Difference between revisions of "2007 AIME I Problems/Problem 6"
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We divide it into 3 stages. The first occurs before the frog moves past 13. The second occurs before it moves past 26, and the last is everything else.  We divide it into 3 stages. The first occurs before the frog moves past 13. The second occurs before it moves past 26, and the last is everything else.  
−  For the first stage the possible paths are <math>0,13</math>, <math>0,3,13</math>, <math>0,3,6,13</math>, <math>0,3,6,9,13</math>, <math>0,3,6,9,12,13</math>, and <math>0,3,6,9,12</math>. That is a total of 6.  +  For the first stage the possible paths are <math>(0,13)</math>, <math>(0,3,13)</math>, <math>(0,3,6,13)</math>, <math>(0,3,6,9,13)</math>, <math>(0,3,6,9,12,13)</math>, and <math>(0,3,6,9,12)</math>. That is a total of 6. 
−  For the second stage the possible paths are <math>26</math>, <math>15,26</math>, <math>15,18,26</math>, <math>15,18,21,26</math>, <math>15,18,21,24,26</math>, and <math>15,18,21,24</math>. That is a total of 6.  +  For the second stage the possible paths are <math>(26)</math>, <math>(15,26)</math>, <math>(15,18,26)</math>, <math>(15,18,21,26)</math>, <math>(15,18,21,24,26)</math>, and <math>(15,18,21,24)</math>. That is a total of 6. 
−  For the second stage the possible paths are <math>39</math>, <math>27,39</math>, <math>27,30,39</math>, <math>27,30,33,39</math>, and <math>27,30,33,36,39</math>. That is a total of 5.  +  For the second stage the possible paths are <math>(39)</math>, <math>(27,39)</math>, <math>(27,30,39)</math>, <math>(27,30,33,39)</math>, and <math>(27,30,33,36,39)</math>. That is a total of 5. 
However, we cannot jump from <math>12 \Rightarrow 26</math> (this eliminates 5 paths) or <math>24 \Rightarrow 39</math> (this eliminates 6 paths), so we must subtract <math>6 + 5 = 11</math>.  However, we cannot jump from <math>12 \Rightarrow 26</math> (this eliminates 5 paths) or <math>24 \Rightarrow 39</math> (this eliminates 6 paths), so we must subtract <math>6 + 5 = 11</math>. 
Revision as of 04:33, 11 February 2011
Problem
A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, is a move sequence. How many move sequences are possible for the frog?
Solution
Solution 1
Let us keep a careful tree of the possible number of paths around every multiple of .
From , we can end at either (mult. of 3) or (mult. of 13).
 Only path leads to
 Continuing from , there is way to continue to
 There are ways to reach .
 There are ways to reach .
 Continuing from , there are ways to get to
 There are ways (the first 1 to make it inclusive, the second to also jump from ) to get to .
Regrouping, work from
 There are ways to get to
 Continuing from , there are ways to continue to .
 There are ways to reach .
 Continuing from , there are (note that the 1 is not to inclusive, but to count ).
In total, we get .
In summary, we can draw the following tree, where in , represents the current position on the number line, and represents the number of paths to get there:


Again, this totals .
Solution 2
We divide it into 3 stages. The first occurs before the frog moves past 13. The second occurs before it moves past 26, and the last is everything else.
For the first stage the possible paths are , , , , , and . That is a total of 6.
For the second stage the possible paths are , , , , , and . That is a total of 6.
For the second stage the possible paths are , , , , and . That is a total of 5.
However, we cannot jump from (this eliminates 5 paths) or (this eliminates 6 paths), so we must subtract .
The answer is
See also
2007 AIME I (Problems • Answer Key • Resources)  
Preceded by Problem 5 
Followed by Problem 7  
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15  
All AIME Problems and Solutions 