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Omni-MATH / Roger the ant is traveling on a coordinate plane, starting at (0,0). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal pro…

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problem

Roger the ant is traveling on a coordinate plane, starting at (0,0)(0,0). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point PP for which he could have reached PP more quickly had he taken a different route. For example, if he goes from (0,0)(0,0) to (1,0)(1,0) to (1,1)(1,1) to (1,2)(1,2) to (0,2)(0,2), he stops at (0,2)(0,2) because he could have gone from (0,0)(0,0) to (0,1)(0,1) to (0,2)(0,2) in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as ab\frac{a}{b}, where aa and bb are relatively prime positive integers. Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Roger the ant is traveling on a coordinate plane, starting at (0,0). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point P for which he could have reached P more quickly had he taken a different route. For example, if he goes from (0,0) to (1,0) to (1,1) to (1,2) to (0,2), he stops at (0,2) because he could have gone from (0,0) to (0,1) to (0,2) in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as (a)/(b), where a and b are relatively prime positive integers. Compute 100a+b.
Original LaTeX notation
Roger the ant is traveling on a coordinate plane, starting at $(0,0)$. Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point $P$ for which he could have reached $P$ more quickly had he taken a different route. For example, if he goes from $(0,0)$ to $(1,0)$ to $(1,1)$ to $(1,2)$ to $(0,2)$, he stops at $(0,2)$ because he could have gone from $(0,0)$ to $(0,1)$ to $(0,2)$ in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

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Source and history

task_id
b528ef23-fd8a-598a-ae02-5d9e1c643265
task_key
test--b528ef23-fd8a-598a-ae02-5d9e1c643265
task_revision_id
1
config / split
/ test

Official source

initial import