# snakes.R
# snakes and ladders
# Build the transition matrix P
P <- matrix(rep(0,101^2),nrow=101)
r <- function(row,inp) 
for (i in 1:6) {
	P[row+1,inp[i]+1] <<- P[row+1,inp[i]+1] + 1/6 }
 r(0,c(38,2,3,14,5,6))
 r(1,c(2,3,14,5,6,7))
 r(2,c(3,14,5,6,7,8))
 r(3,c(14,5,6,7,8,31))
 r(4,c(5,6,7,8,31,10))
 r(5,c(6,7,8,31,10,11))
 r(6,c(7,8,31,10,11,12))
 r(7,c(8,31,10,11,12,13))
 r(8,c(31,10,11,12,13,14))
 r(9,c(10,11,12,13,14,15))
 r(10,c(11,12,13,14,15,6))
 r(11,c(12,13,14,15,6,17))
 r(12,c(13,14,15,6,17,18))
 r(13,c(14,15,6,17,18,19))
 r(14,c(15,6,17,18,19,20))
 r(15,c(6,17,18,19,20,42))
 r(16,c(17,18,19,20,42,22))
 r(17,c(18,19,20,42,22,23))
 r(18,c(19,20,42,22,23,24))
 r(19,c(20,42,22,23,24,25))
 r(20,c(42,22,23,24,25,26))
 r(21,c(22,23,24,25,26,27))
 r(22,c(23,24,25,26,27,84))
 r(23,c(24,25,26,27,84,29))
 r(24,c(25,26,27,84,29,30))
 r(25,c(26,27,84,29,30,31))
 r(26,c(27,84,29,30,31,32))
 r(27,c(84,29,30,31,32,33))
 r(28,c(29,30,31,32,33,34))
 r(29,c(30,31,32,33,34,35))
 r(30,c(31,32,33,34,35,44))
 r(31,c(32,33,34,35,44,37))
 r(32,c(33,34,35,44,37,38))
 r(33,c(34,35,44,37,38,39))
 r(34,c(35,44,37,38,39,40))
 r(35,c(44,37,38,39,40,41))
 r(36,c(37,38,39,40,41,42))
 r(37,c(38,39,40,41,42,43))
 r(38,c(39,40,41,42,43,44))
 r(39,c(40,41,42,43,44,45))
 r(40,c(41,42,43,44,45,46))
 r(41,c(42,43,44,45,46,47))
 r(42,c(43,44,45,46,47,26))
 r(43,c(44,45,46,47,26,11))
 r(44,c(45,46,47,26,11,50))
 r(45,c(46,47,26,11,50,67))
 r(46,c(47,26,11,50,67,52))
 r(47,c(26,11,50,67,52,53))
 r(48,c(11,50,67,52,53,54))
 r(49,c(50,67,52,53,54,55))
 r(50,c(67,52,53,54,55,53))
 r(51,c(52,53,54,55,53,57))
 r(52,c(53,54,55,53,57,58))
 r(53,c(54,55,53,57,58,59))
 r(54,c(55,53,57,58,59,60))
 r(55,c(53,57,58,59,60,61))
 r(56,c(57,58,59,60,61,19))
 r(57,c(58,59,60,61,19,63))
 r(58,c(59,60,61,19,63,60))
 r(59,c(60,61,19,63,60,65))
 r(60,c(61,19,63,60,65,66))
 r(61,c(19,63,60,65,66,67))
 r(62,c(63,60,65,66,67,68))
 r(63,c(60,65,66,67,68,69))
 r(64,c(65,66,67,68,69,70))
 r(65,c(66,67,68,69,70,91))
 r(66,c(67,68,69,70,91,72))
 r(67,c(68,69,70,91,72,73))
 r(68,c(69,70,91,72,73,74))
 r(69,c(70,91,72,73,74,75))
 r(70,c(91,72,73,74,75,76))
 r(71,c(72,73,74,75,76,77))
 r(72,c(73,74,75,76,77,78))
 r(73,c(74,75,76,77,78,79))
 r(74,c(75,76,77,78,79,100))
 r(75,c(76,77,78,79,100,81))
 r(76,c(77,78,79,100,81,82))
 r(77,c(78,79,100,81,82,83))
 r(78,c(79,100,81,82,83,84))
 r(79,c(100,81,82,83,84,85))
 r(80,c(81,82,83,84,85,86))
 r(81,c(82,83,84,85,86,24))
 r(82,c(83,84,85,86,24,88))
 r(83,c(84,85,86,24,88,89))
 r(84,c(85,86,24,88,89,90))
 r(85,c(86,24,88,89,90,91))
 r(86,c(24,88,89,90,91,92))
 r(87,c(88,89,90,91,92,73))
 r(88,c(89,90,91,92,73,94))
 r(89,c(90,91,92,73,94,75))
 r(90,c(91,92,73,94,75,96))
 r(91,c(92,73,94,75,96,97))
 r(92,c(73,94,75,96,97,78))
 r(93,c(94,75,96,97,78,99))
 r(94,c(75,96,97,78,99,100))
 r(95,c(96,97,78,99,100,95))
 r(96,c(97,78,99,100,96,96))
 r(97,c(78,99,100,97,97,97))
 r(98,c(99,100,98,98,98,98))
 r(99,c(100,99,99,99,99,99))
 r(100,c(100,100,100,100,100,100))

init <- c(1,rep(0,100))
## simulate the game 
markov(init,P,50,0:100)

simlist <- replicate(1000,which(markov(init,P,500,0:100) == 100)[1])
 mean(simlist)
# fundamental matrix
f <- solve(diag(100) - M[1:100,1:100])
## absorbtion probabilities
f %*% rep(1,100)