Alan's 3N Game

Number Theory Level 4

Alan and Bob are playing with numbers. Starting with the number \(N=1\) they take turns performing arithmetic manipulations with \(N\). Alan goes first and always multiplies the current number by \(3\). Bob adds \(1\) or \(2\) to the current number, depending on a coin toss. A number is called reachable, if it can possibly appear in this game. How many reachable numbers are there between \(1\) and \(1000\) (inclusive)?

Details and assumptions

Number \(1\) is considered reachable.

Alan's first step would be to multiply 3 to 1, getting 3, which becomes the current number. Bob then adds either 1 or 2 to the current number (which is 3), depending on the coin toss.


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