49), (57, 49), (55, 48), (56, 48), (57, 48), (55, 47), (56, 47), (57, 47)]

The neighbors of state (55, 49) are [(54, 50), (55, 50), (56, 50), (54, 49), (55, 49), (56, 49), (54, 48), (55, 48), (56, 48)]

The neighbors of state (54, 50) are [(53, 51), (54, 51), (55, 51), (53, 50), (54, 50), (55, 50), (53, 49), (54, 49), (55, 49)]

The neighbors of state (53, 50) are [(52, 51), (53, 51), (54, 51), (52, 50), (53, 50), (54, 50), (52, 49), (53, 49), (54, 49)]

The neighbors of state (52, 50) are [(51, 51), (52, 51), (53, 51), (51, 50), (52, 50), (53, 50), (51, 49), (52, 49), (53, 49)]

The neighbors of state (51, 50) are [(50, 51), (51, 51), (52, 51), (50, 50), (51, 50), (52, 50), (50, 49), (51, 49), (52, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 103.58824

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 113.29 seconds*

The neighbors of state (53, 86) are [(51, 88), (53, 88), (55, 88), (51, 86), (53, 86), (55, 86), (51, 84), (53, 84), (55, 84)]

The neighbors of state (51, 84) are [(49, 86), (51, 86), (53, 86), (49, 84), (51, 84), (53, 84), (49, 82), (51, 82), (53, 82)]

The neighbors of state (51, 82) are [(49, 84), (51, 84), (53, 84), (49, 82), (51, 82), (53, 82), (49, 80), (51, 80), (53, 80)]

The neighbors of state (51, 80) are [(49, 82), (51, 82), (53, 82), (49, 80), (51, 80), (53, 80), (49, 78), (51, 78), (53, 78)]

The neighbors of state (51, 78) are [(49, 80), (51, 80), (53, 80), (49, 78), (51, 78), (53, 78), (49, 76), (51, 76), (53, 76)]

The neighbors of state (51, 76) are [(49, 78), (51, 78), (53, 78), (49, 76), (51, 76), (53, 76), (49, 74), (51, 74), (53, 74)]

The neighbors of state (51, 74) are [(49, 76), (51, 76), (53, 76), (49, 74), (51, 74), (53, 74), (49, 72), (51, 72), (53, 72)]

The neighbors of state (51, 72) are [(49, 74), (51, 74), (53, 74), (49, 72), (51, 72), (53, 72), (49, 70), (51, 70), (53, 70)]

The neighbors of state (51, 70) are [(49, 72), (51, 72), (53, 72), (49, 70), (51, 70), (53, 70), (49, 68), (51, 68), (53, 68)]

The neighbors of state (51, 68) are [(49, 70), (51, 70), (53, 70), (49, 68), (51, 68), (53, 68), (49, 66), (51, 66), (53, 66)]

The neighbors of state (51, 66) are [(49, 68), (51, 68), (53, 68), (49, 66), (51, 66), (53, 66), (49, 64), (51, 64), (53, 64)]

The neighbors of state (51, 64) are [(49, 66), (51, 66), (53, 66), (49, 64), (51, 64), (53, 64), (49, 62), (51, 62), (53, 62)]

The neighbors of state (51, 62) are [(49, 64), (51, 64), (53, 64), (49, 62), (51, 62), (53, 62), (49, 60), (51, 60), (53, 60)]

The neighbors of state (51, 60) are [(49, 62), (51, 62), (53, 62), (49, 60), (51, 60), (53, 60), (49, 58), (51, 58), (53, 58)]

The neighbors of state (51, 58) are [(49, 60), (51, 60), (53, 60), (49, 58), (51, 58), (53, 58), (49, 56), (51, 56), (53, 56)]

The neighbors of state (51, 56) are [(49, 58), (51, 58), (53, 58), (49, 56), (51, 56), (53, 56), (49, 54), (51, 54), (53, 54)]

The neighbors of state (51, 54) are [(49, 56), (51, 56), (53, 56), (49, 54), (51, 54), (53, 54), (49, 52), (51, 52), (53, 52)]

The neighbors of state (51, 52) are [(49, 54), (51, 54), (53, 54), (49, 52), (51, 52), (53, 52), (49, 50), (51, 50), (53, 50)]

The neighbors of state (51, 50) are [(49, 52), (51, 52), (53, 52), (49, 50), (51, 50), (53, 50), (49, 48), (51, 48), (53, 48)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 118.2

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 113.0 seconds*

The neighbors of state (11, 5) are [(10, 6), (11, 6), (12, 6), (10, 5), (11, 5), (12, 5), (10, 4), (11, 4), (12, 4)]

The neighbors of state (12, 6) are [(11, 7), (12, 7), (13, 7), (11, 6), (12, 6), (13, 6), (11, 5), (12, 5), (13, 5)]

The neighbors of state (13, 7) are [(12, 8), (13, 8), (14, 8), (12, 7), (13, 7), (14, 7), (12, 6), (13, 6), (14, 6)]

The neighbors of state (14, 8) are [(13, 9), (14, 9), (15, 9), (13, 8), (14, 8), (15, 8), (13, 7), (14, 7), (15, 7)]

The neighbors of state (15, 9) are [(14, 10), (15, 10), (16, 10), (14, 9), (15, 9), (16, 9), (14, 8), (15, 8), (16, 8)]

The neighbors of state (16, 10) are [(15, 11), (16, 11), (17, 11), (15, 10), (16, 10), (17, 10), (15, 9), (16, 9), (17, 9)]

The neighbors of state (17, 11) are [(16, 12), (17, 12), (18, 12), (16, 11), (17, 11), (18, 11), (16, 10), (17, 10), (18, 10)]

The neighbors of state (18, 12) are [(17, 13), (18, 13), (19, 13), (17, 12), (18, 12), (19, 12), (17, 11), (18, 11), (19, 11)]

The neighbors of state (19, 13) are [(18, 14), (19, 14), (20, 14), (18, 13), (19, 13), (20, 13), (18, 12), (19, 12), (20, 12)]

The neighbors of state (20, 14) are [(19, 15), (20, 15), (21, 15), (19, 14), (20, 14), (21, 14), (19, 13), (20, 13), (21, 13)]

The neighbors of state (21, 15) are [(20, 16), (21, 16), (22, 16), (20, 15), (21, 15), (22, 15), (20, 14), (21, 14), (22, 14)]

The neighbors of state (22, 16) are [(21, 17), (22, 17), (23, 17), (21, 16), (22, 16), (23, 16), (21, 15), (22, 15), (23, 15)]

The neighbors of state (23, 17) are [(22, 18), (23, 18), (24, 18), (22, 17), (23, 17), (24, 17), (22, 16), (23, 16), (24, 16)]

The neighbors of state (24, 18) are [(23, 19), (24, 19), (25, 19), (23, 18), (24, 18), (25, 18), (23, 17), (24, 17), (25, 17)]

The neighbors of state (25, 19) are [(24, 20), (25, 20), (26, 20), (24, 19), (25, 19), (26, 19), (24, 18), (25, 18), (26, 18)]

The neighbors of state (26, 20) are [(25, 21), (26, 21), (27, 21), (25, 20), (26, 20), (27, 20), (25, 19), (26, 19), (27, 19)]

The neighbors of state (27, 21) are [(26, 22), (27, 22), (28, 22), (26, 21), (27, 21), (28, 21), (26, 20), (27, 20), (28, 20)]

The neighbors of state (28, 22) are [(27, 23), (28, 23), (29, 23), (27, 22), (28, 22), (29, 22), (27, 21), (28, 21), (29, 21)]

The neighbors of state (29, 23) are [(28, 24), (29, 24), (30, 24), (28, 23), (29, 23), (30, 23), (28, 22), (29, 22), (30, 22)]

The neighbors of state (30, 24) are [(29, 25), (30, 25), (31, 25), (29, 24), (30, 24), (31, 24), (29, 23), (30, 23), (31, 23)]

The neighbors of state (31, 25) are [(30, 26), (31, 26), (32, 26), (30, 25), (31, 25), (32, 25), (30, 24), (31, 24), (32, 24)]

The neighbors of state (32, 26) are [(31, 27), (32, 27), (33, 27), (31, 26), (32, 26), (33, 26), (31, 25), (32, 25), (33, 25)]

The neighbors of state (33, 27) are [(32, 28), (33, 28), (34, 28), (32, 27), (33, 27), (34, 27), (32, 26), (33, 26), (34, 26)]

The neighbors of state (34, 28) are [(33, 29), (34, 29), (35, 29), (33, 28), (34, 28), (35, 28), (33, 27), (34, 27), (35, 27)]

The neighbors of state (35, 29) are [(34, 30), (35, 30), (36, 30), (34, 29), (35, 29), (36, 29), (34, 28), (35, 28), (36, 28)]

The neighbors of state (36, 30) are [(35, 31), (36, 31), (37, 31), (35, 30), (36, 30), (37, 30), (35, 29), (36, 29), (37, 29)]

The neighbors of state (37, 31) are [(36, 32), (37, 32), (38, 32), (36, 31), (37, 31), (38, 31), (36, 30), (37, 30), (38, 30)]

The neighbors of state (38, 32) are [(37, 33), (38, 33), (39, 33), (37, 32), (38, 32), (39, 32), (37, 31), (38, 31), (39, 31)]

The neighbors of state (39, 33) are [(38, 34), (39, 34), (40, 34), (38, 33), (39, 33), (40, 33), (38, 32), (39, 32), (40, 32)]

The neighbors of state (40, 34) are [(39, 35), (40, 35), (41, 35), (39, 34), (40, 34), (41, 34), (39, 33), (40, 33), (41, 33)]

The neighbors of state (41, 35) are [(40, 36), (41, 36), (42, 36), (40, 35), (41, 35), (42, 35), (40, 34), (41, 34), (42, 34)]

The neighbors of state (42, 36) are [(41, 37), (42, 37), (43, 37), (41, 36), (42, 36), (43, 36), (41, 35), (42, 35), (43, 35)]

The neighbors of state (43, 37) are [(42, 38), (43, 38), (44, 38), (42, 37), (43, 37), (44, 37), (42, 36), (43, 36), (44, 36)]

The neighbors of state (44, 38) are [(43, 39), (44, 39), (45, 39), (43, 38), (44, 38), (45, 38), (43, 37), (44, 37), (45, 37)]

The neighbors of state (45, 39) are [(44, 40), (45, 40), (46, 40), (44, 39), (45, 39), (46, 39), (44, 38), (45, 38), (46, 38)]

The neighbors of state (46, 40) are [(45, 41), (46, 41), (47, 41), (45, 40), (46, 40), (47, 40), (45, 39), (46, 39), (47, 39)]

The neighbors of state (47, 41) are [(46, 42), (47, 42), (48, 42), (46, 41), (47, 41), (48, 41), (46, 40), (47, 40), (48, 40)]

The neighbors of state (48, 42) are [(47, 43), (48, 43), (49, 43), (47, 42), (48, 42), (49, 42), (47, 41), (48, 41), (49, 41)]

The neighbors of state (49, 43) are [(48, 44), (49, 44), (50, 44), (48, 43), (49, 43), (50, 43), (48, 42), (49, 42), (50, 42)]

The neighbors of state (50, 44) are [(49, 45), (50, 45), (51, 45), (49, 44), (50, 44), (51, 44), (49, 43), (50, 43), (51, 43)]

The neighbors of state (50, 45) are [(49, 46), (50, 46), (51, 46), (49, 45), (50, 45), (51, 45), (49, 44), (50, 44), (51, 44)]

The neighbors of state (50, 46) are [(49, 47), (50, 47), (51, 47), (49, 46), (50, 46), (51, 46), (49, 45), (50, 45), (51, 45)]

The neighbors of state (50, 47) are [(49, 48), (50, 48), (51, 48), (49, 47), (50, 47), (51, 47), (49, 46), (50, 46), (51, 46)]

The neighbors of state (50, 48) are [(49, 49), (50, 49), (51, 49), (49, 48), (50, 48), (51, 48), (49, 47), (50, 47), (51, 47)]

The neighbors of state (50, 49) are [(49, 50), (50, 50), (51, 50), (49, 49), (50, 49), (51, 49), (49, 48), (50, 48), (51, 48)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 61.38298

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 228.44 seconds*

The neighbors of state (14, 76) are [(13, 77), (14, 77), (15, 77), (13, 76), (14, 76), (15, 76), (13, 75), (14, 75), (15, 75)]

The neighbors of state (15, 75) are [(14, 76), (15, 76), (16, 76), (14, 75), (15, 75), (16, 75), (14, 74), (15, 74), (16, 74)]

The neighbors of state (16, 74) are [(15, 75), (16, 75), (17, 75), (15, 74), (16, 74), (17, 74), (15, 73), (16, 73), (17, 73)]

The neighbors of state (17, 73) are [(16, 74), (17, 74), (18, 74), (16, 73), (17, 73), (18, 73), (16, 72), (17, 72), (18, 72)]

The neighbors of state (18, 72) are [(17, 73), (18, 73), (19, 73), (17, 72), (18, 72), (19, 72), (17, 71), (18, 71), (19, 71)]

The neighbors of state (19, 71) are [(18, 72), (19, 72), (20, 72), (18, 71), (19, 71), (20, 71), (18, 70), (19, 70), (20, 70)]

The neighbors of state (20, 70) are [(19, 71), (20, 71), (21, 71), (19, 70), (20, 70), (21, 70), (19, 69), (20, 69), (21, 69)]

The neighbors of state (21, 69) are [(20, 70), (21, 70), (22, 70), (20, 69), (21, 69), (22, 69), (20, 68), (21, 68), (22, 68)]

The neighbors of state (22, 68) are [(21, 69), (22, 69), (23, 69), (21, 68), (22, 68), (23, 68), (21, 67), (22, 67), (23, 67)]

The neighbors of state (23, 67) are [(22, 68), (23, 68), (24, 68), (22, 67), (23, 67), (24, 67), (22, 66), (23, 66), (24, 66)]

The neighbors of state (24, 66) are [(23, 67), (24, 67), (25, 67), (23, 66), (24, 66), (25, 66), (23, 65), (24, 65), (25, 65)]

The neighbors of state (25, 65) are [(24, 66), (25, 66), (26, 66), (24, 65), (25, 65), (26, 65), (24, 64), (25, 64), (26, 64)]

The neighbors of state (26, 64) are [(25, 65), (26, 65), (27, 65), (25, 64), (26, 64), (27, 64), (25, 63), (26, 63), (27, 63)]

The neighbors of state (27, 63) are [(26, 64), (27, 64), (28, 64), (26, 63), (27, 63), (28, 63), (26, 62), (27, 62), (28, 62)]

The neighbors of state (28, 62) are [(27, 63), (28, 63), (29, 63), (27, 62), (28, 62), (29, 62), (27, 61), (28, 61), (29, 61)]

The neighbors of state (29, 61) are [(28, 62), (29, 62), (30, 62), (28, 61), (29, 61), (30, 61), (28, 60), (29, 60), (30, 60)]

The neighbors of state (30, 60) are [(29, 61), (30, 61), (31, 61), (29, 60), (30, 60), (31, 60), (29, 59), (30, 59), (31, 59)]

The neighbors of state (31, 59) are [(30, 60), (31, 60), (32, 60), (30, 59), (31, 59), (32, 59), (30, 58), (31, 58), (32, 58)]

The neighbors of state (32, 58) are [(31, 59), (32, 59), (33, 59), (31, 58), (32, 58), (33, 58), (31, 57), (32, 57), (33, 57)]

The neighbors of state (33, 57) are [(32, 58), (33, 58), (34, 58), (32, 57), (33, 57), (34, 57), (32, 56), (33, 56), (34, 56)]

The neighbors of state (34, 56) are [(33, 57), (34, 57), (35, 57), (33, 56), (34, 56), (35, 56), (33, 55), (34, 55), (35, 55)]

The neighbors of state (35, 55) are [(34, 56), (35, 56), (36, 56), (34, 55), (35, 55), (36, 55), (34, 54), (35, 54), (36, 54)]

The neighbors of state (36, 54) are [(35, 55), (36, 55), (37, 55), (35, 54), (36, 54), (37, 54), (35, 53), (36, 53), (37, 53)]

The neighbors of state (37, 53) are [(36, 54), (37, 54), (38, 54), (36, 53), (37, 53), (38, 53), (36, 52), (37, 52), (38, 52)]

The neighbors of state (38, 52) are [(37, 53), (38, 53), (39, 53), (37, 52), (38, 52), (39, 52), (37, 51), (38, 51), (39, 51)]

The neighbors of state (39, 51) are [(38, 52), (39, 52), (40, 52), (38, 51), (39, 51), (40, 51), (38, 50), (39, 50), (40, 50)]

The neighbors of state (40, 50) are [(39, 51), (40, 51), (41, 51), (39, 50), (40, 50), (41, 50), (39, 49), (40, 49), (41, 49)]

The neighbors of state (41, 50) are [(40, 51), (41, 51), (42, 51), (40, 50), (41, 50), (42, 50), (40, 49), (41, 49), (42, 49)]

The neighbors of state (42, 50) are [(41, 51), (42, 51), (43, 51), (41, 50), (42, 50), (43, 50), (41, 49), (42, 49), (43, 49)]

The neighbors of state (43, 50) are [(42, 51), (43, 51), (44, 51), (42, 50), (43, 50), (44, 50), (42, 49), (43, 49), (44, 49)]

The neighbors of state (44, 50) are [(43, 51), (44, 51), (45, 51), (43, 50), (44, 50), (45, 50), (43, 49), (44, 49), (45, 49)]

The neighbors of state (45, 50) are [(44, 51), (45, 51), (46, 51), (44, 50), (45, 50), (46, 50), (44, 49), (45, 49), (46, 49)]

The neighbors of state (46, 50) are [(45, 51), (46, 51), (47, 51), (45, 50), (46, 50), (47, 50), (45, 49), (46, 49), (47, 49)]

The neighbors of state (47, 50) are [(46, 51), (47, 51), (48, 51), (46, 50), (47, 50), (48, 50), (46, 49), (47, 49), (48, 49)]

The neighbors of state (48, 50) are [(47, 51), (48, 51), (49, 51), (47, 50), (48, 50), (49, 50), (47, 49), (48, 49), (49, 49)]

The neighbors of state (49, 50) are [(48, 51), (49, 51), (50, 51), (48, 50), (49, 50), (50, 50), (48, 49), (49, 49), (50, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 91.71053

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 226.65 seconds*

The neighbors of state (96, 41) are [(95, 42), (96, 42), (97, 42), (95, 41), (96, 41), (97, 41), (95, 40), (96, 40), (97, 40)]

The neighbors of state (95, 42) are [(94, 43), (95, 43), (96, 43), (94, 42), (95, 42), (96, 42), (94, 41), (95, 41), (96, 41)]

The neighbors of state (94, 43) are [(93, 44), (94, 44), (95, 44), (93, 43), (94, 43), (95, 43), (93, 42), (94, 42), (95, 42)]

The neighbors of state (93, 44) are [(92, 45), (93, 45), (94, 45), (92, 44), (93, 44), (94, 44), (92, 43), (93, 43), (94, 43)]

The neighbors of state (92, 45) are [(91, 46), (92, 46), (93, 46), (91, 45), (92, 45), (93, 45), (91, 44), (92, 44), (93, 44)]

The neighbors of state (91, 46) are [(90, 47), (91, 47), (92, 47), (90, 46), (91, 46), (92, 46), (90, 45), (91, 45), (92, 45)]

The neighbors of state (90, 47) are [(89, 48), (90, 48), (91, 48), (89, 47), (90, 47), (91, 47), (89, 46), (90, 46), (91, 46)]

The neighbors of state (89, 48) are [(88, 49), (89, 49), (90, 49), (88, 48), (89, 48), (90, 48), (88, 47), (89, 47), (90, 47)]

The neighbors of state (88, 49) are [(87, 50), (88, 50), (89, 50), (87, 49), (88, 49), (89, 49), (87, 48), (88, 48), (89, 48)]

The neighbors of state (87, 50) are [(86, 51), (87, 51), (88, 51), (86, 50), (87, 50), (88, 50), (86, 49), (87, 49), (88, 49)]

The neighbors of state (86, 50) are [(85, 51), (86, 51), (87, 51), (85, 50), (86, 50), (87, 50), (85, 49), (86, 49), (87, 49)]

The neighbors of state (85, 50) are [(84, 51), (85, 51), (86, 51), (84, 50), (85, 50), (86, 50), (84, 49), (85, 49), (86, 49)]

The neighbors of state (84, 50) are [(83, 51), (84, 51), (85, 51), (83, 50), (84, 50), (85, 50), (83, 49), (84, 49), (85, 49)]

The neighbors of state (83, 50) are [(82, 51), (83, 51), (84, 51), (82, 50), (83, 50), (84, 50), (82, 49), (83, 49), (84, 49)]

The neighbors of state (82, 50) are [(81, 51), (82, 51), (83, 51), (81, 50), (82, 50), (83, 50), (81, 49), (82, 49), (83, 49)]

The neighbors of state (81, 50) are [(80, 51), (81, 51), (82, 51), (80, 50), (81, 50), (82, 50), (80, 49), (81, 49), (82, 49)]

The neighbors of state (80, 50) are [(79, 51), (80, 51), (81, 51), (79, 50), (80, 50), (81, 50), (79, 49), (80, 49), (81, 49)]

The neighbors of state (79, 50) are [(78, 51), (79, 51), (80, 51), (78, 50), (79, 50), (80, 50), (78, 49), (79, 49), (80, 49)]

The neighbors of state (78, 50) are [(77, 51), (78, 51), (79, 51), (77, 50), (78, 50), (79, 50), (77, 49), (78, 49), (79, 49)]

The neighbors of state (77, 50) are [(76, 51), (77, 51), (78, 51), (76, 50), (77, 50), (78, 50), (76, 49), (77, 49), (78, 49)]

The neighbors of state (76, 50) are [(75, 51), (76, 51), (77, 51), (75, 50), (76, 50), (77, 50), (75, 49), (76, 49), (77, 49)]

The neighbors of state (75, 50) are [(74, 51), (75, 51), (76, 51), (74, 50), (75, 50), (76, 50), (74, 49), (75, 49), (76, 49)]

The neighbors of state (74, 50) are [(73, 51), (74, 51), (75, 51), (73, 50), (74, 50), (75, 50), (73, 49), (74, 49), (75, 49)]

The neighbors of state (73, 50) are [(72, 51), (73, 51), (74, 51), (72, 50), (73, 50), (74, 50), (72, 49), (73, 49), (74, 49)]

The neighbors of state (72, 50) are [(71, 51), (72, 51), (73, 51), (71, 50), (72, 50), (73, 50), (71, 49), (72, 49), (73, 49)]

The neighbors of state (71, 50) are [(70, 51), (71, 51), (72, 51), (70, 50), (71, 50), (72, 50), (70, 49), (71, 49), (72, 49)]

The neighbors of state (70, 50) are [(69, 51), (70, 51), (71, 51), (69, 50), (70, 50), (71, 50), (69, 49), (70, 49), (71, 49)]

The neighbors of state (69, 50) are [(68, 51), (69, 51), (70, 51), (68, 50), (69, 50), (70, 50), (68, 49), (69, 49), (70, 49)]

The neighbors of state (68, 50) are [(67, 51), (68, 51), (69, 51), (67, 50), (68, 50), (69, 50), (67, 49), (68, 49), (69, 49)]

The neighbors of state (67, 50) are [(66, 51), (67, 51), (68, 51), (66, 50), (67, 50), (68, 50), (66, 49), (67, 49), (68, 49)]

The neighbors of state (66, 50) are [(65, 51), (66, 51), (67, 51), (65, 50), (66, 50), (67, 50), (65, 49), (66, 49), (67, 49)]

The neighbors of state (65, 50) are [(64, 51), (65, 51), (66, 51), (64, 50), (65, 50), (66, 50), (64, 49), (65, 49), (66, 49)]

The neighbors of state (64, 50) are [(63, 51), (64, 51), (65, 51), (63, 50), (64, 50), (65, 50), (63, 49), (64, 49), (65, 49)]

The neighbors of state (63, 50) are [(62, 51), (63, 51), (64, 51), (62, 50), (63, 50), (64, 50), (62, 49), (63, 49), (64, 49)]

The neighbors of state (62, 50) are [(61, 51), (62, 51), (63, 51), (61, 50), (62, 50), (63, 50), (61, 49), (62, 49), (63, 49)]

The neighbors of state (61, 50) are [(60, 51), (61, 51), (62, 51), (60, 50), (61, 50), (62, 50), (60, 49), (61, 49), (62, 49)]

The neighbors of state (60, 50) are [(59, 51), (60, 51), (61, 51), (59, 50), (60, 50), (61, 50), (59, 49), (60, 49), (61, 49)]

The neighbors of state (59, 50) are [(58, 51), (59, 51), (60, 51), (58, 50), (59, 50), (60, 50), (58, 49), (59, 49), (60, 49)]

The neighbors of state (58, 50) are [(57, 51), (58, 51), (59, 51), (57, 50), (58, 50), (59, 50), (57, 49), (58, 49), (59, 49)]

The neighbors of state (57, 50) are [(56, 51), (57, 51), (58, 51), (56, 50), (57, 50), (58, 50), (56, 49), (57, 49), (58, 49)]

The neighbors of state (56, 50) are [(55, 51), (56, 51), (57, 51), (55, 50), (56, 50), (57, 50), (55, 49), (56, 49), (57, 49)]

The neighbors of state (55, 50) are [(54, 51), (55, 51), (56, 51), (54, 50), (55, 50), (56, 50), (54, 49), (55, 49), (56, 49)]

The neighbors of state (54, 50) are [(53, 51), (54, 51), (55, 51), (53, 50), (54, 50), (55, 50), (53, 49), (54, 49), (55, 49)]

The neighbors of state (53, 50) are [(52, 51), (53, 51), (54, 51), (52, 50), (53, 50), (54, 50), (52, 49), (53, 49), (54, 49)]

The neighbors of state (52, 50) are [(51, 51), (52, 51), (53, 51), (51, 50), (52, 50), (53, 50), (51, 49), (52, 49), (53, 49)]

The neighbors of state (51, 50) are [(50, 51), (51, 51), (52, 51), (50, 50), (51, 50), (52, 50), (50, 49), (51, 49), (52, 49)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 121.58333

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 230.21 seconds*

The neighbors of state (37, 32) are [(35, 34), (37, 34), (39, 34), (35, 32), (37, 32), (39, 32), (35, 30), (37, 30), (39, 30)]

The neighbors of state (39, 34) are [(37, 36), (39, 36), (41, 36), (37, 34), (39, 34), (41, 34), (37, 32), (39, 32), (41, 32)]

The neighbors of state (41, 36) are [(39, 38), (41, 38), (43, 38), (39, 36), (41, 36), (43, 36), (39, 34), (41, 34), (43, 34)]

The neighbors of state (43, 38) are [(41, 40), (43, 40), (45, 40), (41, 38), (43, 38), (45, 38), (41, 36), (43, 36), (45, 36)]

The neighbors of state (45, 40) are [(43, 42), (45, 42), (47, 42), (43, 40), (45, 40), (47, 40), (43, 38), (45, 38), (47, 38)]

The neighbors of state (47, 42) are [(45, 44), (47, 44), (49, 44), (45, 42), (47, 42), (49, 42), (45, 40), (47, 40), (49, 40)]

The neighbors of state (49, 44) are [(47, 46), (49, 46), (51, 46), (47, 44), (49, 44), (51, 44), (47, 42), (49, 42), (51, 42)]

The neighbors of state (51, 46) are [(49, 48), (51, 48), (53, 48), (49, 46), (51, 46), (53, 46), (49, 44), (51, 44), (53, 44)]

The neighbors of state (51, 48) are [(49, 50), (51, 50), (53, 50), (49, 48), (51, 48), (53, 48), (49, 46), (51, 46), (53, 46)]

The neighbors of state (51, 50) are [(49, 52), (51, 52), (53, 52), (49, 50), (51, 50), (53, 50), (49, 48), (51, 48), (53, 48)]

We have calcuated the expectation of the set P and obtained the following results: Ep(f) = 87.72727

As we would expect from Markov chain theory, our result Ep(f) converges to 100.

*Time of program run = 226.99 seconds*