Solution: Mindsport / Endgame 30 October, Sunday Times Of India (STOI)

Today's question is actually pretty simple or may be its not that simple I just got lucky. Mukul does not update his site regularly so I cannot post a link to the question. End

Answer:
Snakes -- 13 to 32, 45 to 66. Rest are ladders.

Solution:
Total number of moves made / squares covered because of 6 on the die = 12 * 6 = 72
Total number of moves required to be made to reach 100 = 100 - 1 = 99
This means that 27 (99 - 72) moves / squares to be covered with the help of snakes and ladders.
They are 13 to 32 ie 19 squares
25 to 48 ie 23 squares
35 to 54 ie 19 squares
45 to 66 ie 21 squares
63 to 88 ie 25 squares
79 to 84 ie 15 squares.
There are three options possible when one is present at these position.
a) She goes up ie she is at the foot of the ladder. (a factor of +1)
b) She remains at the same place ie she is at the top of a ladder or at the tail of a snake. (a factor of 0)
c) She goes down ie she is at the head of a snake.(a factor of -1)
So the question will be reduced to:
19*n1 + 23*n2 + 19*n3 + 21*n4 + 25*n5 + 15*n6 = 27
where ni is such that it belongs to {-1,0,1}.
and i is such that it belongs to (1,2,3,4,5,6}.
The above equations gives us a unique solution which has n1 = n3 = n6 = 0, n2 = 1, n4 = -1, n5 = 1
A simple checking reveals that all n1 is a snake and n3, n6 are ladders.
n2, n5 have to be ladders because their coefficients are 1.
n4 has to be a snake.