Difference between revisions of "1985 AIME Problems/Problem 12"
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== Solution 1 (One-Variable Recursion) == | == Solution 1 (One-Variable Recursion) == | ||
− | For all positive integers <math>k,</math> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <math>P(7).</math> | + | For all positive integers <math>k,</math> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <math>p=P(7).</math> |
Note that after <math>k-1</math> moves: | Note that after <math>k-1</math> moves: | ||
<ol style="margin-left: 1.5em;"> | <ol style="margin-left: 1.5em;"> | ||
− | <li></li><p> | + | <li>If the bug is at vertex <math>A,</math> then the probability that it crawls to vertex <math>A</math> on the next move is <math>0.</math></li><p> |
− | <li></li><p> | + | <li>If the bug is not at vertex <math>A,</math> then the probability that it crawls to vertex <math>A</math> on the next move is <math>\frac13.</math></li><p> |
</ol> | </ol> | ||
+ | Therefore, the recursive formula for <math>P(k)</math> is | ||
+ | <cmath>P(k) = \begin{cases} | ||
+ | 0 & \mathrm{if} \ k=1 \\ | ||
+ | \frac13(1-P(k-1)) & \mathrm{if} \ k\geq2 | ||
+ | \end{cases}.</cmath> | ||
+ | Two solutions follow from here: | ||
+ | |||
+ | === Solution 1.1 (Recursion) === | ||
+ | |||
+ | |||
+ | ~Azjps (Fundamental Logic) | ||
+ | |||
+ | ~MRENTHUSIASM (Reconstruction) | ||
+ | |||
+ | === Solution 1.2 (Closed Form of P(n)) === | ||
+ | |||
+ | ~MRENTHUSIASM | ||
== Solution 2 == | == Solution 2 == |
Revision as of 22:47, 22 July 2021
Contents
Problem
Let ,
,
and
be the vertices of a regular tetrahedron, each of whose edges measures
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
meters. Find the value of
.
Solution 1 (One-Variable Recursion)
For all positive integers let
be the probability that the bug is at vertex
when it has crawled exactly
meters. We wish to find
Note that after moves:
- If the bug is at vertex
then the probability that it crawls to vertex
on the next move is
- If the bug is not at vertex
then the probability that it crawls to vertex
on the next move is
Therefore, the recursive formula for is
Two solutions follow from here:
Solution 1.1 (Recursion)
~Azjps (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 1.2 (Closed Form of P(n))
~MRENTHUSIASM
Solution 2
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.
![${3\choose1}f(2)f(2)f(3) + {2\choose1}f(2)f(5) + {2\choose1}f(3)f(4) + f(7)$](http://latex.artofproblemsolving.com/c/8/1/c812b183c3cf604b67ca9204ba051a695042977d.png)
![$= 3(3)(3)(2\cdot3) + 2(3)(2^33) + 2(2\cdot3)(2^23) + (2^53)$](http://latex.artofproblemsolving.com/8/7/9/87916ffa4c9bf4f5e0216dda2aa6471dd91befa9.png)
![$= 546$](http://latex.artofproblemsolving.com/b/c/b/bcb42277d86dcd7fef7988f6a45f711227bcb48c.png)
Finally, this is a probability question, so we divide by :
.
Solution 3
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
Solution 5
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
.
~ Nafer
Solution 7 (Multivariable Recursion)
Denominator
There are ways for the bug to make
independent crawls without restrictions.
Numerator (Recursion)
Let denote the number of ways for the bug to crawl exactly
meters starting from vertex
and ending at vertex
where
and
is a positive integer. We wish to find
Since the bug must crawl to vertex 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
Recall that
By
and
we rewrite
recursively:
Answer
The requested probability is from which
~MRENTHUSIASM
See also
- 2003 AIME II Problems/Problem 13 - very similar problem with an equilateral triangle
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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