Lava and Land
On the origin of the Cartesian plane stands a Calvin Lin. He is at a corner of a square grid with corners at and .
The line marks the boundary between the land half of the grid and the lava half of the grid.
Calvin Lin will randomly walk along the grid edges, but he will only walk up or to the right. If he walks into lava, he will not be able to do math for the rest of his life. His home is on the opposite side of the grid.
All Calvin wants to do is to get back to his house. But Calvin has a bad sense of direction, along with his disability of only going in two directions. His legs will randomly move up or to the right with equal probability.
Given that Calvin cannot walk off of the grid, what is the reciprocal of the probability that when Calvin arrives at his house, he will still be able to do math again?
The answer is 82452452623423526.
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1 solution
Notice those outrageous numbers! This probably means they have no importance at all!
Recall the great Catalan numbers! Notice how this problem is just another way to express the Catalan numbers!
It is known that the n t h Catalan number is n + 1 1 ⋅ ( n 2 n ) .
The total number of ways Calvin can walk is ( n 2 n ) .
Therefore the probability is n + 1 1
Thus we are done with an answer of 8 2 4 5 2 4 5 2 6 2 3 4 2 3 5 2 6