1985 AIME Problems/Problem 12
Let , , and be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let be the probability that the bug is at vertex when it has crawled exactly 7 meters. Find the value of .
Let denote the probability that the bug is at after it has crawled meters. Since the bug can only be at vertex if it just left a vertex which is not , we have . We also know , so we can quickly compute , , , , , and , so the answer is . One can solve this recursion fairly easily to determine a closed-form expression for .
We can find the number of different times the bug reaches vertex before the 7th move, and use these smaller cycles to calculate the number of different ways the bug can end up back at .
Define to be the number of paths of length which start and end at but do not pass through otherwise. Obviously . In general for , the bug has three initial edges to pick from. From there, since the bug cannot return to by definition, the bug has exactly two choices. This continues from the 2nd move up to the th move. The last move must be a return to , so this move is determined. So .
Now we need to find the number of cycles by which the bug can reach at the end. Since , cannot be used since on the 7th move the bug cannot move from to . So we need to find the number of partitions of 7 using only 2,3,4,5, and 7. These are , , , and . We can calculate these and sum them up using our formula. Also, order matters, so we need to find the number of ways to arrange each partition.
Finally, this is a probability question, so we divide by : .
There exists a simple heuristic method to arrive at the answer to this question, due to Simon Rubinstein-Salzedo, as follows: after a couple of moves, the randomness of movement of the bug and smallness of the system ensures that we should expect its probability distribution to be very close to uniform. In particular, we would expect to be very close to for decently-sized , for example . (In fact, from looking at the previous solution we can see that it is already close when , and in fact, the earlier values are also the best possible approximations given the restraints on where the bug can be.) Since we know the answer is of the form , we realize that must be very close to , as indeed it is.
Solution 4 (Cheap Solution)
Only do this if you don't know how to solve the problem and want to make a good guess. Since there are 4 vertices of a tetrahedron, there is approximately a 1/4 probability of coming back to A after 7 moves. Dividing 729 by 4 gives a number between 182 and 183. If the ant continuously alternates his location from A to some other vertice, in the end, it will not be at A. Therefore we chose the smaller number, 182. - Rocket123
Let denotes the number of ways that the bug arrives at after crawling meters, then we have .
Notice that there is respectively way to arrive at for each of the different routes after the previous crawls, excluding the possibility that the bug ends up at after the th crawl (as it will be forced to move somewhere else.). Thus we get the recurrence relation Quick calculations yield Thus .
Draw an organized table and keep track of how many ways are there to get to each vertex after 1, 2, 3, ..., 7 steps. Thus you get the answer of . -pi_is_3.141
Solution 7 (Multivariable Recursion)
There are ways for the bug to make independent crawls without restrictions.
Let denote the number of ways for the bug to crawl exactly meters starting from and ending at where and is a positive integer. We wish to find
Since the bug must crawl to or on the first move, we have where
More generally, we get For note that
- Base Case:
- Recursive Case:
Clearly, is a geometric sequence with the first term and the common ratio Therefore, its explicit formula is By and we rewrite recursively: Answer
The requested probability is from which
- 2003 AIME II Problems/Problem 13 - very similar problem with an equilateral triangle
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