AIME I Problem 6 2007

[zefuri]

A frog is placed at the origin on a number line, and moves according to the following rule: in a given move, the frog advanced to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer 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, 0, 3, 6, 13, 15, 26, 39 is a move sequence. How many move sequences are possible for the frog?

[stupidityismygam]

casework: Case 1: goes through no multiples of 13 (except 39)
Case 2: goes through 1 multiple of 13 (except 39)
Case 3: goes through 2 multiple of 13 (except 39)

if you count correctly and dont miss it by 2...then you will get

169 :)

[zefuri]

125+20+20+4=169 i think it was.