2016 AIME I Problems/Problem 13

Problem

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$ . A fence is located at the horizontal line $y = 0$ . On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$ , with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$ . Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

Solution

Notice that we don't really care about what the $x$ -coordinate of the frog is. We will let $f(y)$ denote the expected number of times Freddy will jump starting at a $y$ -coordinate of $y$ until he reaches the line $y = 24$ . We want to find $f(21)$ .

We have $f(24) = 0$ . Suppose Freddy is at $y = n$ . He has a $\frac{1}{2}$ probability of moving horizontally, $\frac{1}{4}$ chance of moving up and $\frac{1}{4}$ chance of moving down. So we have $f(n) = 1 + \frac{1}{2} f(n) + \frac{1}{4} f(n-1) + \frac{1}{4} f(n+1)$ So we get the recursion $2f(n) - f(n-1) - f(n+1) = 4$ . Rearranging we see $f(n) - f(n-1) = f(n+1) - f(n) + 4$ . That means the difference between consecutive terms goes down by $4$ each time. So for convenience let's let $z = f(0)$ and $d = f(1) - f(0)$ . So that means $f(k) = z + d + (d-4) + (d-8) + \cdots + (d - 4(k-1)) = z + kd - 2k(k-1)$ Yes, this is a quadratic. Now, notice that since there is a boundary, we have to give special care to $f(0)$ . We have $f(0) = 1 + \frac{2}{3}f(0) + \frac{1}{3} f(1)$ so this turns into $\frac{1}{3}z = 1 + \frac{1}{3}(z+d)$ and this results in $d = -3$ . So now we know $f(k) = z - k - 2k^2$ Now, we also have that $f(24) = 0$ so that gives us $f(24) = z - 24 - 2 \cdot 576 = 0$ so $z = 1176$ . So now we know $f(k) = 1176 - k - 2k^2$ so plugging in $k = 21$ we get $f(21) = 1176 - 21 - 882 = \boxed{273}$ -fclvbfm934

EDIT: Why switch variables? We don't need to introduce a new variable $n$ instead of $y$ . . . .

2016 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2016 AIME I Problems/Problem 13

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