安迪的清晨散步
Andy's Morning Stroll
题目详情
中文译文待补充。
Andy the ant has spent most of his days living on a strange land consisting of white hexagons that are surrounded by alternating black pentagons and white hexagons (three of each), and black pentagons surrounded by five white hexagons. To us this land is familiar as the classic soccer ball we see above on the left. Due to Andy’s tiny size and terrible eyesight, he doesn’t notice the curvature of the land and avoids the black pentagons because he suspects they may be bottomless pits.
Every morning he wakes up on a white hexagon, leaves some pheromones to mark it as his special home space, and starts his random morning stroll. Every step on this stroll takes him to one of the three neighboring white hexagons with equal probability. He ends his stroll as soon as he first returns to his home space. As an example, on exactly 1/3 of mornings Andy’s stroll is 2 steps long, as he randomly visits one of the three neighbors, and then has a 1/3 probability of returning immediately to the home hexagon.
This morning, his soccer ball bounced through a kitchen with an infinite (at least practically speaking…) regular hexagonal floor tiling consisting of black and white hexagons, a small part of which is shown above on the right. In this tiling every white hexagon is surrounded by alternating black and white hexagons, and black hexagons are surrounded by six white hexagons. Andy fell off the ball and woke up on a white hexagon. He didn’t notice any change in his surroundings, and goes about his normal morning routine.
Let p be the probability that his morning stroll on this new land is strictly more steps than the expected number of steps his strolls on the soccer ball took. Find p, rounded to seven significant digits.
2022-07-08 update: Sorry about the lateness of this post! Much like Andy, our humble puzzle site found itself waking up in a strange new land of the wider Jane Street website update… happy puzzling!
解析
中文解析待补充。
Original Explanation
Andy normally strolls across the hexagons of a soccer ball. To find the average length of this stroll, which turns out to be exactly 20, we could set up a series of equations, but the following intuitive argument may be more satisfying.
Imagine Andy took an extremely long stroll on these hexagons without stopping and we chopped that stroll into segments starting and ending at his home square. The symmetry of the ball suggests, in the limit of the length of the stroll going to infinity, exactly 1/20 of the hexagons Andy visits will be his home hexagon. This means the sum of the lengths of all the segments, divided by the number of segments, will approach 20. But this limiting ratio is exactly the average length of a stroll starting and ending at his home hexagon!
So now we need to compute the probability that his stroll on the kitchen floor will strictly exceed twenty steps. This was a bit of a computational exercise, we can construct a transition matrix showing that this probability, to seven significant digits, is 0.4480326…
Congrats to this month’s solvers!