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MMLU-Pro / 9025 / Consider a horizontal strip of N+2 squares in which the first and the last…

Problem

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question

Consider a horizontal strip of N+2 squares in which the first and the last square are black and the remaining N squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let w(N)w(N) be the expected number of white squares remaining. What is the limit of w(N)/Nw(N)/N as NN goes to infinity?
Plain-text mathematical notation (without MathML)
Consider a horizontal strip of N+2 squares in which the first and the last square are black and the remaining N squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let w(N) be the expected number of white squares remaining. What is the limit of w(N)/N as N goes to infinity?
Original LaTeX notation
Consider a horizontal strip of N+2 squares in which the first and the last square are black and the remaining N squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. What is the limit of $w(N)/N$ as $N$ goes to infinity?

category

math

options

  1. 0.32192809
  2. 0.4567890
  3. 0.36787944
  4. 0.22313016
  5. 0.2789456
  6. 0.5
  7. 0.13533528
  8. 0.69314718
  9. 0.5789098
  10. 0.60653065

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
9025
task_id
b0253983-0748-55ab-a352-c117265f9ed7
task_key
default--test--9025
task_revision_id
1
config / split
default / test

Official source

initial import